THE NATURE OF PHYSICS
THE NATURE OF PHYSICS
A Introduction
What do you mean by Physics? Science in general? etc...? "Science is the ever unfinished quest to discover all facts, the relationships between things, and the laws by which the world runs." (by Gerald Holton).
"Physics" - is the science, which seeks to understand the properties of inanimate matter, the laws of motion, and the processes of converting energy.
- is once called nature philosophy, is the discipline of science most directly concerned with the fundamental laws of nature.
According to one of definitions, physics is the study of matter and motion. Neither this nor any other sentence definition can adequately reflect the mixture of created error, accumulated knowledge, unifying ideas, mathematical equations, philosophical impact, and practical application that comprise physics. The modern physicist has generalized the idea of master to include the distributed energy wave fields and the transistor energy of unstable particles; also as shall frequently emphasize, he is as much concerned with the unchanging aspects of nature as he is with motion and change. Yet it is true that the material world and the interaction of one part of it with another remain at the heart of physics. To encompass as much as possible of the behavior of matter with the simplest possible array of ideas and equations is the primary goal of the physicist. (Physics: the study of matter and its interactions.)
In some ways the beginning student of physics faces a more formidable task than does the advanced student. In surveying physics for the first time at the college level, you must do more than learn facts, laws, equations, and problem - solving techniques. You must also seek to grasp the whole of physics, appreciate its generality, see the interconnections of its parts, and perceive its boundaries. You must learn to distinguish between theory and application, between general law and specific fact, between physical ideas and mathematical tools. The dual goals of study of physics: to gain general insights and power to use physics for practical purposes. But these goals are achieve only through your own dedicated effort.
A law of physics is a statement of fact, usually about a restricted range of phenomena. The word "theory" is used more broadly to mean anything from an untested hypothesis to a firmly established set of ideas capable of accounting for many laws. Physics is concerned mainly with the theories that have progressed well beyond the stage of hypothesis, theories that deserve being called the great theories of physics. Each is a structure of ideas and equations thoroughly tested by experiment. Each is remarkable for its small number of basic concepts and its large number of applications. The great theories of physics are:
1. MECHANICS (sometimes called Newtonian mechanics or classical mechanics): the theory of the motion of material objects.
2. THERMODYNAMICS the theory of heat, temperature, and the behavior of large arrays of particles.
3. ELECTROMAGNETISM: the theory of electricity, magnetism, and electromagnetic radiation.
4. RELATIVITY: the theory of in variance in nature and the theory of high-speed motion.
5. QUANTUM MECHANICS: the theory of the mechanical behavior of the submicroscopic world.
Even these five (above) are closely interrelated. Every one of the myriad of phenomena in the physical world that is understood can be explained in terms of one or more of these few theories. Quantum mechanics relativity, and electromagnetism, for example, governs the behavior of single atoms. To describe a collection of many atoms also requires the theories of mechanics and thermodynamics.
About these five theories we can say with confidence that none will ever be completely overthrown. If history is a reliable guide, then we can say with equal confidence that none will prove to be entirely correct. Mechanics is already known to be "incorrect". Relativity "overthrew" mechanics when it showed Newton's laws of mechanics to be incorrect for describing ultra high-speed motion. Latter, quantum mechanics showed classical mechanics to be incorrect for describing the internal motions within atoms. Each of these 20th century developments proved mechanics to be wrong. Why then do we doggedly list mechanics as one of the great theories of physical science? Because over what is still a vast domain of sizes and speeds, mechanics is so extremely accurate that it is for all practical purposes completely correct. It is the best and simplest tool for describing nature in a certain domain. It is better to say that relativity and quantum mechanics have chipped away at the boundaries of mechanics, reducing it from an infinite to a finite domain, than it is to say that either theory has over-thrown mechanics.
At the limits of the very large and very small, our theoretical base is least secure. To describe nature of these limits, new general theories may be necessary. Physics is an active and still-evolving science.
1 WHY STUDY PHYSICS?
Everything that we can see, hear or feel is subject matter belonging to Physics because of the reasons "why" the natural events occur. We study Physics because it is important in the study of other subjects as biology, chemistry, astronomy, engineering and others. We also study physics because it is valuable in giving discipline to the mind and because it affects our daily life.
2 Uses of Physics
Physics is a science which we use everyday. It is a science we deal with in our everyday life knowingly and unknowingly. There is physics in cooking food, in ironing clothes, in writing letters or in looking at mirrors. There is physics in running automobiles, calluses and trains. There is physics in the flight of airplanes and jet planes. Physics is present in the construction of roads, bridges, and buildings. Laws and principles of physics are used in practically every machine and everything we do. Physics plays an important role in transportation, communications, amusements, sports, industry and the home.
Physics has raised our standard of living. The modern facilities and appliances in the homes like refrigerators, air conditioning units, electric mixers, percolators and others have reduced housework. Radios, television sets and high fidelity radios have made home life enjoyable.
Physics protect us from accidents by means of signal lights in the streets, in light houses and ships in distress; through safety devices like fuses and lightning arresters in homes and factories and Geiger counters in laboratories. Physics helps in prolonging life with useful instruments needed in hospitals and clinics and by providing range that kill germs.
Physics makes us understand our environment. We learn why rain falls and a storm occurs, why we have day and night, why seasons change and why tides are sometimes high and sometimes low.
Physics teaches us the manipulation and operation of the many complicated and simple devices that are necessary in our modern life. Some such devices are the cameras, projectors, automobiles, pumps, engines and motors.
In transportation physics has given us the modern automobiles, locomotives, airplanes, jets and rockets. It has given us the luxury liners and atomic power submarines. In communication, physics has given us the telephone, telegraph systems (with or without wires), Teletype and telecast system.
There is perhaps no more need to mention specific machines that physics has made available for industry. One can just watch the big machines in some of our bottling companies as the Pepsi-Cola and the Coca-Cola plants, in glass factory of San Miguel and in soap and lard sections of Philippine Manufacturing Company. All these giant machines could not have been produced without physics.
Physics provides avenues for a life career or profession. One can be an engineer, architect, mechanics, doctor, teacher, agriculturist or a scientist if he studies physics. Physics teachers are very much in demand in schools. Laboratory technicians too, who have a knowledge of physics are needed in our fast developing industries.
Knowledge of physics makes us appreciate the modern discoveries and inventions in science like the radar, the rocket, the atom ships, rain making radioisotopes and many others.
Physics has transformed the world into its modern farm. It has supplied methods of learning and solving problems. It has explained many truths that were once inexplicable. It has provided tools for the engineers, chemist, geologist astronomer, biologists and agriculturists.
In a broad sense Physics is the branch of knowledge, which describes and explains the material world and its phenomena, which uses the resulting understanding to create new areas of human experience.
At present it is customary to use physics in a more restricted sense. Aspects of nature which are ordinarily regarded in the domain of physics are:
1. Mechanics - deals primarily with motion of bodies, the concept of force, the effect of forces on motion and the form or shape of bodies; energy, momentum, work and power, properties of solids liquids and gases, and plasma.
2. Heat - deals with temperature scales and measurement, the concept of heat, thermal expansion, heat capacities of substances, changes of state, heat transfer and thermodynamics.
3. Sound leads to the consideration of waves and wave motion - deals with different sources of sound its transmission through various media, acoustics and hearing.
4. Electricity and Magnetism - deals with concepts of electrical changes, the flow of electrical changes known as current, electrical instruments, electrical and magnetic properties of matter and electronics.
5. Optics - deals with fundamental concepts of electromagnetic waves, absorption and transmission of light, reflection and refraction, optical instruments, interference, diffraction and polarization.
6. Atomic and Nuclear - study of radiation, photo-electric effect, X-rays, structure of the atom, radioactivity, nuclear disintegration and properties of nuclei.
3 Frontiers of Physics
1. Biophysics - deals with the application of physics in the study of our life and other living matter. It includes the study of the eye and color vision, the ear as an organ of hearing and balance, body heat and the effects of radiation to human body.
2. Geophysics - Physics of the earth-includes the study of the structure of the earth and its motion, the earthquakes and the layers of the atmosphere. It bridges the gap between physics and geology.
3. Astrophysics - deals with the physical constitution of the stars application of the new method of observation of the heavenly bodies with makes use of such devices like spectroscope, thermocouple and photoelectric.
4. Medical physics - work and application of chemistry and biochemistry dealing with the use of radioactivity materials - most important applications of radioisotopes is in medical field and agriculture.
5. Engineering physics - sometimes called applied or industrial physics - study of the different applications of physics in the form of mechanical gadget.
6. Nuclear physics - structure and properties of atoms and their nuclei.
4 Suggested way of solving the Physical Problems
The ability to solve problems is a mark of an effective and efficient scientist or engineer. Through practice in the solution of problems commensurate with one's knowledge, one attains ability and confidence in independent thinking.
In problem solving, the following systematic approach is recommended. First, read the statement of the problem carefully, and decide exactly what is required. Then: DID-DeSCO...
1. Draw a suitable diagram, and list the data given.
2. Identify the type of problem, and write physical principles, which seem relevant to it solution. These maybe expressed concededly as algebraic equations.
3. Determine if the data is adequate. If not, decide what is missing and how to get it. This may involve consulting a table, making a reasonable assumption, or drawing upon your general knowledge for such information.
4. Decide whether in the particular problem it is easier to substitute numerical values immediately or first to carry out an algebraic solution. Some quantities may cancel.
5. Substitute numerical data in the equations obtained from physical principles. Include the units for each quantity, making sure that they are all in the same system in any one problem.
6. Compute the numerical value of the unknown. Determine the units in which the answer is expressed. Examine the reasonableness of the answer. Can it be obtained by an alternative method to check the result?
Note: If possible do not use solved unknown to solve for another unknown in the same problem.
An orderly procedure aids clear thinking, helps to avoid errors, and usually serve time. Most important, it enables a student to analyze and eventually solve these move complex problems whose solution is not immediately or intuitively apparent.
"The ability to analyze a problem involves a combination of inherent insight and experience. The former, unfortunately, cannot be learned, but depends on the individual. However, the latter is of equal importance, and can be gained with patient study".
By Vedal S. Arpaci
*GENERAL NOTE IN PHYSICS*
"Anyone who tries to memorize without understanding will most likely run into difficulty.*
5 Measurement
"The most important thing for a young man to acquire from his first course in physics", the late Prof. William S. Franklin frequently said, "is as appreciation of the necessity for precise ideas".
In dealing with physical quantities, the question "HOW LARGE?" or "HOW MUCH?" is usually asked and this leads to the process of MEASUREMENT. In measuring anything is simply a comparison with some given standard. To carry out accurate measurements, we have to establish a system of units and a system of standards to fix and preserve the sizes of the units. A UNIT is a value or quantity in terms of which other values or quantities expressed. In general, a unit is fixed by definition (for example, one meter) and is independent of such physical conditions as temperature. A STANDARD is the physical embodiment of a unit. In general, a standard is a true embodiment of the unit only under specified conditions. A standard meter bar has the length of one meter only when at one particular temperature and when supported in a certain manner.
Methods of Measurement
1. Direct - placing directly the standard value over the thing to be measured. (Fundamental Quantities)
Example. Measuring the length and width of a rectangular table with a standard measuring device, say stick or a foot rule.
2. Indirect - value is determined by computation (Derived Quantities)
Example. Measuring the area of the rectangular table (in. no. 1) is done through computations by the use of a formula.
Table 1 Fundamental/Basic SI Unit
|Quantity and |Name of Unit |Definition of Base Unit of International System of units |
|Symbol |and Symbol | |
|length |Meter |The meter is the length equal to 1 651 763.73 wavelength in vacuum of the radiation |
|l |m |corresponding to the transition between the levels 2p10 and 5d3 of krypton – 86 atom. |
|mass |Kilogram |The kilogram is the mass of the international prototype of the kilogram. The International |
|m |Kg |prototype of the kilogram is a particular cylinder of platinum dridium alloy, which is |
| | |preserved in a fault at Seyres, France, by the International Bureau of Weights and Measures.|
|time |Second |The second is the duration of 9 192 631 770 periods of the radiation corresponding to the |
| | |transition between the two hyperfine levels of the round state of caesium-133 atom. |
|Electric current |Ampere |The ampere is that constant current, which if maintained in two straight parallel conductors|
| | |of infinite length, of negligible circular cross-section, and placed 1 metre apart in |
| | |vacuum, would produce between these conductors, a force equal to 2 x 10-3 newton per meter |
| | |length. |
|thermodynamic |Kelvin |The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic|
|temperature |K |temperature of the triple point of water. |
|T | | |
|liminous |Candela |The candela is the luminous intensity, in the perpendicular direction, of a surface of 1/600|
|intensity |Cd |square metre of a black body at the temperature of freezing platinum under a pressure is 101|
|Iv | |325 pascal. |
|amount of substance |Mole |The mole is the amount of substance in a system which contains as many elementary entities |
| |Mol |as there are atoms in 01012 kg of carbon 12. |
Table 2 Derived Units
|Quantity |Name of Unit |Symbol |Quantity |Name of Unit |Symbol |
|Length |Meter |m |Power |Watt |W or J/s |
|Mass |Kilogram |kg |Quantity of electricity |Coulomb |C or A.S |
|Time |Second |S |Potential Diff. EMF |Volt |V or W/A |
|Electric current |Ampere |A |Electric Field Strength |Volt per meter |V/m |
|Thermodynamic Team |Kelvin |K | | | |
|Luminous Intensity |Candela |cd |Electric Resistance |ohm |Q or V/A |
|Amount of Substance |Mole |mole |Capacitance |farad |F or A.s/V |
|Area |square meter |m2 |Magnetic flux |Weber |Wb or V.S |
|Volume |cubic meter |m3 |Inductance |henry |H or V.8/A |
| |hertz |Hz or s-1 |Magnetic flux density |tala |T or wb/m2 |
|Frequency Quantity |Name of Units |Symbol |Quantity |Name of Unit |Symbol |
| | | | | | |
|Mass Density |Kilogram per cubic | |Magnetic field strength |ampere per |A/m or 1 m cd,.st |
|(Density) |meter |kg/m3 | |meter | |
| | | | | | |
|Speed, velocity |meters per second |m/s |Luminous flux |lumen |cd/m2 |
| | | | |Candela per square | |
| |meter per sq.sec | |Luminance |meter |m-1 |
|Acceleration | |m/s2 |Illuminance |lux | |
| |radian per second | | |1/meter jour per | |
|Angular velocity | |rad/s |Wave number |kelvin | |
| |radian per second | | | | |
| |squared | |Entropy | |J/K |
|Angular acceleration | |rad/s2 | |Joule/per kilogram |J/(kg.k) |
| |newton | |Specific heat capacity |kelvin | |
| | | | | | |
|Force |pascal |kg.m/s2 | |Watt per |W/m.k |
| | |or N |Thermal conductivity |meter-kelvin | |
|Pressure (Mechanical) Stress | |Pa | | | |
| |square meter per |or n/m2 | |Ampere radian |A |
|Kinematic Viscosity |second | |Magnetomotive force |steradian |rad |
| | | | | |sr |
| |newton-second per |m2/s | | | |
|Dynamic Viscosity |square meter | |Plane angle | | |
| | | | | | |
| |Joule | | | | |
|Work, Energy, Quantity of Heat | | |Solid angle | | |
| | | | | | |
|Radiant Intensity |Watt per steradian |J or N.m | | | |
|Radioactive source) | | | | | |
| | | | | | |
| | |W/sr | | | |
| | | | | | |
| | | | | | |
Note: A suitable unit must be chosen to each of the fundamental/derived physical quantities, such as metric or english system. The choice is purely a matter of convention.
Systems of Measurement
CGS = centimeter – gram - second
Metric system
MKS = meter – kilogram – second
British/English -FPS = foot-pound-second
The British gravitational system of measurement is not an easy system to work with since there are no convenient or predictable ratios between units. For example, there for some nonscientific reason(s) for 12 inches in 1 foot, 3 feet in 1 yard, and 5280 feet in 1 mile. These units are largely based on no reproducible standards and traditions. For example, it is said that King Henry I established the Yard by measuring the distance between the tip of his finger and the tip of his nose. An INCH is the length of three dry and round barleycorns laid end to end (side by side), by pronouncement of King Edward II. Interestingly, the system of shoe sizes we used today is based upon that definition. The shoemakers of King Edward found that the longest foot on these days was 39 barleycorns, or 13 inches long. They called this size 13, and traded sizes downward by one barleycorn to a size.
The MILE comes from the Latin mile or thousand and was determined by the thousand double steps of the average Roman soldier.
In Noah's time, carpenters had a measurement called the CUBIT. This was the length of the forearm-from the tip of the middle finger to the elbow. Assuming several carpenters worked on the same project, it is a wonder that the Ark floated (this was commented by Edward Teller - a US scientist, a great nuclear physicist).
As of this time, Metric system became more popular and sooner or later all nations would be using the units in the metric system.
PD 187 of May 10, 1973
Only metric system of weights and measures will be allowed for use in all business and legal transactions of products, materials, commodities, utilities and services effective January 1, 1975.
However, we can not totally abolish the English System since equipment and apparatus in factories and in schools are in the British system.
Based from the three systems of measurement, there must be agreement of units, thus there is a need to convert from one unit to another.
Scientific Notation
Measured values are sometimes very small and very big, thus, numbers are written in scientific notation in the form M x 10n
where: M = is the number having a single non-zero digit to the left of the decimal point
n is + if decimal is moved to the left
- if moved to the right.
Example:
Velocity of light (v) = 300,000,000 m/s
Mass of earth (Mo) = 6 000 000 000 000 000 000 000 000 kg
Mass of electron (Me) = 0.000 000 000 000 000 000 000 000 000,910,953 (/kg
Acceleration due to gravity (g) = 980.665 m/s2
Significant figures = digits which indicate the number of units we are reasonably some of having counted in making a measurement. It includes digit in a number that is known with certainty plus one digit that is uncertain.
Example: 365.8 0.00435
3 0000 16 000000
0.2800
Rules in Determining Significant Figures:
1. All nonzero digits are significant: 112.8oC have four significant figures.
2. All zeros between two nonzero digits are significant: 108.005 m has six significant figures.
3. Zero to the right of a nonzero digit, but to the left of an understood decimal point, are not significant unless specifically indicated to be significant. The rightmost a bar placed above it indicates such, zero who is significant,: 109,000 km contains three significant figures: 109,000 contains five significant figures.
4. All zeros to the right of a decimal point but to the left of a nonzero digit are not significant: 0.000647 kg has three significant figures.
5. All zeros to the right of a decimal point and following a nonzero digit are significant: 0.07080 cm and 20.00 cm each has four significant figures.
Rounding Off Numbers
Numbers written in scientific notation are written to three significant figures thus other digits are to be dropped.
Example:
6235 to the nearest tens
29.045 to the nearest hundredth
8262 to the nearest thousands
2.0046 to the nearest thousandths
Rule for rounding. If the first digit to be dropped in rounding is 4 or less, the preceding digit is not changed; if it is 6 or more, the preceding digit is raised by 1. If the digits to be dropped in rounding are a 5 followed by digits other than zeros, 1 raises the proceeding digit. If the digits to be dropped in rounding are a 5 followed by zeros (or if the digit is exactly 5), the preceding digit is not changed if it is even; but if it is odd, it is raised by 1.
Conversion of units using Greek Prefix/Metric Units
|Prefix |Symbol |Decimal Number |Power of Ten |
|yotta |Y |1 000 000 000 000 000 000 000 000 |1024 |
|zetta |Z |1 000 000 000 000 000 000 000 |1021 |
|Exa |E |1 000 000 000 000 000 000 |1018 |
|Peta |P |1 000 000 000 000 000 |1015 |
|Tera |T |1,000,000,000,000 |1012 |
|Giga |G |1,000,000,000 |109 |
|Mega |M |1,000,000 |108 |
|Kilo |k |1,000 |103 |
|Hecto |h |100 |102 |
|Deka |da |10 |101 |
|BASE UNIT | |1 |100 |
|Deci |d |0.1 |10-1 |
|Centi |o |0.01 |10-2 |
|Milli |n |0.001 |10-3 |
|Micro |[pic] |0.000001 |10-6 |
|Nano |n |0.000000001 |10-9 |
|Pico |p |0.000000000001 |10-12 |
|Femto |F |0.000000000000001 |10-15 |
|Atto |a |0.000000000000000001 |10-18 |
|zepto |z |0.000000000000000000001 |10-21 |
|yocto |y |0.000000000000000000000001 |10-24 |
Example:
1723 mg ( kg
0.8206 MW ( KW
17.28 x 105 (f ( f
1723 mg ( kg
10-3 ( 103
1723 x 10-6 kg
Rule: Subtract the exponent of the Greek prefix to get the exponent of the converted value.
Conversion of units using conversion factors
Example:
124 in ( ft
6.5 tons ( lb
30 mi/hr ( ft/s
62.4 lb/ft3 ( g/cm3
[pic]
6.5 tons x 2000 lb/ton = 13000 lb
Thus: Conversion factor has same units divide if different we multiply
1.5.d.b Conversion of Units to SI
Length
1 inch = 0.0254 m = 2.54 cm
1 foot = 0.3048 m = 30.48 cm
1 yard = 0.9144 m
1 mile = 5280 ft = 1.60934 km
1 nautical mile = 6080 ft
1 light year = 9.461x1015m
1 [pic]=10-10 m
Area
1 square inch = 6.45 x 10-4m2
1 square foot = 9.29 x 10-2m2
1 square yard = 0.8636 m2
1 square mile = 2.59 x 166m2 = 2.59 km2
1 acre = 4.047 x 103m2 = 0.4047 ha
1 hectare = 104m2 = 2.47 acres
Volume
1 cubic inch = 1.6387 x 10-5m3 = 0.00164 litre
1 cubic foot = 0.0283 m3 = 28.3 litres
1 gallon (UK) = 454609 x 10-3m3 = 4.546 litres
1 cubic yard = 0.746 m3
1 bushel = 8 gallons = 36.37 litres
Mass
1 lb = 0.4536 kg = 454 g
1 metric ton = 1,000 kg.
1 slug = 14.59 kg
Density
1 lb/cu.ft = 16.02 kg m-3
1 lb/cu inch = 2.768 x 104 kg m-3
1 lb/gallon = 0.0998 kg/litre
Force
1 pound force = 4.448 N
1 dyne = 10-5N
1 poundal = 0.138 N
Pressure
1 psi = 6.895 x 103 N m-2
1 atmosphere = 1.01325 x 105 N m-2
1 bar = 105 N m-2
Note (1 a atmosphere = 760 mm Hg = 29.92 inches Hg)
1 N m-2 = 1 Pascal (Pa)
(1 metre head of water = 9.81 x 103 N m-2)
Velocity
1 miph = 0.4472 mps
1 foot/sec = 0.3048 mps
1 knot = 0.5144 mps
Work and Energy
1 kilowatt hour (KWh) = 3.6 x 106J
1 Btu = 1055 J
1 erg = 10-7J
1 electron volt or eV = 1.602 x 10-19J
1 ft .lb = 1.356 J
1 calorie = 4.186 J (based on 150 calorie)
Power
1 Horsepower (HP) = 746 W
1 ft lb/sec = 1.356 W
1 J/sec = 1 W
Other Combined Units
1 cu t/acre = 125.53 kg/ha
Other useful equivalent
12 in = 1 ft 1 m = 3.28 ft
3 ft = 1 yd 1 mi = 1.609 km
5280 ft = 1 mi 1 acre = 43560 ft2
1 lb = 16 oz 1 ha = 404m2
1 ton = 2000 lb 1 lb = 1000 cc
1 kg = 2.2 lb
6 Practice Exercise: Measurement
I. How many significant figures are there in the following measurements:
1. 823456 6. 93[pic]000
2. 725.00 7. 10-7
3. 0.0000029 8. 0.8050
4. 634 x 1012 9. 179243
5. 43000001 10. 2.007
II. Round off as indicated:
11. 782 ( tens 16. 1650763 ( thousands
12. 13.0745 ( thousandths 17. 19.72500 ( hundredths
13. 67678 ( thousands 18. 273.16 ( tens
14. 0.095 ( tenth 19. 169929 ( ten-thousands
15. 89.60555 ( ten-thousandths 20. 231 ( hundreds
III. Write the following is scientific notation:
16. shortest electric wave is 2200000.0 A
17. shortest ultra violet wave is 0.00760 (
18. speed of light is 186000 mi/s
19. radius of the earth is 6370000 m
20. acceleration due to gravity is 980.665 cm/s2
21. Avogadro’s number is 602.200,000,000,000,000,000,000 particles/mole
22. electron charge is 0.000,000,000,000,000,000,160219 coul
23. Coulomb constant is 8987550000 N-m2/coul2
IV. Convert the following as indicated:
24. velocity of sound in air is 1090 ft/s to m/s
25. density of mercury is 13.6 g/cm3 to lb/ft3
26. maximum speed of man is 28 mi/hr to m/s
27. highest mountain in the world is Mount Everest at 8848 m to ft
28. distance of the moon from the earth is 238900 mi to ft
29. distance from the earth to the sun is 1.5 x 1011 m to mi
30. average human head weighs 6.35 kg to lb
31. average weight of a baby at birth is 7.25 lb to Mg
V. Problem:
32. The unit measure in the metric system is the liter, which is equal to 103 cm3, while the unit of liquid measure in the Ux-S is the gallon, which is equal to 231 in3. How many liters are there in a gallon? How many gallons are there in a liter?
33. A “boardfoot” is a unit of lumber measure that corresponds to the volume of a piece of wood 1 ft square and 1 in thick. How many in3 are there in a boardfoot? How many ft3? How many m3?
34. A stick is 20 cm long. What is the area of the surface it will describe? a) when it moves parallel of 10 cm? b) when it rotates in a plane about one end?
35. How many tons of waterfall on 1 acre (640 acres = 1 mi2) of land during a 1 in rain if 1 ft3 of water weighs 62.4 lb?
36. The earth goes around the sun once a year. The distance of the earth and the sun is 9.3 x 107 mi. What is the circumference of the earth’s orbit around the sun assuming it to be circular. What is the speed of the earth around the sun in m/s.
B VECTORS and VECTORS ADDITION
Physical quantities can be classified into two categories:
1. Scalar – specified by their magnitude (number and unit)
2. Vector – both magnitude and direction
Scalar quantities are added by ordinary algebraic method.
Vectors are added by geometric methods.
a) (b)
Figure 1. (a) Vector [pic]is the displacement from point P1 to P2. (b) A displacement is always a straight-line segment directed from the starting point to the end point, even if the actual path is curved. When a path ends at the same place where it started, the displacement is zero.
[pic] [pic] [pic]
Figure 2. The displacement from P3 to P4 is equal to that from P1 to P2. The displacement [pic] from P5 to P6 has the same magnitude as [pic]and [pic]but opposite direction; displacement [pic]is the negative of displacement [pic].
1 Addition of Vectors:
[pic]
[pic] [pic] [pic] [pic] [pic] [pic]
[pic] [pic]
Figure 3. Vector [pic]is the vector sum of vectors [pic]and [pic]. The order in vector addition doesn’t matter; vector addition is commutative.
In figure 3, supports;
[pic]
[pic]
then [pic]
[pic]
[pic]
[pic]
[pic] [pic] [pic] [pic]
[pic]
(a) (b)
[pic] [pic]
[pic] [pic] [pic] [pic] [pic]
[pic] [pic]
(c) (d)
[pic]
[pic] [pic] [pic]
(e)
Figure 4. Several constructions for finding the vector sum [pic].
[pic] [pic]
Figure 5. (a) Vector [pic]and vector [pic]. (b) Vector [pic]and vector [pic]. (c) The vector difference [pic] is the sum of vectors [pic]and [pic]. The tail of [pic] is placed at the head of [pic]. (d) To check: ([pic])+[pic]=[pic].
Example 1. A cross-country skier skis 1.00 km north and then 2.00 km east on a horizontal snow field. a) How far and in what direction is she from the starting point? b) What are the magnitude and direction of her resultant displacement?
Solution:
[pic]
[pic]
2 Components of Vectors
Figure 5. Vectors [pic]and [pic]are the rectangular component vectors of [pic]in the directions of the x- and y-axes. For the vector [pic]shown here, the components [pic] and [pic] are both positive.
[pic] and [pic]
[pic] and [pic]
Example 2. Components a) What are x- and y-components of vector [pic]in figure below? The magnitude of the vector is D=3.00m, and the angle (=450. b) What are the x- and y-components of vector in figure? The magnitude of the vector is E=50 m, and the angle (=370.
Figure:
(a) (b)
Dx = D cos( = 3.00m cos (-450) = +2.1 m,
Dx = D sin ( = 3.00m sin (-450) = -2.1 m.
Ex = E cos( = 4.50m cos (370) = +2.71 m,
Ex = E sin( = 4.50m sin (370) = +3.59 m.
USING COMPONENTS
Problem-Solving Strategy in Vector Addition
1. First draw the individual vectors being summed and the coordinate axes being used. In your drawing, place the tail of the first vector at the origin of coordinates, place the tail of the second vector at the head of the first vector, and so on. Draw the vector sum [pic] from the tail of the first vector to the head of the last vector.
2. Find the x- and y-components of each individual vector and record your results in a table. If a vector is described by its magnitude A and its angle (, measured from the +x-axis towards the +y-axis, then the components are given by
Ax = A cos (, Ay = A sin (,
Some components may be positive and some may be negative, depending on how the vector is oriented (that is, what quadrant ( lies in). You can use this sign table as a check:
|Quadrant |I |II |III |IV |
|Ax |+ |- |- |+ |
|Ay |+ |+ |- |- |
| | | | | |
If the angles of the vectors are given in some other way, perhaps using a different reference direction, convert them to angles measured from the +x-axis as described above. Be particularly careful with signs.
3. Add the individual x-components algebraically, including signs, to find Rx, the x-component of the vector sum. Do the same for the y-components to find Ry.
4. Then the magnitude R and direction ( of the vector sum are given by
[pic] [pic]
Remember that the magnitude R is always positive and that ( is measured from the positive x-axis. The value of ( that you find with a calculator may be the correct one, or it may be off by 1800. You can decide by examining your drawing.
Example 3. Adding vectors with components. The three finalists in a contest are brought to the center of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements:
72.4 m, 320 east of north;
57.3 m, 360 south of west;
17.8 m straight south.
The three displacements lead to the point where the keys to a new Porsche are buried. Two contestants start measuring immediately, but the winner first calculates where to go. What does she calculate?
Solution:
Ax = A cos (A = 72.4 m cos 580 = 38.37 m,
Ay = A sin (A = 72.4 m sin 580 = 61.40 m.
|Distance |Angle |x-component |y-component |
|A=72.4 m |580 |38.37 m |61.40 m |
|B = 57.3 m |2160 |-46.36 m |-33.68 m |
|C = 17.8 m |2700 |0 m |-17.80 |
| | |Rx=-7.99 m |Ry= 9.92 m |
[pic]
[pic]
3 Unit Vectors
A unit vector:
• It has a magnitude of 1, with no units.
• Its only purpose is to point, that is, to describe a direction in space.
• It provides a convenient notation for many expressions involving components of vectors.
• It has a caret or “hat” (^) in the symbol for a unit vector to distinguish it from ordinary vectors whose magnitude may or may not be equal to 1.
[pic]
[pic]
VECTOR ADDITION
When two vectors [pic] and [pic]are represented in terms of their components, we can express the vector sum [pic]using unit vectors as follows:
[pic]
If the vectors do not all lie in the xy-plane, then we need a third component. We introduce a third unit vector [pic] that points in the direction of the positive z-axis. The generalized forms of equations,
[pic] [pic][pic]
[pic]
Example: Given the two displacements, [pic] and [pic], find the magnitude of the displacement [pic].
Solution: Let [pic], we have
[pic]
[pic]
[pic] answer
SCALAR PRODUCT
The scalar product of two vectors [pic] and [pic] is denoted by [pic]. Because of this notation, the scalar product is also called the dot product
[pic]
[pic]
We define [pic] to be the magnitude of [pic] multiplied by the component of [pic] parallel to [pic].
[pic]
[pic]
VECTOR PRODUCT
The vector product of two vectors [pic] and [pic], also called the cross product, is denoted by,[pic].
[pic]
[pic]
4 Practice Exercises
1. Hearing rattles from a snake, you make two rapid displacements of magnitude 8.0 m and 6.0 m. Draw sketches, roughly to scale, to show how your two displacements might add to give a resultant of magnitude a) 14.0 m; b) 2.0 m; c) 10.0 m.
2. A postal employee drives a delivery truck along the route shown in Figure below. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram and by component method. Answer: 7.8 km, 380 north of east
Figure
3. For the vectors [pic] and [pic] in Figure below, use a scale drawing to find the magnitude and direction of a) the vector sum [pic]; b) the vector difference [pic]. From your answers to parts (a) and (b), find the magnitude and direction of c) [pic]; d) [pic].
Figure:
4. Use a scale drawing to find the x- and y-components of the following vectors. In each case the magnitude of the vector and the angle, measured counterclockwise, that it makes with the +x-axis are given. a) magnitude 7.40 m, angle 300; b) magnitude 15 km, angle 2250; c) magnitude 9.30 cm, angle 3230.
5. Compute the x- and y-components of each of the vectors [pic], [pic], and [pic] in Figure.
Figure:
Answer: 7.2m, 9.6m : 11.5m, -9.6m : -3m, -5.2m
6. For the vectors [pic] and [pic] in figure below, use the method of components to find the magnitude and direction of a) the vector sum [pic]+[pic]; b) the vector sum [pic]+[pic]; c) the vector difference [pic]-[pic]; d) the vector difference [pic]-[pic].
Answer: a)11.1m; 77.60, b) 11.1m, 77.60
c)28.5m, 202.30 d) 28.5m, 22.30
7. Find the magnitude and direction of the vector represented by each of the following pairs of components:
a) Ax = 5.60 cm, Ay = -8.20 cm;
b) Ax = -2.70 m, Ay = -9.45 m;
c) Ax = -3.75 km, Ay = 6.70 km.
8. Vector [pic] has components Ax= 3.40 cm, Ay= 2.25 cm; vector [pic] has components Bx = -4.10 cm, By=3.75 cm. Find a) the components of the vector sum [pic]; b) the magnitude and direction of [pic]; c) the components of the vector difference [pic]; d) the magnitude and direction of [pic];.
9. A disoriented physics professor drives 4.25 km south, then 2.75 km west, then 1.50 km north. Find the magnitude and direction of the resultant displacement, using the method of components. Draw a vector addition diagram, roughly to scale, and show that the resultant displacement found from your diagram agrees with the result you obtained using the method of components. Answer: 3.89 km, 450 west of south
10. An explorer in the dense jungles of equatorial Africa leaves her hut. She takes 80 steps southeast, then 40 steps 600 east of north, then 50 steps due north. Assume her steps all have equal length. a) Draw a sketch, roughly to scale, of the three vectors and their resultant. b) Save her from becoming hopelessly lost in the jungle by giving her the displacement vector calculated by using the method of components that will return her to her hut.
11. A cross-country skier skis 7.40 km in the direction 450 east of south, then 2.80 km in the direction 300 north of east, and finally 5.20 km in the direction 220 west of north.
a) Show these displacements on a diagram. b) How far is the skier from the starting point? Answer: b) 5.79 km
12. On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa; then to St. Joseph, Missouri; then to Manhattan, Kansas. The directions are shown relative to north: 00 is north, 900 is east, 1800 is south, and 2700 is west. Use the method of components to find a) the distance she has to fly from Manhattan to get back to Lincoln; b) the direction (relative to north) she must fly to get there. Illustrate your solution with a vector diagram.
13. Find the magnitude of the single displacement that is equivalent to successive displacements of 30 m and 50 m, the direction of the second displacement being perpendicular to that of the first.
14. A rope attached to a sled makes an angle of 40o with the ground. With what force must the rope be pulled to produce a horizontal component of 100 Nt? What will then be the vertical component of the force.
15. A farmer plowing in the contour of his land plows 150 m on a bearing of 315o and then turns and plow 50 m on a bearing of 200o. Determine the distance and bearing of the farmer’s present position from his starting point.
16. A ferryboat goes straight across a river in which there is a current of 3 km/hr. If the speed of the boat relative to the water is 10 km/hr, find the direction in which it is pointed. What is the velocity relative to the earth? (17.5o, 9.54 kph)
17. Two forces, one of 30 lb and the other unknown, act at the same point to produce a resultant force of 36 lb. If the angle between the two forces is 120oi, find the magnitude of the unknown force.
18. A janitor holds the handle of his mop such that it makes 60o with the floor. If he pushes with a force of 80 lbs what force will drive the mop against the floor?
19. Three football players participating simultaneously in a tackle exert the, following forces in the ball carrier 100 lb due E, 120 lb 20o N of E and 80 lb 35o W of N. Find the resultant of these forces.
20. A web page designer creates an animation in which a dot on a computer screen has a position of [pic][pic].
a) Find the magnitude and direction of dot’s average velocity between t =0 and t = 3.0 s.
b) Find the magnitude and direction of the instantaneous velocity at t =0, t =2, and t =3 s.
c) Sketch the dot’s trajectory from t =0 to t =3 sec and show the velocities calculated in part (b).
KINEMATICS
(Purely descriptive study of motion)
A Motion Along a Straight Line
Kinematics is a motion of an object without considering outside factors which causes their motion.
1 Motion
Motion - denotes a change in position of a body with respect to some fixed point or reference point.
Speed - distance which a body traverse per unit time (scalar quantity)
Velocity - displacement of a body per unit time (vector quantity)
[pic] (displacement)
[pic] (average velocity)
[pic] (instantaneous velocity)
Speed is the magnitude of velocity.
It the bodies move equal displacement in equal intervals of time then the body is said to be moving with uniform motion.
V = constant
Instantaneous velocity = velocity of object at a particular instant or at particular point
2 Acceleration
When the velocity of a moving body changes continuously as the motion proceeds, the body is said to move with accelerated motion. Three possible ways in which velocity may change:
1. magnitude - direction of acceleration is parallel to the direction of motion.
2. direction - acceleration is at right angles to the direction of motion.
3. both magnitude and direction - acceleration is in any direction.
Average acceleration = change in velocity/time elapsed.
[pic] (average acceleration)
[pic] (instantaneous acceleration)
If magnitude of velocity is increasing, acceleration is positive; if decreasing, acceleration is negative; if decreasing, acceleration is negative (deceleration).
B Uniformly Accelerated Motion
If the rate of change of velocity is uniform (constant acceleration) then the average velocity in any time interval is 1/2 the sum of the velocities at the beginning and the end of the interval.
[pic]
[pic]
[pic]
When a body starts from rest, vo is zero and acceleration is positive. If the body decreases in velocity, acceleration is negative, velocity becomes smaller than vo and when the body stops v = 0.
C Practice Exercise
1. How far does an automobile move while its speed increases uniformly from 15 mi/hr to 45 mi/hr. in 10 s?
2. An airplane requires a speed of 80 mi/hr to be airborne. It start from rest on a runway 1600 ft long. a) What must be the minimum safe acceleration of the airplane? b) With this acceleration, how many seconds will it take for the plane to acquire its needed speed for take off? (80 mi/hr - 117.3 ft/s).
3. A car starts from rest and accelerates 6 m/s2 for 5 s after which it travels with a constant velocity for 9 s. The brakes are then applied so that it decelerates at
4 m/s2. Find the total distance traveled by the car.
4. An object starts from rest and accelerates 4 m/s2. a) How far will it travel after 2s? b) How far will it travel during the third second?
5. A freight train is travelling with a velocity of 15 m/s. at the instant if passes through a station a passenger train at rest starts to accelerate 3 m/s2 in the same direction as the velocity of the freight train? a) In how many seconds will the passengers train overtake the freight train? b) How far will the passenger train travel before it overtakes the freight train?
6. The brakes of a car are capable of producing an acceleration of 20 m/s2. How far will the car go in the course of slowing down from 90 m/s to 30 m/c (180 m)
7. At the instant the traffic lights turn green, an automobile that has been waiting at an intersection starts ahead with a constant acceleration of 2 m/s2. At the same instant a truck traveling with a constant velocity of 10 m/s overtakes and passes the automobile. a) How far beyond its starting point will the automobile overtakes the truck b) How fast will it be traveling. (100 m, 20 m/s).
8. A sporting car starting from rest accelerates 40 km/hr2 for 30 min after which it travels with a constant velocity of 1 hr. When the brakes wire applied it slow down at 2 km/hr2 until it stops. Find the total distance covered. (3.5 km)
9. A truck starts from rest and rolls down a hill with constant acceleration. It travels a distance of 400 m in the first 20 s. Find the acceleration and the speed of the truck after 20 sec. (2 m/s2; 40 m/s)
10. What velocity is attained by an object which is accelerated at 0.3 m/s2 from a distance of 50 m if its initial velocity is 0.5 m/s. (5.5 m/s)
11. The brakes of an automobile traveling with a velocity of 50 ft/s are suddenly applied. If the automobile comes to a stop after 5 s what is its acceleration?
D Freely Falling Bodies
The most common example of uniformly accelerated translation is that of body falling under the action of its own weight. (freely falling body).
In the absence of air resistance it is found that all bodies regardless of their size or weight; fall with the same acceleration at the same point on the earth surface and if the distance covered is small compared to the radius of the earth, the acceleration remains constant throughout the fall.
Actual acceleration of object fall depends on:
1. location of the earth
2. size and shape of object
3. density
4. state of atmosphere
5. rotation of the earth
Thus Galileo's conclusion is an idealization of reality for these factors are being neglected. (Idealized motion of free fall)
It should be kept in mind that direction of "g" is always downward no matter whether we are dealing with a dropped object or one which is initially thrown upward.
At or near the earth's surface computation purposes
g = 32.17 ft/s2 g = 32 ft/s2
= 9.806 m/s2 = 9.8 m/s2
= 980.6 cm/s2 = 980 cm/s2
On the surface of the moon the acceleration of gravity is due to the attractive force on a body by the moon rather than on the earth.
g = 1.67 ms-2
= 5.47 ft/s-2
near surface of the sun
g = 274 m/s2
Since freely falling body is uniformly accelerated translation, then the equations for freely falling body are the same as those of the equations for acceleration bodies with a replaced by "g".
[pic]
[pic]
[pic]
[pic]
Acceleration of gravity is positive for bodies that are falling and negative for bodies that are thrown upward.
E Motion in Two or Three Dimensions
[pic] (position vector)
[pic] (average velocity vector)
[pic] (instantaneous velocity)
[pic] (instantaneous velocity)
1 Projectile motion
A body that moves through space usually has a curved path rather than a perfectly straight one.
Projectile any body that is given an initial velocity and then allowed to move under the influence of gravity.
The path followed by a projectile is called its trajectory.
If we neglect air resistance and the variation of “g” with altitude, we consider only trajectories which are sufficiently short range (Idealized model)
The motion is best referred to a set of rectangular coordinate axis. Horizontal component of acceleration is zero and vertical component is downward and equal to that of a freely falling body. Since zero acceleration means constant velocity, the motion can be described as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.
v = [pic]
Sx = vxt
R = vot
Projectile thrown at an angle
Since initial vertical velocity is upward, then body is decelerated, hence velocity along y-axis is:
vy = voy – gt
= vo sin ( - gt
Horizontal component vox remains constant during flight
vx = vox = vo cos (
Time for projectile to reach its maximum height, vertical velocity becomes zero thus
vy = vo sin ( - gta
ta = [pic] time to reach max. height
Maximum height
[pic] = [pic]
[pic]
Maximum range
R = Voxt
= [pic]
General equation of projectile motion: [pic]
F Practice Exercise: Freely falling bodies and motion
1. From what height must water fall from a dam to strike the turbine wheel with a speed of 120 ft/s?
2. A stone is thrown upward with an initial velocity of 50 ft/s. What will its maximum height be? when will it strike the ground? where will it be in 1 1/8 s?
3. If an object is thrown vertically down with a velocity of 20 ft/s. Find its velocity after 3 s and the distance the stone falls during these 3 s.
4. With what initial velocity will a body moving along a vertical line have to be thrown, if after 5 sec it is to be 50 ft above its starting place.
5. A boy on a bridge throws a stone horizontally with a speed of 25 m/s releasing the stone from a point 19.6 m above the surface of the river. How far from a point directly below the boy will the stone strikes the water?
6. A mango falls from its tree, how high is the tree if it takes 3 seconds for the fruit to reach the ground.
7. A pebble is thrown vertically downward with a speed of 20 ft/s from the roof of a building 60 ft high a) How long will it take the ball to reach the ground b) What will its speed be when it strikes the ground.
8. A girl throws a ball vertically upward with a speed of 20 ft/s from the roof of a building 60 ft high a) How long will it take the ball to reach the ground b)What will its speed be when it strikes the ground.
9. An object is dropped from rest at a height of 300 ft:
a. Find its velocity after 2 seconds
b. Find the time it takes for the object to reach the ground
c. With what velocity does it hit the ground
10. A stone is thrown horizontally from bridge 122.5 m above the level of the water. If the speed of the stone was 5 m/s what horizontal distance will the stone travel before striking the water.
11. A pistol that fires a signal flare gives the flare an initial speed (muzzle speed) of 120 m/s. a) If
12. A tennis ball rolls off the edge of a table top 0.75 m above the floor and strikes the floor at a point 1.40 m horizontally from the edge of the table. Ignore air resistance.
a). Find the time of flight.
b). Find the magnitude of the initial velocity.
c). Find the magnitude and direction of the velocity of the ball just before it strikes the floor.
13. A projectile is launched with speed vo at an angle [pic]above the horizontal. The launched point is a height h above the ground. Show that if air resistance is neglected, the horizontal distance that the projectile travels before striking the ground is [pic]. Verify that if the launch point is at ground level so that h =0, this is equal to the horizontal range R at y=0. (10 pts)
14. In an action-adventure film the hero is supposed to throw a grenade from his car, which is going 90 km/hr, to his enemy’s car, which is going 110 km/hr. The enemy’s car is 15.8 m in front of the hero’s when lets go of the grenade. If the hero throws the grenade so its initial velocity relative to him is at an angle of 450 above the horizontal, what should be the magnitude of the initial velocity? The cars are both traveling in the same direction on the level road. Ignore air resistance. Find the magnitude of the velocity both relative to the hero and relative to the earth. (10 pts)
Motion in a circle
Uniform circular motion
When a particle moves in a circle with constant speed, the motion is called uniform circular motion.
[pic] or [pic]
[pic]
[pic]
[pic] (uniform circular motion)
Non-uniform circular motion
If the speed varies, we call the motion non-uniform circular motion.
[pic] (radial or centripetal acceleration)
[pic] (tangential acceleration)
[pic] (acceleration)
Relative Velocity
[pic]
[pic]
[pic] (relative velocity along a line)
[pic]
[pic] (relative velocity in space)
DYNAMICS
(Relates Motion to Forces)
A System of Force
Our primitive concept of force is that it is a push or a pull exerted by our muscles. Changes in the motion of bodies are caused by some kind of interaction between them called force. Therefore, force maybe regarded as an action of one body on another. If the interaction between bodies does not produce motion, it means that the forces neutralize each other. If forces are not neutralized a change in motion of the body or system will result.
1 Classification of Forces
a. Concurrent - forces that act at a point or whose line of action converges or intersects at a common point.
b. Nonconcurrent - forces whose line of action does not converge at a common point.
c. External force - force that a body exerts on another body.
d. Internal force - forces exerted by one part of a body on other parts of same body.
e. Co-planar - forces acting on one plane.
f. Non-coplanar - forces acting in more than one plane.
2 Units of Force
MKS 6 Newton = force that will give a mass of 1 kg an acceleration of 1 m/s2.
CGS 6 dyne = force that will give a mass of 1 g an acceleration of 1 cm/s2.
FRS 6 poundal = force that will give a mass of 1 slug an acceleration of 1 ft/s2.
1 kg force = 9.8 N 1 lb force = 32 poundal
1 g force = 980 dynes 1 slug mass = 32 lb
B Newton’s Laws of Motion
1 Law of Inertia
There is no change in the motion of a body unless a resultant force is acting upon it.
If the body is at rest, it will continue at rest. If it is in motion it will continue in motion with constant speed in straight line unless there is net external force acting.
The mass of the body is a measure of its inertia.
Inertia refers to the property of tending to resist changes in their state of rest or uniform motion.
2 Law of Acceleration
If a net external force acts on a body, the body accelerates. The direction of acceleration is the same as the direction of the net force. The net force vector is equal to the mass of the body times the acceleration of the body.
The constant ratio of the net force to acceleration is a measure of inertia and is the mass of the body.
[pic]
1
Whenever, a net or unbalanced force acts on a body, it produces an acceleration in the direction of the resultant force, an acceleration that is directly proportional to the resultant force and inversely proportional to the mass of the body.
[pic]
2
F = Kma if k = unity
then F = ma
Consistent System of Units for Newton's Second Law
---------------------------------------------------------------------------------------
System Force Mass Acceleration
---------------------------------------------------------------------------------------
MKS (abs) N kg m/s2
CGS (abs) dyne g cm/s2
CGS (Grav'l) g-force m = w/g cm/s2
British (abs) poundal lb ft/s2
British (grav'l) pound slug ft/s2
---------------------------------------------------------------------------------------
C Procedure in the solution of Problems Involving Newtons 2nd Law
1. Make a sketch showing the conditions of the problem. Indicate dimensions or other given data.
2. Select for consideration the one body whose motion is to be studied. Construct force vector diagram.
3. From vector diagram find resultant force acting on the body.
4. Find unknown quantity. If weight is given compute m from m = w/g
Free-Body Diagrams
Drawing a correct free-body diagram is the first step in analyzing almost any physics or engineering problem involving the motion of a body.
Exercises. Construct the Free-Body-Diagram of the following figures below. All objects are objects are at rest.
|Figure 1. With friction. | |
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|Figure 2. With friction | |
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|Figure 3. With friction | |
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|Figure 4. With frictions, between blocks A&B, and between floor | |
|and block B | |
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|Figure 5. Frictionless | |
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|Figure 6. Weightless strut. | |
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|Figure 7. | |
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D Practice Exercise 7: Newton's Laws of Motion
1. A force of 60 dynes acts upon a mass of 15g a) What acceleration is imparted to the body, b) What velocity will the body acquire in 8s? c) What distance will the body cover in these 8s?
2. A 10-kg box starting from rest is pulled by means of a rope which make an angle of 30o with the horizontal. If it travels a distance of 10 km in 2s, what is the magnitude of the force exerted by the rope.
3. A horizontal cord is attached to a 6.0-kg body in a horizontal table. The cord passes over a pulley at the end of the table and to this end is hung a body of mass 8 kg. Find the distance the two bodies will travel after 2s, if they start from rest. What is the tension in the cord.
4. A string in a double inclined plane connects two bodies. If the 6.0 kg mass starts from the top of the plane and the length BC is 8 m. Find the velocity of the 6 kg body when it reaches the bottom of the plane.
5. The brakes of a 1000 kg automobile can exert a retarding force of 3000 N. Find the distance the car will move before stropping if it is travelling at the rate of 24 m/s when the brakes are applied (96 m).
6. An elevator weighs 2500 lb; a cable, which can sustain a maximum tension of 6000 lb, supports it. IF the elevator is going down with a velocity of 9 ft/s find the minimum safe distance it can travel before coming to a stop. (.9 ft)
7. A 6.0 kg body rests on a smooth horizontal table top. A horizontal cord attached to the body passes over a light frictionless pulley at the edge of the table to 1 2.0 kg body hanging freely. Find the acceleration and the tension in the cord when the system is released (2.45 m/s2, 14.7 N)
8. Two bodies, one of mass 4 slugs and the other of mass m is fastened to the two ends of a string. The string passes over a smooth pulley so that the two bodies hang vertically and the 4 slug mass is 4 ft from the floor. One second after the bodies are released, the 4 slug mass reaches the floor. Find the mass of the other body (2.4 slugs)
9. One side of a double inclined plane makes an angle of 30o with the horizontal, the other makes an angle of 53o. A 100 lb weight and a 50 lb weight are attached to the ends of the string which passes over a pulley at the top of the smooth double plane with the 50 lb load on the steeper side. Find the velocity of the 50 lb load when it reaches the bottom of the plane. Length of the steeper plane is 16.4 ft.
(2.16 ft/8.4 ft)
10. A body starting from rest acquires a velocity of 18 cm/s in 3 s. If the body has a mass of 15 g. What force is exerted on the body? (90 days)
11. A box weighing 15 lb is pulled by a force of 10 lb along a plane inclined 30o with the horizontal; the force being parallel to the plane. Starting from rest how far will the box travel after 5 sec? (66.7 ft).
E Law of Action - Reaction (Inter-Action)
If body A exerts a force on body B (an “action”), then body B exerts a force on body A (a “reaction”). These two forces have the same magnitude but are opposite in direction. These two forces act on different bodies.
Experience shows that the action of every force involves two bodies.
1. Acting force - force exerted by the body.
2. Reacting force - force exerted by another body which is equal in magnitude but opposite in direction on the first body.
Thus:
a. Forces always appears in pairs
b. Mutual action of two bodies upon each other are always equal in magnitude and opposite in direction
c. To every action there is an equal and opposite reaction
Momentum - product of the mass and the velocity of a body, Newton called it as the quantity of motion
[pic]
Impulse - product of the force and the time during which the first acts.
Impulse = Ft
[pic]
[pic]
From Newtons 2nd Law
[pic]
Hence impulse of a force is equal to the change in momentum
F Friction
Friction - refers to actual forces that are exerted to oppose motion
- a resistance that opposes every effort to slide or roll a body over another
Causes of Friction:
1. Interlocking of minute irregularities of the bodies
2. Adhesion
Laws of Friction:
1. The friction between surface sliding in one another depends upon
a. the nature of the substance,
b. condition of the surfaces,
c. the normal force pressing the surface together.
2. Friction between solids is independent of the area of the surfaces in contact and of the speed.
3. Static friction is greater than kinetic friction.
4. Kinetic friction is greater than rolling friction.
Static friction - force that will just start the body.
Kinetic friction - force that will pull the body uniformly.
[pic]
3
Coefficient of friction - the ratio of the force necessary to move one surface ever the other with uniform velocity to the normal force pressing the two surfaces to other.
Angle of Repose - when a body rests on an incline it is subject to the action of three forces; the weight which acts vertically; the reaction of the plane or the component of the weight normal to the plane and the component of the weight parallel to the plane, that tends to slide the Body downhill. By increasing gradually the inclination we will reach an angle at which the body just begins to slide. At this angle the component parallel to the plane will give the force necessary to start motion while the normal component will give the force pressing bodies together. This also is called the angle of repose the angle at which the body just begins to slide.
[pic]
4
Rolling friction - Friction force are such smaller when a wheel or circular object is pulled along a surface. Rolling friction varies inversely as to that radius of the wheels and directly to the hardness of the surface.
Fluid friction - friction encountered by solid objects passing through liquids and gases of liquid objects passing through gases. The frictional resistance experienced by a body moving through a fluid depends in a) size, b) shape c) speed of the moving object as well as on, d) the nature of the fluid itself "streamlining" is an attempt to reduce frictional forces.
Viscosity or internal friction - property of a fluid by which it resists flow. It due to the frictional forces between the molecule when an fluid flows over horizontal surface, the layer of fluid that is in constant with the surface remains stationary because of adhesion; but each successive layer of fluid moves with respect to the layer directly below it. The speed of each layer increased with its distance from the solid.
G Practice Exercises 8
1. What applied horizontal force is required to accelerate a 5 kg dv along a horizontal surface. With an acceleration of 2 m/s2 if the coefficient of friction is 0.15.
2. A 6.0 lb box is pulled along horizontal floor by a rope that makes an angle of 30o above the horizontal. The coefficient of kinetic friction between box and floor is 0.10. If the tension in the rope is 1.0 lb find the acceleration of the box.
3. A block of mass 3.0 kg slides with uniform velocity down a plane inclined 25o with the horizontal. If the angle of inclination is increased to 40o, what will be the acceleration of the block (2.7 m/s2).
4. An object traveling with a speed of 10 m/s slides on a horizontal floor. How far will it travel before coming to rest if the coefficient of friction is 0.30?
5. A stockroom worker pushes a box with mass 11.2 kg on a horizontal surface with a constant speed of 3.5 m/s. The coefficient of kinetic friction between the box and the surface is 0.20. a) What horizontal force must be applied by the worker to maintain the motion? b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest? Answer: a) 22 Ñ, b) 3.1 m
6. Consider the system shown in figure. Block A has weight wA and block B has weight wB. Once block B is set into downward motion, it descends at a constant speed. a) Calculate the coefficient of kinetic friction between block A and the table top. b) A cat, also of weight wA, falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration (magnitude and direction)?
7. Two crates connected by a rope lie on a horizontal surface. Crate A has mA, and mB. The coefficient of kinetic friction between each crate and the surface is (k. The crates are pulled to the right at constant velocity by a horizontal force F. In terms of mA, mB, and (k, calculate a) the magnitude of the force F; b) the tension in the rope connecting the blocks. Include the free-body diagram or diagrams you used to determine each answer.
Answer: [pic]
8. Block A, B, and C are C are placed as in figure and connected by ropes of negligible mass. Both A and B weigh 25 Ñ each, and the coefficient of kinetic friction between each block and the surface is 0.35. Block C descends with constant velocity. a) Draw two separate free-body diagram showing the forces acting on A and on B. b) Find the tension in the rope connecting blocks A and B. c) What is the weight of block C? d) If the rope connecting A and B were cut, what would be the acceleration of C?
Answer: b) 8.75Ñ, c) 30.8Ñ, d) 1.54m/s2
9. Block A, with weight 3w, slides down an inclined plane S of slope angle 36.90 at a constant speed while plank B, with weight w, rest on top of A. The plank is attached by a cord to the top of the plane. a) Draw a diagram of all the forces acting on block A. b) If the coefficient of kinetic friction is the same between A and B and between S and A, determine its value. Answer: b) 0.45
Dynamics of uniform Circular Motion
- a body moving in a horizontal circle with uniform speed
s ♠v
--- = ----
r v
vt ♠v
---- = ----
r v
v2 ♠v
---- = ----
r t
v2
---- = a - centripetal or radial acceleration
r
F = ma
mv2
F = -----
r
Centripetal force = force acting in the body directed towards the center.
Centripetal force = force acting of the body away from the center.
Conical Pendulum Motion in a vertical circle
1 Banking Curves
When a car turns around a curve there must be a centripetal force acting on it. On a horizontal surfaces, the centripetal force is furnished by frictional force between the tires and the road. IF the car turns round with a high speed and if the radius of curvature is small, then the necessary centripetal force is rather large and the frictional force may not be enough; as a result, the car may "skid" or it fails to make the necessary turn round the curve. This is especially so when the road is wet and very slippery so that the centripetal force will not be dependent on frictional force alone, most curves are "banked", the outer surface is elevated so that the road or surface is inclined.
v = [pic] maximum velocity of a car moving around a track, without skidding off the track.
A body whose base is small and whose center of gravity is high maintains its equilibrium in moving with a high speed around a track by inclining itself inward. The bigger the speed, the greater is the angle of inclination from the vertical.
[pic]
For equilibrium
[pic]
2 Gravitation
Laws of motion of planets is discovered by Johann Kepler:
1. Planets move around the sun in elliptical paths, with the sun in one of the force.
2. Any planet moves in such a manner that the radius joining the sun and the planet sweeps over equal areas in equal intervals of time.
3. The square of the period of revolution of any plant around the sun is proportional to the cube of its average distance from the sun.
Sir Isaac Newton generalized into on a law the laws discovered by Kepler and is now known as the Law of Universal Gravitation which states:
Any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
[pic]
[pic]
= 3.41 x 10-8 ft3/slug – sec2
Mass - quantity of matter is a body and therefore, it is the same over where mass is identical to inertia so that inertia is the quantitative or relative measure of mass.
Weight- force of attraction of the earth (gravitation) on a body. The weight of the body at a particular place is proportional to the mass, hence, one of the convenient methods of determining the mass of a body is to determine the weight.
I Practice Exercises 9
1. A body weighing 12 oz tied at the end of a string 3 ft long revolves around a vertical circle at the rate of 2 rps.
a) What is the tension of the string when the body is at the top of the circle?
b) What is the tension when the body is at the bottom of the circle.
c) What is the tension when the body is at the horizontal diameter.
2. A level track has a radius of curvature of 100 ft. What must be the coefficient of friction between the tires and the read for the circle have a safe speed of 20 mi per hour.
3. A 1 kg mass is attached to a cord 60 cm long and made to move as a conical pendulum. If the cord makes an angle of 30o with the vertical, find the time it takes for the mass to make one complete revolution. Find the tension in the cord for this configuration.
4. The moon has a mass of 7.32 x 1022 kg and a radius of 1609.4 km. Calculate the value of "g" at the surface of the moon.
5. How large must the coefficient of friction be between the tires and the road if a car
is to round a level curve of radius 62 m at a speed of 55 km/h?
6. A child moves with a speed of 1.50 m/s when 7.8 m from the center of a merry-go-
round. Calculate a) the centripetal acceleration of the child, and b) the net horizontal
force exerted on the child (mass = 25 kg).
7. A ball on the end of a string is revolving at a uniform rate in a vertical circle or
radius 96.5 cm as shown in the figure below. If its speed is 3.15 m/s and its mass is
0.335 kg, calculate the tension in the string when the ball is a) the top of its path ,
and b) at the bottom of its path.
mg
T
T
mg
5. Calculate the acceleration due to gravity at the surface of the moon. The moon’s radius is about 1.7x106 m and its mass is 7.4x 1022 kg.
6. Four 8.0-kg spheres are located at the corners of the square of side 0.50 m. Calculate the magnitude and direction of the gravitational force on one sphere due to the other three.
7. At what distance from the earth will a spacecraft on the way to the moon experience zero net force because the earth and moon pull with equal and opposite forces?
8. A coin is place 12.0 cm from the axis of the rotating turntable of variable speed. When the speed of the turntable is slowly increase, the coin remains fixed on the turntable until a rate of 58 rpm is reached, at which point the coin slides off. What is the coefficient of static friction between the coin and the turntable?
9. Calculate the force of gravity on a spacecraft 12,800 km above the earth’s surface if its mass is 850 kg.
10. What minimum speed must a roller coaster be travelling when upside down at the top of a circle if the passengers are not to fall out? Assume a radius of curvature of 8.0 m.
11. Calculate the centripetal acceleration of the earth in its orbit around the sun and the net force exerted on the earth. What exerts this force on the earth? Assume the earth’s orbit is a circle of radius 1.49 x 10 11 m.
12. A 1200-kg car rounds a curve of radius 65 m banked at an angle of 140. If the car is travelling at 80 km/h, will a friction force be required? If so, how much and in what direction?
13. A ball of mass ‘M’ is revolved in a vertical circle at the end of a cord of length ‘L’ .
What is the minimum speed ‘v’ needed at the top of the circle if the cord is to
remain taut?
17. If a curve with a radius of 60 m is properly banked for a car travelling 60 km/h,
what must be the coefficient of static friction for a car not to skid when travelling at
90 km/h?
18. How far above the earth’s surface will the acceleration of gravity be half what it is
on the surface?
19. How fast in (rpm) must a centrifuge rotate if a particle 9.0 cm from the axis of
rotation is to experience an acceleration of 110,000 g’s?
20. Given: Figure:
m = 10 kg
h = 5 m
Ø = 4 m
L =3 m
S1 = 2 m
Ø =30o
[pic] coef. of kinetic friction
Note:
❖ Particle “m” is release from rest at pt. A and moves
to pt. B, then to pt.C, and finally to pt.D.
❖ Neglect the effect of the change in velocity direction at pt.B.
❖ The same value of coef. friction, [pic] from pt.A to pt.C.
❖ Projectile motion from pt.C to pt.D.
Required (20 pts)
a. Free-body diagram of the particle at inclined plane AB.
b. Free-body diagram of the particle at horizontal plane BC.
c. Unbalanced force of the particle along the inclined plane.
d. Unbalanced force of the particle along the horizontal plane BC.
e. acceleration, a1 of the particle along the inclined plane.
f. acceleration, a2 of the particle along the horizontal plane BC.
g. velocity of the particle at pt.B.
h. velocity of the particle at pt.C.
i. Range, (S2)
j. Time of travel of particle from pt A to pt. D.
2. Two masses are connected by a light string that passes over a frictionless pulley. If the incline has a coef. of friction equal to 0.1 and if m1=2 kg., m2=6 kg, and [pic], find (a) the acceleration of the masses, (b) the tension in the string, and (c) the speed of each mass 2sec after released from rest.
3. A penny of mass 3.10g rests on a small 20g block supported by a spinning disk. If the coefficients of friction between block and disk are 0.750 (static) and 0.640 (kinetic) while those for the penny and block are 0.450 (kinetic) and 0.520 (static), what is the maximum rate of rotation (in revolution per minute) that the disk can have before either the block or the penny starts to slip?
STATICS
Study on body that is at rest or in equilibrium
1 Resultant and component of forces
Since forces are vector quantities, the process of finding the resultant of this or more forces can be done by any method of vector addition.
Composition of forces = process of finding a single force which will produce an effect the same as the effect produced by the given force.
Resolution of forces = a single force is broken into separate forces called the components of the force. The component of the force in a given direction is its effective value in that direction.
Components of forces can be determined by rectangular components, however, components are not necessarily be along the horizontal and vertical direction.
P = tends to pull down along (or || to) the plane
N = pushes the body against (or ( to) the plane.
2.5 Equilibrium of Particles
A particle which has no net force acting to it is said to be in equilibrium.
Equilibrant - single force that holds two or more forces in equilibrium. It prevents the motion of the body which is equal to the resultant but in opposite direction.
Equilibrium – state in which there is no change in the motion of the body.
a. Static equilibrium - resultant of all the forces is zero and the body is at rest.
b. Dynamic equilibrium - resultant of all forces is zero, velocity of body is constant thus body moves in uniform motion.
B First condition for equilibrium
In order that the translational motion of a body will not change, the vector sum of all the forces acting on it, must be equal to zero.
(F = 0
If the resultant of all the forces acting on a body is zero, the sum of the rectangular components of these forces along any axis must be zero.
Usually, but not always, it is desirable to choose vertical and horizontal axes.
1. The sum of all upward force components is equal to the sum of all downward force components.
2. The sum of all force components to the right is equal to the sum of all force components to the left.
3. The sum of all force components upward is equal to the sum of all components downward.
C Practice Exercises: First Condition for Equilibrium:
1. A pendulum bob with a weight of 20 N hangs from a cord. A horizontal force sufficient to bring the cord to an angle of 25o with the vertical is applied to the bob. Find the horizontal force and the tension in the cord. (9.3 N; 22.1 n)
2. A tightly stretched high wire is60 m long and sags 3.2 m when a 60-kg tightrope walker stands at its center. What is the tension in the wire?
3. A uniform beam 10 ft long and weighing 10 lb is hinged at one end to a vertical wall. The beam is supported in a horizontal position by a rope tied to the free end. The rope is attached to the wall and makes an angle of 45o with the vertical. What is the tension in the rope and the force of the hinge on the beam.
4. A 100-kg man sits on a hammock whose ropes makes 30o with the horizontal. What is the tension on each part of the rope?
5. A car is stuck in the mud. To set it out; the driver ties one end of a rope to the car and the other end to a tree 100 ft away. He then pulls sideways on the rope at its midpoint. If he exerts a force of 120 lb. how much force is applied to the car when he has pulled the rope 5 ft to one side?
6. Two strings support a lamp weighing 12 lb. If one string makes an angle of 30o with the horizontal and the other string makes an angle of 45o with the horizontal find the tension of the two ropes. (8.78 lb; 10.76 lb)
7. An object weighing 50 lb is set on the surface of a plane inclined 40o with the horizontal. What force applied parallel to the plane, is required to keep the object in equilibrium. Neglect friction. (31 lb)
8. A frictionless car standing in an inclined plane that makes an angle of 15o with the horizontal is kept from rolling downhill by a force of 12 N. applied in a direction parallel to the plane. What is the weight of the car? What is the normal force exerted on the car to the plane. (46 N; 44 N).
9. A 30-kg traffic light is supported by two wires one of which makes an angle of 20o with the horizontal while the second makes an angle of 10o. Find the tension in each. (60 kg; 58kg).
D Second condition for equilibrium
When vector sum of concurrent forces acting on a body is zero, then there is no change in its translational motion but when there are several non-current forces acting on a body, there is in general a change in the state of rotational motion.
In order that the rotational motion of a body will not change the sum of the torques about any axis acting on the body must be equal to zero.
(M = 0
Clockwise torque = counterclockwise torque
1 Torque or moment of force
- measures the effectiveness of the force in changing rotation about the chosen axis.
- turning effect of the force
The effect of a force on a rotational motion of a body depends on:
1. Magnitude of the force
2. Torque arm or moment arm = perpendicular distance from the axis of rotation (fulcrum) to the line of action of the force.
To change the state of rotational motion of a body, the point of application of the force is taken into consideration.
The effect of a given force upon the rotational motion of a body is greater, the further the line of action of the force if from the axis of rotation.
If the line of action of the applied force passes through the axis of rotation then this force will not produce any change in the rotational motion of the body.
2 Center of gravity
- point about which the sum of gravitational torques is equal to zero.
- point of application of the resultant of the attraction that the earth exerts upon all the particles of a body.
- a point in which the total weight of the body is concentrated.
- point where object balances
The center of gravity of a body may be situated outside of the body. Center of gravity determines the stability of the body.
When a force whose line of action passes through the C-G, this force will affect only the translational motion of the body but when line of action does not pass through the C-G both translational and rotational motion of the body is affected.
3 Determination of C-G
d.3.1 Uniform body
d.3.2 Non-uniform body
E Types of Equilibrium
1. Stable - G.G. at its lowest position, object displaced slightly will return to original position.
2. Unstable - object when displaced slightly will change position completely.
3. Neutral - object when displaced location of center of gravity remains the same.
Illustration
[pic]
F = F1 + F2 + F3
[pic]
LF = F1L1 + F2L2 + F3L3
F Practice Exercise: Second Condition for Equilibrium
1. A bar 10 ft long is acted upon by a force of 20 lb that makes an angle of 60o with the bar. Calculate the torque due to this force about an axis perpendicular to the bar and a) through the near end of the bar, b) through the middle of the bar, c) through the far end of the bar.
2. A piece of wooden bar 4 ft long and weighing 500 g has its center of gravity 18 in from one end. Where must a 300 g weight be hung so that the bar can be suspended at the middle?
3. A uniform bar weighs 50 lb and is 12 ft long. At one end, a load of 16 lb is attached and on the other end a load of 32 lb. Determine the force to be applied to the other end so that it remains in a horizontal position where will this force be applied?
4. A non-uniform bar rests across two supports that are 20 ft apart when loads of 200 lb, 4 ft from end A and 150 lbs 6 ft from end B are on the bar. End A supports 233 lb and end B supports 197 lb. Find the weight of the bar and the position of C-G.
5. A ladder is 25 ft long, has its center of gravity 8 ft from the bottom and weighs 60 lb. A man weighing 160 lb stands halfway up the ladder, which makes an angle of 20o with the vertical. Find the force exerted on the ladder by the smooth wall and the horizontal and vertical components of the force exerted on the ladder by the ground. (36 lb; 36 lb; 220 lb)
6. Two vehicles are crossing a bridge 60 ft long. A passenger car weighing 3000 lb is 10 ft from one end. A truck weighing 9000 lb is 20 ft from the same end. If the bridge is symmetrical with respect to the center and weighs 50 tons, what are the forces on the two supports at the ends of the bridge? (58,500 lbs, 53 500 lbs)
7. A seesaw is 10 ft long. A boy weighing 80 lb sits at one end of the seesaw. At what point on the other side of the seesaw must a man weighing 175 lb sit in order to balance the boy.
WORK, ENERGY AND POWER
A Work
In ordinary language, work may mean any form of physical or mental activity. In physics and engineering, work is accomplished only when a force acts on a body and this force is able to move the body.
Work is defined is either of two ways:
1. It is the product of the force and the component of the displacement in the direction of the force or
2. The product of the displacement and the component of the force in the direction of the displacement.
Work is positive if the applied force is in the same direction as the displacement, negative if force is in opposite direction.
In general work may be done on a system in three ways:
1. If the force is just enough to impart uniform motion on a body, the first of friction has done the same amount of work.
2. In changing the position or configuration of the body or system as in raising the body or in compressing a spring.
3. Imparting acceleration to the body or system.
In (1) work spent is converted to heat (wasted work) while in (2) and (3) work done is stored in the body in the form of conserved energy.
Units of work:
------------------------------------------------------------------------------------------------
Force (F) Displacement (S) Work (W)
------------------------------------------------------------------------------------------------
CGS (absolute) dyne cm erg
(gravitational) grm force cm g-cm
MKS (absolute) Newton meter joule
(gravitational) kg force m kg-m
Eng'g (absolute) poundal ft foot poundal
(gravitational) lb force ft ft-lb
-----------------------------------------------------------------------------------------------
1 joule = N – m = 105 dynes (102 cm) = 107 ergs
B Energy
Energy is often associated with work. When work is done on the body, a change is produced in the body.
1. Change in its motion or inertia
2. Change in position
3. Change in temperature (work to overcome friction)
The change produced in the body by the application of work is often times called energy so that energy may be thought of as the ability or capacity to do work.
An agent is said to possess energy if it is also to do work. Energy being the maximum work, then the units of energy is the same as the units of work.
There are many forms of energy, mechanical, chemical, electrical, heat or thermal electromagnetic, nuclear, etc. Our daily observations make us realize that transformations occur from one form of energy to another.
Forms of mechanical energy:
1. Potential – energy at rest
2. Kinetic - energy in motion
Kinds of P-E:
1. Gravitational P-E - energy possessed due to its position or elevation.
2. Elastic P-E - due to its state of strain or configuration/
3. Chemical P-E - due to its composition
f = P = mg sin ( W = wh
w = (mg sin () L but sin ( = h/L = mgh = P.E
= (mg h/L)L
W = mgh = P-E
Thus P-E at the top of the plane is the product of the weight multiplied by the vertical height irrespective of the path taken by the body.
Kinetic energy:
A moving body is capable of doing work the amount of work it can do depends on its mass and velocity.
W = FS but F = ma
from v2 = 2as + v2o
= [pic] as = [pic]
In general if the force results only in the increase of the velocity of a body of mass m from a certain initial velocity to a final velocity then the work done is equal to the increase in K-E of the body.
W = FS = [pic]
C Power
Given enough time one can do much work as the other so that the difference in two machines is in the amount of work it can do per unit time or the rate at which work is done called power.
[pic]
5
Units of power
----------------------------------------------------------------------------------------------
P W t
---------------------------------------------------------------------------------------------
CGS erge/s ergs sec
MKS watt joule sec
FPs ft-lb/s ft-lb sec
---------------------------------------------------------------------------------------------
Traditional machine, power is expressed in horsepower.
ft-lb
1 Hp = 550 --------
s
ft-lb
= 33000 --------
min
= 745.7 watts = 746 watts
= 3/4 kw
= 2545 Btu/hr.
W = Pt =kw-hr
1 Kw = work done in one hour by a system working at a constant rate of 1 Kw.
1 kw = 1000 j/s.
Total work done in one hour with a power of 1 kw is:
j
1 kw - hr = 1000 ---- x 3600 s
s
= 3.6 x 106 joules
D Practice Exercises
1. 1. A force of 8 lbs. pulls a body along a horizontal surface to a distance of 10 ft. a) How much work is done, b) If the force acts at an angle of 30o above the horizontal, how much work is done?
2. A 100-g object is dragged with a uniform velocity along a plane inclined 30o with the horizontal by a force parallel to the inclined. If the coefficient friction between the object and the plane is 0.2, how much work is done when the object is moved a distance of 40 cm along the plane?
3. A man weighing 120 lbs, climb up a stairway inclined 45o consisting of 20 steps each step 6" high. What is his potential energy at the top.
4. A body a mass 10 slugs is thrown with a velocity of 6 ft/s. along a horizontal floor. The coefficient of friction between the body and the floor is 0.2.
Find a) K-E and velocity of the body after travelling a distance of 2 ft. b) How far will the body travel before it comes to rest.
5. A body weighing 64 lb slides down from rest at the top of a plane 18 ft long and inclined 30o above the horizontal. The coefficient of friction is 0.1. Find the velocity of the body as it reaches the bottom of the plane.
6. A 20 Hp engine is used to lift gravel from the ground to the top of a building 60 ft high. Neglecting loss of energy due to friction how many tons of gravel can be lifted in 50 minutes.
7. What weights can a 6 Hp engine pulls along a level road at 15 mi/hr if the coefficient of friction between the weight and the road is 0.2?
8. A rock of mass 2.0 kg is dropped from a bridge. After it has fallen 6.5 m, a) how much potential energy has it lost; b) how much kinetic energy has it gained; c) from your answer to (b), how fast is it going?
9. 3.0 kg cart is pushed on a horizontal frictionless surface. It is pushed 2.5 m with a force 12 N, and the force then changes to 18 N and pushes it another 1.8 m (a) How much work has on it? (b) What is its kinetic energy? (c) How fast is it going?
10. A 35-kg crate slides from rest down a rough inclined plane, going a vertical distance of 2.5 m. When it reaches the bottom, it is going 6.2 m/sec. (a) How much potential energy has it lost? (b) How much kinetic energy has it gained?
11. A crate is pulled for a distance of 10 meters by means of a rope that makes an angle of 450 with the ground. If the force exerted on the rope is 300 N, how much work is done?
12. A force of 200 N is exerted in lifting a 10 kg mass straight up to a height of 5 m (a) How much work is done? (b) What are the kinetic and potential energy of the object when it gets to that height?
13. If it takes a force of 1 N to depress a typewriter key through a distance of 1 cm in a time of 0.1 sec, how much average power does it take?
14. A bullet is shot straight up with a muzzle velocity of 600 m/sec. Find out how high it will rise by equating its original kinetic energy to the potential energy it has at the highest point. Notice that we have not specified the mass of the bullet.
15. A tennis ball with a mass of 60 grams is dropped to the floor from a height of 1 m and bounces back to a height of 0.8 m. Using the conservation of energy law, find: (a) its velocity just before if struck the floor and just after it started up again, (b) the energy lost in the collision.
16. A man shoves a box with a mass of 50 kg across the floor with a force of 100 N through a distance of 5 m. He then shoves it up a 300 inclined to a height of 1 m by exerting a force equal to 3/5 its weight. What is the final potential energy of the box?
17. What is the escape energy necessary to free a mass of 1 kg from the earth?
18. What is the escape energy necessary to free a mass of I kg from the moon’s influence, starting from the surface of the moon?
19. A bullet is fired straight up. It is given enough kinetic energy so that as it rises, the loss of kinetic energy is just sufficient to supply the potential energy needed for it to escape the earth’s gravitation, and it will never come back. At what speed must is the fired?
20. What is the potential energy for an 800-kg elevator at the top of Chicago’s Sear Tower, 440 m above street level? Let the potential energy be zero at street level.
Ans. a) 3.45x106 J
21. A baseball is thrown from the roof of a 22.0-kg tall building with an initial velocity of magnitude 12.0 m/s and directed at an angle of 53.1o above the horizontal. A) What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. B) What is the answer for part (a) if the initial velocity is at an angle of 53.1o below the horizontal? C) If the effect of air resistance are included, will part (a) or (b) give the higher speed?
Ans. a) 24m/s; b) 24 m/; c) part (b)
22. A force of 800 N stretches a certain spring a distance of 0.200 m. a) What is the potential energy of the spring when it is stretched 0.200 m? b) What is its potential energy when it is compressed 5.00 cm?
Ans. a) 80 J, b) 5 J
23. A spring of negligible mass has force constant k=1600 N/m. a) How far must the spring be compressed for 3.20 J of potential energy to be stored in it? b) You place the spring vertically with one end on the floor. You then drop a 1.20-kg book onto it from a height of 0.80 m above the top of the spring. Find the maximum distance the spring will be compressed.
Ans. a) 6.32 cm, b) 12 cm
MOMENTUM, IMPULSE, AND COLLISIONS
Momentum And Impulse
[pic]
[pic] (definition of momentum)
[pic] (Newton’s second law in terms of momentum)
[pic]
[pic] (definition of impulse)
[pic] (impulse-momentum theory)
[pic] (general definition of impulse
Conservation of Momentum
[pic]
[pic]
[pic]
[pic]
Principle of Conservation of Momentum: “If the vector sum of the external forces on a system is zero, the total momentum of the system is constant”
• Elastic Collision: If the forces between the bodies are also conservative, so that no mechanical energy is lost or gained in the collision, the total kinetic energy of the system is the same after the collisions before.
• Inelastic Collision: If a collision in which the total kinetic energy after the collision is less than that before the collision.
Completely inelastic Collision: If an inelastic collision in which the colliding bodies stick together and move as one body after the collision.
Remember the following rule: In any collision in which external forces can be neglected, momentum is conserved and the total momentum before and after equals the total momentum after; in elastic collisions only, the total kinetic energy before equals the total kinetic energy after.
Example. Collision in a horizontal plane. Figure below two chunks of ice sliding on a frictionless frozen pond. Chunk A, with mass mA=5 kg, moves with initial velocity vA1=2 m/s parallel to the x-axis. It collides with chunks B, which has a mass mB=3 kg and is initially at rest. After the collision, the velocity of chunk A is found to be vA2 = 1m/s in a direction. What is the final velocity of chunk B?
Figure:
FLUID MECHANICS
Fluid Statics
Density defined as its mass per unit volume. [pic] units: [g/cc or kg/m3 or lb/ft3]
Density of water is constant. 1 g/cm3 = 1000 kg/m3
Specific gravity of a material is the ratio of its density to the density of water at 4.00C. It is pure number.
Pressure in a Fluid
• Pressure p at that point as the normal force per unit area, that is that ratio of [pic] to [pic]:
[pic]
units: 1 pascal = 1 Pa = 1 N/m2 1 bar = 105 Pa 1 mbar = 100 Pa
• Atmospheric pressure pa is the pressure of the earth’s atmosphere, the pressure at the bottom of this sea of air in which we live. This pressure varies with weather changes and with elevation. Normal atmospheric pressure at sea level (an average value) is 1 atmosphere (atm), defined exactly 101,325 Pa.
[pic]
Pressure in a fluid of uniform density: [pic]
Pascal’s Law: “ Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of a containing vessel.
• Gauge pressure is the excess pressure above atmospheric pressure.
• Absolute pressure is total pressure of atmospheric and gauge pressure.
Archimedes’ Principle states: “when a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.”
Fluid Dynamics
• Flow line – a path of an individual particle in a moving fluid.
• Steady Flow – if the overall flow pattern does not change with time.
• Streamline – a curve whose tangent at any point is in the direction of the fluid velocity at that point.
• Flow Tube – the flow lines passing through the edge of an imaginary element of area form a tube.
• Laminar flow – flow patterns in which adjacent layers of fluid slide smoothly past each other and the flow is steady.
• Turbulent flow – sufficiently high flow rates, or when boundary surfaces cause abrupt changes in velocity, the flow can become irregular and chaotic.
• Continuity Equation – the mass of a moving fluid doesn’t change as it flows.
[pic] (continuity equation, incompressible fluid)
[pic] (volume flow rate)
[pic] (continuity equation, compressible fluid)
[pic] (Bernoulli’s equation)
Let: [pic] (unit weight of fluid)
Then: [pic] (Bernoulli’s energy equation)
Where: [pic] is velocity head (m)
[pic] is pressure head (m)
[pic] is elevation head (m)
Thus: The total heads at any point of a moving fluid is always constant
Answer the questions
1. As you walk past a cup of coffee sitting on a desk, the coffee has velocity relative to you. Would you describe the coffee with fluid statics or fluid dynamics? Why?
2. Why do jet airplanes usually fly at altitudes above 30,000 ft, though it takes a lot of fuel to climb that high?
3. An old question is “Which weighs more, a pound of feathers or a pound of lead?” If the weight in pounds is the gravitational force, will a pound of feathers balance a pound of lead on opposite pans of an equal-arm balance? Explain, taking into account buoyant forces.
4. You are told, “Bernoulli’s equation tells us where there is higher fluid speed, there is lower fluid pressure, and vice versa.” Is this statement always true, even for an idealized fluid? explain. You take an empty glass jar and push it into a tank of water with the open mouth of the jar downward, so that the air inside the jar is trapped and cannot get out. If you push the jar deeper into the water, does the buoyant force on the jar stay the same? If
THERMODYNAMICS
A Temperature and Heat
• Thermodynamics is the study of energy transformation involving heat, mechanical work, and other aspects of energy and how these transformations relate to the properties of matter.
• Temperature is rooted in qualitative idea s of “hot” and “cold” based on our sense of touch.
• Thermometer measures temperature.
• Thermal Equilibrium – if two bodies must have the same temperature. A conducting material between two bodies permits interaction leading to thermal equilibrium; an insulating material prevents or impedes this interaction.
• The Celsius and Fahrenheit temperature scales are based on the freezing temperature (0oC = 32 oF) and the boiling temperature (100oC = 212oF) of water.
[pic]; [pic]; and [pic]
• The Kelvin scale has zero at the extrapolated zero-pressure temperature for a constant-volume gas thermometer, which is -273.15oC. Thus 0 K= -273.15oC, and
[pic]
• Under a temperature change [pic], any linear dimension Lo of a solid body changes by an amount [pic] given approximately by
[pic]
where [pic] is the coefficient of linear expansion. Under a temperature change [pic], the change [pic]in the volume Vo of any solid or liquid material is given approximately by
[pic]
where [pic] is the coefficient of volume expansion.
• Heat is energy in transit from one place to another as a result of a temperature difference. The quantity of heat Q required to raise the temperature of a mass m of material by a small amount [pic] is
[pic]
where c is the specific heat capacity of the material. When the quantity of material is given by the number of moles n, the corresponding relation is
[pic]
where [pic]is the molar heat capacity (M is the molar mass). The number of moles n and the mass m of material are related by m = nM.
• To change a mass m of a material to a different phase at the same temperature (such as liquid to vapor or liquid to solid) requires the addition or subtraction of a quantity of heat Q given by
[pic]
where L is the heat of fusion, vaporization, or sublimation.
• Three mechanics of heat transfer:
1. Conduction is transfer of energy of molecular motion within materials without bulk motion of the materials.
2. Convection involves mass motion from one region to another.
3. Radiation is energy transfer through electromagnetic radiation.
• The heat current H for conduction depends on the area A through which the heat flows, the length L of the heat path, the length L of the heat path, the temperature difference (TH - TC ), and the thermal conductivity k of the material:
[pic]
• The heat current H due to radiation is given by: [pic]
B Thermal Properties of Matter
• The pressure p, volume V, and temperature T of a given quantity of a substance are called State Variables. They are related by an Equation of State. The Ideal-gas equation of state is
[pic],
where n is the number of moles if gas and T is the absolute temperature. The constant R is the same for all gases for conditions in which this equation is applicable.
• A pV-diagram is a set of graphs, called isotherms, each showing pressure as a function of volume for a constant temperature.
• The molar mass M of a pure substance is the mass per mole. The total mass mtot is related to the number of moles n by
[pic]
Avogadro’s number NA is the number of mole. The mass m of an individual molecule is related to M and NA by
[pic]
• The average translational kinetic energy of the molecules of an ideal gas is directly proportional to absolute temperature:
[pic]
By using the boltzmann constant, [pic], this can be expressed in terms of the average translational kinetic energy per molecule:
[pic]
The root-mean-square speed of molecules in an ideal gas is
[pic]
• The molar heat capacity CV at constant volume for an ideal monatomic gas is
[pic]
For an ideal diatomic gas, including rotational kinetic energy,
[pic]
For an ideal monatomic solid
[pic]
• The speeds of molecules in an ideal gas distributed according to the Maxwell-Boltzmann distribution:
[pic]
C The First Law of Thermodynamics
• A thermodynamic system can exchange energy with its surroundings by heat transfer or by mechanical work and in some cases by other mechanisms. When a system at pressure p expands from volume V1 to V2 , it does an amount of work W given by
[pic]
If the pressure p is constant during the expansion,
[pic] (constant pressure only)
• In any thermodynamic process, the heat added to the system and the work done by the system depends not only on the initial and final states but also on the path (the series of intermediate states through which the system passes).
• The first law of thermodynamics states that when heat Q is added to a system while it does work W, the internal energy U changes by an amount
[pic]
In an infinitesimal process,
[pic]
The internal energy of any thermodynamic system depends only on its state. The change in internal energy in any process depends only on initial and final states, not on the path. The internal energy of an isolated system
•
[pic]
Hypothetical setup for studying the behavior of gases
[pic]
PVT-surface
[pic]
Cycle of a four-stroke internal-combustion engine
[pic]
Human cyclic thermodynamic process
Problems in Thermodynamics
Ex. Equation of state. A 20.0-L tank contains 0.225 kg of helium at 18.0oC. The molar mass of helium is 4.00 g/mol. a) How many moles of helium are in the tank? b) What pressure in the tank, in pascals and in atmospheres? (Ans.56.2 mol, 6.81x106Pa=67.2 atm)
Ex. Equation of state. A cylinder tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110m3 of air at a pressure of 3.40 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 m3. If the temperature remains constant, what is the final value of the pressure? (Ans. 0.959 atm )
Ex. First Law of Thermodynamics. A gas in a cylinder is held at a constant pressure of 2.30x105 Pa and is cooled and compressed from 1.70 m3 to 1.20 m3. The internal energy of the gas decreases by 1.40x105 J. a) Find the work done by the gas. b) Find the absolute value [pic] of the heat flow into or out of the gas, and state the direction of heat flow. c) Does it matter whether or not the gas is ideal? (Ans. a) -1.15x105 J, b) 2.55x105J, out of gas, c) no
Ex. Adiabatic process. During an adiabatic expansion the temperature of 0.450 mol of Argon (Ar) drops from 50oC to 10oC. The argon may be treated as an ideal gas. a) Draw a pV-diagram for this process. b) How much work does the gas do? c) Does heat flow into or out of the gas? If so, what is the direction and absolute value of this heat flow? d) What is the change in internal energy of the gas? [Ans. b) 224 J, c) Q=0, d) -224 J]
Heat Engines
Ex. A diesel engine performs 2200 J of mechanical work and discards 4300 J of heat each cycle. a) How much heat must be supplied to the engine in each cycle? b) What is the thermal efficiency of the engine? (Ans. a) 6500J, b) 0.34=34%)
Ex. A gasoline Engine. A gasoline engine takes in 16,100J of heat and delivers 3700J of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of 4.60x104 J/g. a) What is the thermal efficiency? b) How much heat is discard in each cycle? d) If the engine goes through 60 cycles per second, what is its power output in kilowatts? In horsepower?
(Ans. a) 0.23=23%, b) 12,400J, c) 0.350 g)
Ex. What compression ratio r must an Otto cycle have to achieve an ideal efficiency of 65% if [pic] ?
Ex. A refrigerator has a coefficient of performance of 2.10. Each cycle it absorbs 3.40x104 J of heat from the cold reservoir. a) How much mechanical energy is required each cycle to operate the refrigerator? b) During each cycle, how much heat is discarded to the high-temperature reservoir?
[Ans. a)1.62x104J, b) 5.02x104J ]
Molar Heat Capacities of Gases at Low Pressure
|Types of Gas |Gas |CV |Cp |Cp-Cv |[pic] |
| | |(J/mol.K) |(J/mol.K) |(J/mol.K) | |
|Monatomic |He |12.47 |20.78 |8.31 |1.67 |
| |Ar |12.47 |20.78 |8.31 |1.67 |
|Diatomic |H2 |20.42 |28.74 |8.32 |1.41 |
| |N2 |20.76 |29.07 |8.31 |1.40 |
| |O2 |20.85 |29.17 |8.31 |1.40 |
| |CO |20.85 |29.16 |8.31 |1.40 |
|Polyatomic |CO2 |28.46 |36.94 |8.48 |1.30 |
| |SO2 |31.39 |40.37 |8.98 |1.29 |
| |H2S |25.95 |34.60 |8.65 |1.33 |
Second Law of Thermodynamics
It is impossible for any system to undergo a process in which it absorbs heat from a reservoir at a single temperature and converts the heat completely into mechanical work, with the system ending in the same state in which it began.
Engine Problem
A heat engine takes 0.35 mol of a diatomic ideal gas around the cycle shown in the pV-diagram of Figure below. Process 1→2 is at a constant volume, process 2→3 is adiabatic, and process 3→1 is at a constant pressure at 1.00 atm. The value of γ for this gas is 1.40. a)Find the pressure and volume at three points 1,2, and 3. b) Calculate Q, W, and ΔU for each of the three processes. c) Find the net work done by the gas in the cycle. d) Find the net heat flow into the engine in one cycle. e) What is the thermal efficiency of the engine? How does this compare to the efficiency of a carnot-cycle engine operating between the same minimum and maximum temperatures T1 and T2?
ANSWERs:
a) p1= 1 atm, V1=8.62x10-3 m3; p2=2 atm, V2=8.62x10-3 m3
p3=1 atm, V3=1.31x10-2 m3
b) 1→2: Q=2183 J, W=0, ΔU=2183 J
2→3: Q=0, W=1055 J, ΔU= -1055 J
3→1: Q=-1579 J, W= -451 J, ΔU= -1128 J
c) 604 J d) 604 J e) e = 27.7% ecarnot =50%
ELECTROSTATICS (ELECTRICITY AT REST)
A Electrification
Mass is a basic quality of matter. It is the root of the gravitational interaction. Matter has a second basic quality, charge, which is the root of the electric interaction. Similar in many respects to gravitational interaction yet differing from it in significant ways.
We may speak of electric potential energy, just as we speak of gravitational potential energy. Kinetic energy is associated with mass in motion, likewise energy maybe associated with charge in motion. In investigating electricity, the idea of energy serves as a constant guide.
1 Electrical Nature of Matter
Rubbing two different materials will produce static electricity due to friction contact property of substance to attract light objects when rubbed was known by the Greeks as early as 600 B.C. Tholes observed these effect were particularly strong in electron a greek word for amber.
Thus when a body has a acquired the property of attracting light objects we say that it is charged or electrified.
Process of producing a charge on an object is called electrification.
Kinds of electrification:
1. Negative electrification - produced on the rubber rod when stroked with flannel or fur. (Rod is said to be negatively charged).
2. Positive electrification - produced on the glass rod when rubbed with silk.
Term positive and negative were adopted by Benjamin Franklin for convenience.
Originally, these two charges were called vitreous electricity (positive) and resinous electricity (negative) by Charles Dufay.
Uncharged objects contain equal amount of positive and negative electricity.
2 Qualitative law of electricity
1. Charges of same kind repel
2. Charge of opposite kind attracts
3. A charge body always attract a non-charged body
4. Two kinds of charges are produced simultaneously
5. They can be dissociated
Electron Theory an Atomic structure according to the modern theory of atomic structure proposed by Dalton, Sir Erhest Rutherford and Neils Bohr; all matter is composed of atoms.
Atom is pictured as consisting of centrally located nucleus which is composed of:
1. Protons - positive charge
2. Neutrons - neutral or no charge
3. Whirling around the nucleus like planets around the negatively charge electrons which are held in their orbits by electrical force of attraction.
The size of atom, which is the size of the orbit of the electron, is of the order 10-8 cm in diameter.
Diameter of the nucleus is of the order 10-12 cm. Each atom has its own number of electrons and protons.
Normal State
Number of electrons - no. of protons in the nucleus
Therefore atom as a whole is electrically neutral
Atomic number of an atom is the number of electrons and in therefore the number of protons in the nucleus.
The protons of an atom are packed together in the nucleus together with the neutrons.
Number of neutrons is equal to or more than the number of protons.
The mass number of an atom is equal to the number of protons in the nucleus plus the number of neutrons.
3 Free Electrons
Electrons arranged themselves in one or more fixed orbit or shell. Each atom has its own pattern of arrangement of electron.
Shells are lettered from innermost K L M N O P and Q. Each shell has definite maximum capacity for holding electrons. Theoretically number of electrons in each orbit is:
Modern theory used the s p d f shell for electronic configuration.
If outer orbit contains maximum number of electrons the orbit is considered complete. If less number of electrons than its holding capacity, it is considered incomplete or even empty and electrons are considered free.
Free electrons determine the principal chemical and physical properties. They are the ones that carry the charges in solid conductors.
They are the ones that are easily pulled out of the atom when we produce charges by rubbing.
Electrons can jump from one orbit into another or even out of the atom of the atom into another atom. One atom may gain electron and another may lose an electron. Thus if one or more electrons are removed remaining positively charged structure is called positive ion.
Ion - gained one or more extra electron
Ionization - process of losing or gaining electrons
In solids only negative electricity is transferred. These justifies that rubbing objects does not create electricity nor destroy charges but merely changes electrical neutrality of substance in contact - law of conservation of charge also states that algebraic sum of electric charge in any closed system remains constant.
4 Insulators and Conductors
Originally substances are classified according to their electrified nature by William Gilbert as electrics (insulators) and non-electrics (conductors).
Electrics are those which could be electrified by rubbing and non-electrics are those which can not be electrified by rubbing.
Substances that are easily electrified by friction are all insulators because when electricity is produced on an insulator by rubbing it stays there and makes its presence known, but if it is a conductor electricity leaks away at once.
Metals are generally good conductors while non-metals are generally poor conductors or good insulators. In most solid conductors the transfer of charge is by movement of electrons. In liquid conductors the molecules break into two parts and these charged particles are called ions. The conduction takes place by the movement of positively charged ions and negatively charged ions.
Good conductors of heat are also good conductors of electricity. There is no sharp boundary between materials which are insulators and those which are conductors, all materials can conduct electricity to some extent. No conductor is perfect as no insulator is perfect.
Materials that are ordinarily insulators but become conductors under particular conditions are called semi-conductors (materials which are intermediate between conductors and insulators). Germanium in its pure taste is a non-conductor of electricity. When a small amount of impurity (usually antimony and arsenic) is introduced it becomes a conductor. Process is called "doping". Semi-conductors are used in transistor radios.
5 How to Produce charges on a body
1. Contact
2. Induction
6 Distribution of Electric Charges
Electric charges reside on the outer surface of the body. Two balls, therefore, one hallows an the other solid provided they have the same diameter will have the same amount of charge. Charges depend only on the extent of the surface and not on the mass of the body.
Electroscope placed inside a charged metal box will not record presence of charge, hence the screening effect of hollow conductor, thus interior of room is well protected against action of electrical storm.
Electric charges readily escape from pointed ends. Except for spherical surface distribution of charges is not evenly concentrated on corners and parts which have sharp curvature. Density of charge becomes so great that it escapes to surrounding medium.
Electroscope - device used to detect presence and kind of charge and measures intensity of charge.
1. Pith ball
2. Leaf electroscope
Repulsion is a sure sign of electrification since a charged body attracts almost any light objects.
B Coulomb’s Law
A body is charged when it has more electrons or less electrons than its normal number.
- charge = excess of electrons
+ charge = deficiency of electrons
1 Quantity of charge (Q) is expressed as:
1. Statcoulomb (osu) = charge which will repel a similar charge of the same sign with a force of one dyne when the charges are separated at a distance of 1 cm.
2. Coulomb - MKS
1 coul = 2.997 x 109 statc
c = 4.8022 x 10-10 statc
= 1.6019 x 10-19 coul
2 Force between charges
The first quantitative investigation of the law of force between charged bodies was carried out by Charles Agustin de Coulomb thus it is known as Coulomb's Law of Electrostatics which states as follows:
The force between two small charged bodies is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them and is a function of the nature of the medium surrounding the charge.
Force is repulsive if charges are alike in sign and attractive if charges are unlike.
dyne - cm2
CGS: k (air and vacuum) = 1 ------------
statc2
MKS: k (air and vacuum) = 8.98742 x 109
N - m2
= 9 x 109 -------
coul2
However, it has been found convenient to make the substitution
Åo = represent the permitivity of the medium surrounding the charges (property of free space)
When more than two charges are in the same region, the force on any one of them may be calculated by adding vectorially, the forces exerted on it by each of the others.
3 Practice Exercise
1. Find the force between two point charges of 0.01 and -0.02 ìc if they are 8 cm apart in air (-2.81 x 10-4 N).
2. Charge A of 250 statc is placed on a line between two charges B of 50 statc and c of -300 statc. Charge A is 5.0 cm from B and 10 cm from C. What is the force on A. (1250 dynes)
6. Charges, A, B and C of 25, 20 and -8 c, respectively are arranged as shown in the
7. figure. Find the magnitude of the force on change A. (0.17N)
C Electric Field
Every charged body gives to its surrounding region special properties. Another charged body placed in this region will experience a force of repulsion or attraction. This region is called electric field.
Electric field is any region in which electric forces may be detected or electric field exists at a point if a test changed placed at that point experiences a force.
The intensity of the electric field at a point may be defined as the force per unit test charge placed at that point.
Component fields must be added vectorially. The direction of the electric field at a point is the same as the direction of the force on a + test charge which is placed at that point.
1 dyne/statc = 1 esu of elect. field intensity
= 3 x 104 N/coul
The intensity of the electric field at a point may be represented by an arrow and the field around an isolated point charge is represented by arrows. These arrows also represent lines of force.
The concept of lines of force was introduced by Michael Furaday as an aid of visualizing electric (magnetic) fields. Line of force is an imaginary line drawn in such a way that its direction at any point is the same as the direction of the field at that point.
Characteristics of lines of force:
Electric field of a point charge
Qq
The force in the test charge from coulombs law is F = k -----
s2
When more than one charge contributes to the electric field at a point the net field is the vector sum of the fields of individual charges.
1 Practice Exercise
1. The electric field in the space between the plates of a discharge tube is 3.25 x 104 N/coul. What is the first of the electric field on a proton in this field. Compare this force with the weight of the proton if the mass of the proton is 1.67 x 10-27 kg and its charge is 1.6 x 10-19 coul. (5.2 x 10-15 N; 3.17 x 1011).
2. A charge of 30 statc and another charge of 50 statc are 10 cm apart. What is the field intensity at a point 8 cm from 50 and 6 cm from the 30 statc? (1.14 dynes/stat)
3. The force on a small test charge is 2.4 x 10-6 N when the charge is placed in an electric field of intensity 6 x 105 N/c. How many electrons would be required to neutralize this charge. (4 x 10-12 c; 2.5 x 107 e)
4. At the three consecutive corners of a square 10 cm on the side are point charges of 50 x 10-9 c; 100 x 10-9 c and 100 x 10-9 c respectively. Find the electrostatic field at the fourth corner of the square. (95 x 103 N/C).
D Electric Potentials
One of the great unifying concepts of Physics is that of energy. Many difficulties due to the vector nature of electric fields can be avoided by dealing with electric energy and electric potential rather than with force and electric field.
What we call P-E in mechanics is really work done against the force of attraction of earth. In like manner work done in moving a test charge against force of repulsion of similar charge.
Often it happens that we are not interested in the absolute value of the potential at a particular point but only in the difference in potential between two points. This difference in electric potential is sometimes called the voltage between the two points or electromotive force (emf).
A battery is a device that uses chemical means to produce a potential difference between two terminals. A "six volt" battery is one that has a potential difference of 6 volts between its terminal.
When a charge q goes from one terminal of a battery whose PD is V to the other the work W = qV is done on it regardless of the path taken by the charge and regardless whether the actual electric field that caressed the motion of the charge is strong or weak.
To bring like charges nearer, we must do positive work on them (for like charges repel) and so the potential energy of the system increases and is positive. For unlike charges we do negative work on them (they do positive work on us, for they attract each other) and so their potential energy is negative becoming more and more negative, the closer they approach each other. (Field does the work, or no external source of energy is needed).
The amount of work done per unit charge when a unit positive charge is moved from one point to another is called the electric potential.
In a uniform electric field
in a direction parallel tom that of the field.
Positive charge tends to move from a position of high potential to one of lower potential.
When there are a number of charges in a region, the potential at any point is the sum of the potentials due to each charge acting alone.
Units
joule
MKS ------ = volt = 1 joule of work maybe done to move a charge of 1 coul
coul between the points considered.
ergs
CGS ------ = statv
statc
1 erq 3 x 109 statc 1 j
1 statv = ---------- x ---------------- x -----
statc coul 107 ergs
= 300 jouls/coul
1 statv = 300 volts
E Practice Exercises
1. The above figure shows a tube which has a source of electrons at one end and a metal plate at the other. A 100-V battery is connected between the electron source and the metal plate so that there is potential difference of 100V between then. The negative terminal of the battery is connected to the electron source. What is the velocity of the electrons when they arrive at the metal plate. (The tube is evacuated to prevent collision between the electrons and air molecules.
2. Charge A of 8.0ìc situated 1.0 m from charge B of -2.0ìc. What is the potential at point C located at the midpoint between A and B. What is the potential at point D located 80 cm from A and 20 cm from B. How much work would be required to move a charge of 0.03ìc from D to C.
3. Two point charges of 200 x 10-9c and -300 x 10-9c are placed at two corners A and B of an equilateral triangle ABC respectively. The side of the triangle is 20 cm. How much work is needed to transfer a third charge from the third corner to a point exactly midway between A and B.
Supplementary Exercises
1. A charge of 20 x 10-8 coul is 20 cm from another charge of 180 x 10-8 coul
a) Find the force between the two
b) What is the potential at the point which is exactly midway between the two
c) What is the electric field intensity at the same point.
2. The identical charges of 40 x 210-8 coul are 10 cm apart. How much work is needed to bring them to a distance of 5 cm apart.
3. Electrons are released from the cathode of a vacuum tube with zero velocity and are accelerated towards the positively charged plates. If the potential difference between the cathode and the plates is 500V, what is the velocity of the electrons just before reaching the plates.
4. A charge of 0.6ìc is 10 cm from another charge of -0.9ìc. Find the force on a charge of 1 ìc placed at a point which is 8 cm from the negative charge and 6 cm from the potential charge.
5. A point charge of 0.03ìc is placed 0.6 m from a point charge of 0.04ìc, what force is exerted on each charge. Find the electric field strength at the point midway between the charges.
Electrodynamics
(Electricity in Motion)
A Nature of Electric Current
Two regions of unequal pressure are in unbalance equilibrium. There is always a tendency for the two pressures to equalize.
In electrostatics, unequal electric pressure or potential difference produces a discharge but the energy they produce is of very little value.
The supply of electrical energy must be continuous and this can be obtained if there is a constant potential difference to keep the charges in constant motion and there is a circuit to provide a complete path for the moving charge. These flowing charges from a stream of electrons called electric current.
Since the time of Galvain man has learned to produce and develop different ways of making electrons flow.
Electricity is now produced:
1. Chemical action (Electrochem) - dry cells and storage batteries
2. Motion of a conductor across a magnetic field or by the variation of magnetic field (Electromagnet) - dynamos and transformers.
3. Radiant energy falling o some metals like selenium and caesium - solar batteries.
4. Light falling on some metals like potassium - phoelectric cell.
5. By means of heat (thermoelectric cell) - thermocouple
Siebeck effect
--------------
Head electricity
--------------
Peltier effect
6. Application of pressure on certain substances like quarty crystals, tourmaline and rochelle salt. (Piezoelectric effect) - microphones, headphones, pick-ups and sonar equipment.
B Theory of Ionization
Some substances like common table salt or sulfuric acid when dissolved in water conduct electricity readily. These are called electrolytes. (substances whose water solution conducts electric current).
sugar and alcohol are non-electrolyte
Because of its unbalanced structure water can easily dissolved materials into positively and negatively charged particles called ions, (atom or group of atoms which has gained or lost one or more electrons)
The ions are considered the carriers of electricity in solution in the same way that electrons are the carriers of electricity in solid conductors.
C Producing Electricity by Chemical Action
If we dip two dissimilar plates like Cu and Zn in an electrolyte we shall have a simple electric cell - Voltaic Cell (Alessandro Volta)
Plates are called electrodes and serves as terminal in the solutions. When Cu is dipped in the electrolyte no reaction takes place but when Zn is dipped in the electrolyte, the Zn dissolves forming Zn ions which are positively charged. Electrons are left in the Zn plate this making Zn negatively charged.
Zn ions in the solution repels the H2 ion to the Cu plate where it takes electrons and becomes neutral. The result is the formation of H2 gas. Cu now having deficiency in electron, becomes positively charged. In the solution Zn ions combine with Cl ions and form ZnCl2. Chemical reaction in the cell gives the Zn higher electric potential than Cu. Work is done by the cell.
Different combination of electrodes gives different emf. An electrode except C and Zn maybe positive with respect to certain electrodes and negative with respect to others. Different electrolytes gives different emf. Emf of cell will not change even if spacings between electrodes is increased or if they are lifted from electrolyte without pulling them out of the solution.
Experiment shows that emf of a cell is determined only by 1) kind of electrodes 2) kind of electrolyte and not affected by distance or size of plates.
A small dry cell gives same emf as big dry cell of same material. Bigger cells however last longer since they have more fuel.
When the two electrodes are connected so as to form a complete circuit the difference in potential between the electrodes will cause the electrons to flow from the Zn plate to Cu plate in the circuit.
D Direction of Current
The charges which are primarily responsible for the current in metallic conductors are negative electrons. However, early in the 19th century there was no way to know whether it was negative or positive charges (or both) which were in motion.
About 1820 Andie Ampere introduced the convention that the direction of current is the direction in which a positive charge would move under the influence of electric field. This convention is still used by physicist and engineers.
By definition then, conventional current in a wire flows from a point of higher potential to lower potential as thought current represented a movement of positive charge. Actually in metallic conductors the positive nuclei are not free to move and the transfer of charge results from a flow of electrons in a direction opposite that of a conventional current.
In liquid and gaseous conductors, both + and - ions are in motion. In some of the modern high energy accelerators as Van de Graaf generators and cyclotrons, current maybe a movement of positive charges.
Obviously no convention could be most convenient for handling every possible situations.
When a constant potential difference is maintained between two points in a conductor, a constant flow of charge results. The current is always in the same direction and is said to be a direct current (one way).
When flow of charges is first in one direction and then in opposite direction the flow of charge is Alternating Current (AC).
[pic]
6
The magnitude of the current is the charge per unit of a time that passes through any cross section of the wire:
Since 1 coul of electricity consists of 6 x 1018e then:
[pic]
7
E Emf of a cell
When two electrodes, Cu and Zn are again disconnected the chemical reaction continues until Cu begins repelling the H+2 ion reaching it Amg H2 ions in turn begin repelling the Zn+ back to the Zn plate then ionization stops. When this stage is reached the potential difference between the electrodes is maximum.
This maximum PD is called the emf of the cell which is defined as the maximum potential difference between the terminals of a source of electrical energy when the circuit is open; that is when there is no load.
It is equal to the work spent by the electric cell in giving energy to the electrons and is expressed in volts.
Emf of lV = Work done in moving a unit charge of one coul equal to 1 joule
[pic]
8
Terminal potential difference (TPD) of a cell is the difference in potential between the terminals when the switch is closed. (Potential drop is due to the load)
F Ohm's Law
Just as the rate of flow of water between two point depends upon the difference of height between them, the rate of flow of electric current between two points depends upon the difference of potential between then.
A large PD means a large "push" to send a charge around a circuit.
The precise relationship between PD (V) across the ends of conductor and current (I) that flows as a result depends upon the nature of the conductor which was discovered by George Simon ohm which is considered as the basic law for current electricity.
[pic]
9
[pic]
10
Constant is the resistance of the conductor (opposition of the circuit or a portion of the circuit to the passage of the current)
G Factors Determining Resistance
1. Material L
2. Length of conductor R = ρ ----
3. Cross-section A
4. Temperature
ρ = resistivity or specific resistance
= resistance offered by a conductor of unit length and of unit cross-section to the passage of current with the current flowing in a direction perpendicular to the cross section.
Reciprocal of resistance is conductance. The smaller the resistivity the greater the conductivity.
Resistivity is found to depend strongly on temperature. Generally it increase as temperature is increased in case of metals and decreases with a rise in temperature for good insulators.
Rt = Rs + ΔR where ΔR = αRsΔT
ΔR
α = -------
Rt = Rs + αRsΔT RsΔT
Rt = Rs (1 + αΔT) = temperature of coefficient
of resistance
H Measurement of Resistance
1. Voltmeter - Ammeter Method
2. Wheatstone bridge
D is adjusted so that the galvanometer reads zero thus no current flow across CD so C and D have the same potential. The current from the source divides at M such that I, flows through R3 and I2 flows through R.
Since no current passes through CD the same current I1 continues to Rr and current I2 through P2.
The voltage
MC = I2R1 CN = I2R2
MC = I1R1 CN = I1R4
But MC = MD and CN = DN
I2R1 = I1R3 I2R2 = I1R4
I2R1 I1R3
-------- = --------
I2R2 I1R4
R1R4 = R2R3
Thus when the bridge is balanced the products of cross resistances are equal so that unknown resistance placed in any four arms of the bridge can be obtained.
Practice Exercises
1. The ends of a wire of resistance 10Ω are at a potential difference of 4.5 volts. a) How much charge enters one end of the wire in 1 minute? b) How many electrons leave the end of the wire in the same time.
2. A current of 1.5A is maintained in a wire of resistance 3.0Ω for 5 min. a) What energy is taken in during this time, b) what charge flows through a cross section in 1 min?
3. The difference of potential between the terminals of an electric heater is 120V when there is a current of 8A in the heater, what current will be maintained in the heater if the difference of potential is increased to 180V.
4. A 200Ω resistor is to be constructed by winding No. 30 Ag wire in a coil. How much wire is needed. (dia = .01 in).
5. Find the current passing through a platinum wire 4m long and a 0.44 mm in diameter if the voltage across the wire is 6V.
6. The resistance of Cu wire in the armature of a motor at 20oC is 2.46Ω. When the motor is running it is observed that the resistance is increased to 2.98Ω. Find the operating temperature of the armature.
DIRECT CURRENT CIRCUIT
Simplest electric circuit consists of a cell and an external resistance. External circuit - electricity is conducted by electrons in the wire.
Internal circuit - movement of ions
Open circuit voltage - Emf of a cell
Switch closed - reading is lowered
- TPD of the cell
It shows that I has encountered resistance within the cell.
V = emf
R = total resistance of the circuit
emf
I = ---------
(Ri + Re)
Emf = I Ri + IRe
IRi = voltage drop due to internal R
IRe = voltage drop due to external R
Because of the internal resistance of the cell some power is being wasted inside the cell. Generally storage cell which can be recharged have lower internal resistance than dry cells especially when it is freshly charged and plates are new.
An efficient cell is one that has low resistance.
When a complete circuit is to be considered, one must take into account all the emf's in the circuit and all the resistances in the circuit.
Net emf
----------- = total resistance
current
Wherever only a part of a circuit is to be considered the potential difference V1 is the drop in potential across that part and the resistance R1 is the resistance of that part only.
V1
----- = R1
I1
Resistance in series
R3 R2 R1
┌──┬── ───── ────── ┌──────┐
│ │ │ │
│ └────────V ────────┘ │
│ │
└─────────── ───────── A ───┘
V
R = ----
I
I = I1 = I2 = I3
V = V1 + V2 + V3
R = R1 + R2 + R3
Resistors in Parallel
R1
┌───┬───── ─────┬───┐
│ │ │ │
│ │ │ │ V
│ └────── V──────┘ │ R = ---
│ │ I
│ │ │
│ │ │ │ │
└────┼─┼───┤ A─────────┘
I = I1 + I2 + I3
V = V1 = V2 = V3
1 1 1 1
--- = --- + --- + ---
R R1 R2 R3
Connecting additional resistors in series increases the total resistance while connecting additional resistors in parallel decreases the total resistance.
I Cells in Series
A group of cells maybe connected together. Such a grouping of cells is known as a battery.
Cells are said to be connected in series when they are joined end to end so that the same quantity of electricity must flow through each cell.
In ordinary series connection of cells + terminal of one cell is connected to the - terminal of the next, etc.
│ │ │ │ │ │
┌───┤ ├───────┤ ├───┤ ├──┐
│ │ │ │ │ │ │ │
│ │
│ │
│ │
└──────── ─────────────┘
V = V1 + V2 + V3
i = i1 = i2 = i3
r = r1 + r2 + r3
If two cells are connected in series in such a way that both would produce a current in the same direction, emf is the sum of the two emf (series aiding).
If two cells are connected in such a way that they would send currents in opposite direction the net emf is the difference between/the two (series opposing).
When battery is to be charged it must be connected in series opposing with some other source of emf which supplies electrical energy to be transformed into chemical energy.
Cells in Parallel
Cells are connected in parallel when the current is divided between the various cells.
In normal parallel connections of cells all positive terminals are connected together and all negative terminals are connected together. If resistor is connected the cells in parallel that resistor is connected directly to the last cell only.
┌─────── ─────────┐
│ │
│ │
│ │
└─────── ─────────┘
v = v1 = v2 = v3
i = i1 + i2 + i3
1 1 1 1
--- = --- + --- + ---
r r1 r2 r3
The parallel connection is used only when the aim is to get more current that can be supplied by one cell.
A large current through a cell will shorten the life of the cell.
To get maximum current connect cells in series if external resistance is large but if Re is small connect cells in parallel.
If external resistance is low the current furnished by the battery is even less than the current furnished by each cell. When the external resistance is high, cells are connected in series to provide the maximum voltage that will supply the necessary current.
Capacitor is a device that stores electric potential energy and electric charge. Any two conductors separated by an insulator ( or a vacuum) form a capacitor
Capacitance (C) is the ratio between charge (Q) and potential difference (Vab).
[pic]
SI unit of capacitance C = 1 farad = 1F
1 F = 1 farad = 1C/V = 1 coulomb/volt
Capacitance of a parallel-plate capacitor:
[pic]
Capacitors in series
[pic]
Capacitors in parallel
[pic]
Example: The plates of the parallel-plate capacitor are 3.28 mm apart and each has an area of 12.2 cm2. Each carries a charge of magnitude 4.35x10-8C. The plates are in vacuum. a) What is the capacitance? b) What is the potential difference between the plates? c) What is the magnitude of the electric field between the plates?
Ans. a) 3.29 pF, b) 13.2 kV c) 4.02x106 V/m
Example: A cylindrical capacitor has an inner conductor of radius 1.5 mm and an outer conductor of radius 3.5 mm. The two conductors are separated by vacuum, and the entire capacitor is 2.8 m long. a) What is the capacitance per unit length? b) The potential of the inner conductor is 350 mV higher than of the outer conductor. Find the charge (magnitude and sign) on both conductors.
Ans.: a) 6.56x10-11F/m; b) 6.43x10-11C
Example: In the circuit shown, C1=3.00 μF, C2=5.00 μF, and C3=6.00 μF. The applied potential is Vab=+24.0V. Calculate a) the charge on each capacitor; b) the potential difference across each capacitor; c) the potential difference between points a and d.
Ans.: a) Q1=3.08x10 -5 C, Q2=5.13x10-5C, Q3=8.21x10-5C
b) V2=V1=10.3V, V3=13.7 V
c) Vad=10.3 V
Current (I) is any motion of charge from one region to another.
[pic]
unit: 1 C/sec = 1 ampere = 1A
Current density (J) is the ratio between current (I) and cross-sectional area (A).
[pic]
unit: [ A/m2 ]
Resistivity ([pic] ) of a material as the ratio of the magnitudes of electric field and current density.
[pic]
unit: 1 V.m/A = 1 Ω.m
Temp. dependence of resistivity [pic]
where: ρ(T) = resistivity at temp T
α = temp coef of resistivity
T = temp of material
ρo = resistivity at temp To
[pic] relationship between resistance and resistivity
where: R = resistance value (Ω); ρ = resistivity (Ω.m); A = area (m2)
Temp. dependence of resistor [pic]
where: R(T) = resistance at temp T
α = temp coef of resistance
T = temp of material
Ro = resistance at temp To
Resistivities at room temperature [20oC]
| |Substance |ρ(Ω.m) |
|Conductors |Silver |1.47x10-8 |
| |Copper |1.72x10-8 |
| |Gold |2.44x10-8 |
| |Aluminum |2.75x10-8 |
|Semiconductors |Pure carbon |3.5x10-5 |
| |Pure Germanium |0.60 |
| |Pure Silicon |2300 |
|Insulators |Amber |5x1014 |
| |Glass |1010-1014 |
Temp Coef. of resistivity (α)
|Material |α [(Co)-1] |
|Aluminum |0.0039 |
|Brass |0.0020 |
|Carbon |-0.0005 |
|Copper |0.00393 |
Example: Suppose the resistance of the 18-gauge wire is 1.05Ω at a temperature of 20oC. Find the resistance at 0oC and at 100oC.
Ans. 0.97Ω, 1.38 Ω
Resistors in series:
“The equivalent resistance of any number of resistors in series equals the sum of their individual resistances.”
Resistors in parallel:
“For any number of resistors in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of their individual resistances.”
Example. Two identical light bulbs are to be connected to a source with ε = 8V and negligible internal resistance. Each light bulb has a resistance R=2Ω. Find the current through each bulb, the potential difference across each bulb, and the power delivered to each bulb and to the entire network if the bulbs are connected a) in series, b) in parallel, c) Suppose one of the bulbs burn out; that is, its filament breaks and current can no longer flow through it. What happens to the other bulb in the series case? in the parallel case?
Kirchhoff’s Rules
(by German physicist Gustav Robert Kirchhoff, 1824-87)
Kirchhoff’s junction rule: “The algebraic sum of the current into any junction is zero.”
[pic] (valid at any junction)
Kirchhoff’s loop rule: “The algebraic sum of the potential differences in any loop is zero.”
[pic] (valid for any closed loop)
Practice Exercises
1. Three resistances of 4 Ω, 12 Ω, and 8 Ω are available. Find the joint resistance if a) the three are connected in series, b) the three are connected in parallel.
2. The resistance of four rheostat are 10.0 Ω. These are connected in series to a battery which produces a potential differences of 754. Across its terminals. Find the current in each rheostat and voltage across each.
3. ┌─── ───┐
┌──── ───┤ ├───┐
│ └──── ───┘ │
┌─────┤ ├────┐
│ └───── ────────────────┘ │
│ │ │ │ │ │
└──────────────┤ │ │ ├─────────────┘
a. Find the equivalent resistance.
b. Find the current that flows through each resistor if a potential difference of 12V is applied across the set of resistors.
4.
2Ω 7Ω
┌──────── ───────┬─────── ─────────┐
│ │ │
│ │ │
──┴─ │ │
20 V │ 6Ω │ 1Ω solve for I
───┬─ │ │
│ │ │
└──────── ──────┴─────── ─────────┘
8Ω 10Ω
5. The emf of a battery is 4.2 V and its internal resistance is 0.2 Ω. It is connected to an external resistance of 0.9 Ω by means of lead wires of resistance 0.10 Ω.
a. Find the TPD of the battery
b. What is the voltage across the external resistance
6. The emf of a battery is 4.5 V. When it is connected to an external resistance of 12 Ω, the TPD is 4.3 V. What is the internal resistance of the battery?
7. Given 10 storage cells of 2 V and an internal resistance of 0.2 Ω each. A 7-Ω resistor is placed in the circuit
a) find a) current supplied by each cell
b) voltage when cells are connected in series
c) voltage when cells are connected in parallel
d) current supplied when cells are in series
e) current supplied when cells are in parallel.
┌──── ────┐
8. │ │ │ │ Each of the identical cells has an emf
┌───┼──── ────┼─┤ ├───┐ of 1.5 V and an internal resistance of 0.10
│ │ │ │ │ │ Ω. The last cell has an emf of 1.8 V and
│ └──── ────┘ │ an internal resistance of 0.2 Ω.
│ │
└──────── ───────────────┘
External resistance 0.8 Ω.
a) What is the emf of the battery
b) Find the current through each cell and through Re.
9. A battery consists of 8 cells arranged in two rows in parallel, there being and identical cells in series in each row. The emf of each cell is 2 V and the internal resistance of each is 0.2 Ω. The battery is then connected to an external resistance of 2, 4, and 12 Ω in parallel by means of conducting. Wires of resistance 0.2 Ω. Find the current delivered by the battery. What is the current passing through the 12 Ω resistance.
10. Find the resistance between points A & B. What voltage across AB will cause a current of 1.2 A to pass through the 1.5 Ω resistor.
1.5Ω 3Ω
A────────── ─────┬────── ───┐
│ │
│ │
3Ω 4Ω 5Ω
│ │
│ │
B──────────────────┴────── ────┘
1Ω
ELECTRICAL ENERGY AND POWER
When a resistor is connected to a battery we say that the resistor "uses up" the current. However, current is never used up in the resistor since it flows back to some source of emf.
Owing to the resistance that all conductors offer to the flow of current through them work must be done continuously to maintain a current.
Electrical resistance is analogous to friction and so the work that is done in causing a flow of current is dissipated as heat.
By definition electric potential is the work done in transferring a unit charge from one point to another.
ω
V = ---
q
W = Vq work done in carrying the charge against the
resistance which is done by the source of emf
but q = It
w = VIt - energy dissipated
ω V2
P = --- = VI = I2 r = ----
t R
Work actually is the transforming of electrical energy into other forms of energy when current is passed through the appliance.
joule
1 watt = ---------
S
1 j = 1 watt-s
1 kw = 1000 watts
Monthly electrical bills are based on electrical energy. We take during the month in kw-hr.
1 kw hr = 1000 watts (3600 s)
= 3.6 x 106 watt-s
= 3.6 x 106 joules
Ordinary filament lamps = 50-100 watts
Fluorescent = 20-40 w
Electric fan = 80 watts
21-in TV set or Stereo = 200 watts
9 cu ft ref =3/8 Hp = 300 w
Heaters = 600-1000 w
Range = 280-1250 w
Clock = 3-5 w
Heating Effect of Electric Current
When a potential difference produces a current through a conductor, electrical energy is converted to thermal energy
Energy dissipated
W = VIT but V = IR
W = I2 Rt
This was discovered by James Prescott Joule that the amount of thermal energy produced by an electric current is proportional to the square of the current to the resistance and the time (Joule's Law of Heating).
If one wishes to obtain the energy in calories one may use the relation.
1 cal = 4.186 j
1 j = 0.24 cal
H = .24 I2 Rt
Thus it is desirable to have some sort of device to protect electrical machines and appliances from excessive currents.
Most common is the FUSE which consists essentially of a wire that has low melting point when an excessive heat passes through it, the heat generated is sufficient to melt the wire and the circuit in which it is inserted is opened thus prevents the overloading of wiring circuit beyond a certain current. Size of fuse is chose that it melt when current becomes greater than the preselected amount.
Practice Exercises
1. How much current is drawn by a 2 Hp electric motor operated from a 120-V source of battery.
2. A heating coil which draws a current of 8A from 120 V line is used for heating water. The coil is immersed in 5 l of water which is initially at 20oC the water being in a 300-g container of specific heat 0.10.
a. Find the power of the coil.
b. How long will it take the coil to raise the temperature of water to boiling point
c. At P3/kw-hr how much would the process cost?
3. The internal resistance of a 6V storage battery is 0.6 Ω. It is to be charged for 2 hours from a 120-V line. The charging current is to be limited to 4 A. Find:
a. resistance of the rheostat to be placed in series with the battery when it is being charged.
b. Total heat generated in the rheostat
c. The chemical energy stored during 2 hours.
d. Power wasted in the battery
e. Total energy wasted in the process of charging
f. Cost of the process at P3/kw-hr.
MAGNETISM
A Magnets and Magnetic Pole
Magnetism comes from magnesia name of the region in Ancient Asia minor where naturally magnetic pieces of iron oxide (Fe3O3) called lodestones are found.
Magnet is a body endowed with polarities and capable of exerting and experiencing a force called magnetic force. When suitably suspended, it comes to rest in a definite direction relative to the poles of the earth.
Substances are classified as:
1. Ferromagnetic – substances which are strongly attracted by magnets.
2. Paramagnetic – slightly attracted by strong magnets
3. Diamagnetic – instead of being attracted are actually repelled by strong magnets
A bar magnet which is dipped in iron filings attracts the filings very prominently near its ends and these region are called the poles of the magnet.
Poles – sets of centers of the force of attraction and are located inside and near the ends of the magnet.
Magnetic axis – line joining the poles appears in pairs and cannot be dissociated.
North pole actually the north seeking pole (one that points towards the north of the earth).
Actual poles do not appear in a magnetized iron ring, however, poles will appear if ring is split.
Qualitative Laws of Interaction of Magnetic Poles
Like poles repel each other, unlike poles attract each other.
Either poles attracts small fragments of non-magnetized iron. Then the fragment of iron under the action of the approaching magnet becomes also a magnet and therefore subject to any of the previous laws.
Magnetic field and magnetic lines of Force
Magnetic field – region or space within which the influence of a magnet extends or the space around the magnet in which it is possible to detect magnetic forces.
Magnetic lines of force are geometric lines supposed to be drawn around the field. They represent a) the path of a north pole free to move in the field b) the tangential direction of the magnetic force at any point of the field.
The direction has been set to be from N to S outside the magnet and from S to N inside the magnet.
Properties of Magnetic Lines of Force
B Permeability and Retentivity.
If a piece of soft iron is placed across U-shaped magnet, magnetic lines of force instead of passing through the air passes through the soft iron bar which offers less resistance to magnetic flux.
Permeability – capacity of a magnetic material to allow magnetic flux to pass through it. Permeable substance are easily magnetized as well as demagnetized thus substances with high permeability make good temporary magnets.
Anti-magnetic watches have permeable cores to protect their delicate parts from magnetism.
Magnetism shields – takes in lines of force and protects parts of machine.
Magnetic keeper – a piece of soft iron which provides a path for the magnetic lines of force.
Retentivity – property of resisting magnetization or demagnetization
Reluctance – resistance to magnetic flux.
Non-magnetic materials are transparent to magnetism (wood, paper, water, glass, Cu, Zn, and Pb).
Magnetic lines of force pass through as if they are not there.
Magnetization – process of converting a magnetizable substance into an actual magnet.
a. by contact (rubbing or stroking)
b. induction (substance is placed in magnetic field)
c. electric current, when a wire carrying a current is wound around an iron bar
d. heating bar while it is oriented along earth magnetic field
e. hammering while it is aligned along earth’s field
Process of magnetization does not increase indefinitely. For every substance there is a point at which it fails to acquire a higher degree of magnetization no matter how much the magnetizing power is increased.
To make a magnet last longer a magnet maybe dipped in boiling oil for a few minutes after magnetization – aging a magnet.
Coulomb’s law – Quantitative Law of Attraction
Two poles will attract or repel each other with a force that is proportional to the product of their magnetic pole strength and inversely proportional to the square of the distance between them.
[pic]
K depends on the system of units used
MKS K = 10-7 [pic]
CGS k = 1
Magnetic Induction – strength of the magnetic field or the intensity of the magnetic field at a point.
Magnetic induction at any point in space is the force per unit North magnetic placed at that point.
[pic]
B = [pic]
One line passing through a square meter represent a magnetic induction of 1 weber/m2.
Magnetic lines of force are collectively called magnetic flux.
Magnetic flux density – number of lines of force passing perpendicularly through a given area.
1 weber – 108 lines of force
Terrestrial Magnetism
Core of the earth is believed to be composed of an inner solid core that is highly radioactive and outer core that is highly radioactive and outer core of liquid iron. Heat from inner core caused outer core to churn lines crating electric current. Electric current produces magnetic field. Mollen iron moving in the field crated additional current. Combined fields produces the present strong magnetic field of earth.
Magnetic meridian- direction of compass in the magnetic field.
Magnetic poles of earth are not exactly on geographic poles.
Magnetic pole in North – Greenland 1200 MI S, the N geographic pole 73o31’ N lot 95o48’ S long.
Magnetic pole in South – Wilkes land in Antarctic Ocean 72o25’ S lot 155o18’ E long.
Declination – deviation of compass from the North
Isogonic lines – places of same declination.
Agonic lines - places where compass points to the North lines of Zero declination.
Dip – deviation of compass from horizontal position (angle of inclination)
Isoclinic lines – equal dip
Aclinic – no dip (magnetic equator)
William Gilbert – gives us the idea that the earth is a huge magnet interior consisted of permanently magnetic material.
James Ross – magnetic pole of the North.
Ernest Shackleton – magnetic pole of the South
Karl Friedrich Gauss – magnetic field of the earth originate inside the earth.
Walter Elasser – earth’s magnetic field results from currents generated by the flow of matter in the fluid core of the earth.
International Geophysical Year 1957 - 1958 give the 1st most comprehensive theory about earth’s magnetism.
Factors changing magnetic field of earth
1. Sunspot – surface of sun is not smooth but is made of spikes which shoots jets of gas into the atmosphere in definite stream of particles which is ejected during solar flares (solar activity).
Because they are electrically charged their magnetic effect interacts with that of earth and produces magnetic storms. This disrupt radio communication and cause Aurora Borealis – a display of colored lights in Northern hemisphere.
Sunspot maximum occurs every 11 years. Magnetic storms associated with sunspot are violent.
Milder magnetic storm occurs at regular intervals of 27 days a period of one complete rotation of the sun.
2. Moon – tidal pull of the moon creates electric current. Because of the rotation of the earth these currents create additional magnetic field.
Field is found to change slowly over a period of years so that Government must issue new map from time to time.
Modern Optics
OPTICS - deals with the behavior of light and other electromagnetic waves.
The nature of light
• Isaac Newton (1642-1727): light consisted of streams of particles (called corpuscles) emitted by light sources.
• Galileo and others tried (unsuccessfully) to measure the speed of light.
• Around 1665, evidence of wave properties of light began to be discovered.
• Early nineteenth century, evidence that light is a wave had grown very persuasive.
• 1873, James Clerk Maxwell predicted the existence of electromagnetic waves and calculated their speed of propagation.
• 1887, Heinrich Hertz showed conclusively that light is indeed an electromagnetic wave.
• Light waves is packaged in discrete bundles called photons or quanta.
• 1930, with the development of quantum electrodynamics, a comprehensive theory that includes both wave and particle properties.
Propagation of light is the best described by a wave model, but understanding emission and absorption requires a particle approach.
Light
• Light source is coming from a hot matter. (examples are candle flame, hot coals in campfire, etc.
• Light is also produced during electrical discharges through ionized gases. [examples: bluish light of mercury-arc lamp, e.g. fluorescent lamp. Light uses phosphor to convert the ultraviolet radiation from a mercury arc into visible light, (more efficient fluorescent lamp)].
Measurement of the speed of light
• 1849, French scientist Armand Fizeau first who measure speed of light using a reflected light beam interrupted by a notched rotating disk.
• 1983, Jean Foucaults in France and by Albert A. Michelson in the United States, measure speed of light as,
C= 2.99792458x108 m/s (speed of light)
C=3x108 m/s for calculations
Wave front is to describe wave propagation.
Rays is to describe the directions in which the light propagates.
Index of refraction is the ratio of the speed of light c in vacuum to the speed v in the material:
[pic] (index of refraction)
Experiment studies in optics
1. The incident, reflected, and refracted rays and the normal to the surface all lies in the same plane.
2. The angle of reflection (r, is equal to the angle of incidence (a for all wavelengths and for any pair of materials.
[pic] (Law of reflection)
3. The ratio of the sines of the angles (a and (b, where both angles are measured from the normal to the surface, is equal to the inverse ratio of the two indexes of refraction:
[pic]
[pic] (Law of refraction)
Index of refraction for yellow sodium light (λo=589 nm)
|Substance |Index of refraction |
|Ice (H2O) |1.309 |
|Quartz |1.544 |
|Light flint |1.58 |
|Dense flint |1.66 |
|Methanol |1.329 |
|Water (H2O) |1.333 |
|Ethanol |1.36 |
Wavelength: [pic]; [pic]; [pic]
Example: Material ‘a’ is water and material ‘b’ is a glass with index of refraction 1.52. If the incident ray makes an angle of 60O with the normal, find the directions of the reflected and refracted rays.
Example: A beam of light has a wavelength of 650 nm, in vacuum, a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.47? b) What is the wavelength of these waves in the liquid?
Ans. a) 2.04x108m/s, b) 442 nm
[pic]
Reflection and refraction
[pic]
* Reflection and Refraction at a Plane surface
Object is anything from which light radiate.
Point object has no physical extent.
Extended Object is a real object with length, width, and height.
Image is something that can be seen from reflecting surface.
Virtual image is an image form when the outgoing rays don’t actually pass through the image point.
Real image is an image form when the outgoing rays do pass through the image point.
Image formation by a plane mirror
[pic] ; [pic]
where:
s = object distance; s’ = image distance; m = magnification
y’ = image height; y = object height
* Reflection at a Spherical Surface
Center of curvature, C is a center point of the circle.
Vertex, V is the center of surface mirror.
Optical axis is a line that passes through vertex and center of the circle.
Paraxial Approximation is a group of nearly parallel rays to the optical axis.
Focal Point is a point in which the incident parallel rays converge.
Focal Length is the distance from the vertex to the focal point.
[pic] or [pic] (object-image relation, spherical mirror)
[pic] (focal length of a spherical mirror)
[pic] (lateral magnification, spherical mirror)
[pic][pic]
[pic] [pic]
[pic]
[pic] [pic]
[pic]
Prob34-30. After a long day of diving you take a late-night swim in a motel swimming pool. When you go to your room, you realize that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom surface and is directed at the surface horizontal distance of 1.5m from the edge (Fig). If the water here is 4.0 m deep, how far is the key from the edge of the pool?
[pic]
Prob.35-5. An object 0.600 cm tall is placed 16.5 cm to the of the convex of a concave spherical mirror having a radius of curvature of 22.0 cm. a) Draw a principal-ray diagram showing formation of the image. b) Determine the position, size, orientation, and nature (real or virtual) of the image.
Prob.35-8. An object is 24.0 cm from the center of a silvered spherical glass Christmas tree ornament 6.00 cm in diameter. What are the position and magnification of its image?
TABLE OF CONTENTS
COURSE OUTLINE
COURSE NUMBER : PHYSICS 11
COURSE TITLE : General Physics I
COURSE DESCRIPTION: Fundamental concepts on force, work and energy; heat and temperature measurements; properties of matter; electricity and magnetism.
PREREQUISITE : Math 12
6 hrs. a week (3 lec., 3 lab.)
Credit: 4 units
OBJECTIVE OF THE COURSE:
1. To acquire an understanding of the basic concepts, principles and laws of physics to cope with the present mechanized environment.
2. To stimulate critical and analytical thinking among students as basis for making them more intelligent and more responsive members of society.
3. To acquire skill in manipulating measuring instruments and in conducting experiments correctly, thus it will allow students to make operational definitions, formulate questions and hypothesis, gather and interpret data, draw conclusions and design experiments and apparatus.
4. To develop appreciation and sense of gratitude to men and women who haven laboured unselfishly in the pursuit of scientific truths.
5. To show the relationship between physics and the "real world".
6. To develop interest in Physics particularly to young people who possesses talent to pursue careers in science and technology.
7. To be able to discuss intelligently science news and inventions.
I. INTRODUCTION (5 hrs.)
1. Why study Physics?
2. Uses of Physics
3. Frontiers of Physics
4. Suggested way of solving Physical Problems
5. Measurements -
a. Fundamental quantities and units
b. System of units (P.D. 187)
c. Dimensions and dimensional analysis
d. Suggested experiments/demonstrations
e. 1. Measurements of length
2. The vernier and micrometer devices
3. The vernier scales and micrometer screws
6. Vectors
a. Definitions; vector and scalar quantities, representation of vectors
b. Addition of vectors;
1. Graphical
2. Analytical
c. Suggested experiments/demonstration
1. Vectors; graphical methods
2. Vectors; rectangular resolution and polygon theorem
II. KINEMATICS (7 HRS.)
1. Kinematics
2. Definitions: Speed, velocity, acceleration, uniformly accelerated linear motion
3. Equations of uniformly accelerated linear motion
4. Freely falling bodies
5. Suggested experiments/demonstrations
a. Tickets tape timer
b. Uniform velocity apparatus
c. Accelerometer
d. Continuous flow of liquid apparatus
III. DYNAMICS (4 HRS.)
1. Newton's law of motion; units; application problems
2. Friction: static and kinetic friction, coefficient of friction, angle of repose, fluid friction.
3. Uniform circular motion and gravitation (concept only)
a. Control acceleration
b. Centripetal force; centrifugal reaction
c. Gravitation
4. Motion in a vertical circle
5. Banking curve
6. Suggested experiments/demonstrations
a. Newton's second law of motion - at woods machine
b. Kinetic and static friction
c. Centripetal and centrifugal force
IV. WORK AND ENERGY (3 HRS.)
1. Definitions work energy
2. Potential and kinetic energy
3. Transformation from potential to kinetic energy and vice versa
4. Conservative and dessipative forces
5. Conservation of energy principles
6. Suggested experiments/demonstrations
7. 1. The tension and compression spring.
V. POWER; SIMPLE MACHINES (2 hrs.)
1. Definition: power, units
2. Simple machines: lever, inclined plane wheel and axle, jackscrew, etc.
a. Actual and Ideal mechanical advantage
b. Efficiency
3. Suggested experiments/demonstrations
a. Simple machine
b. Mechanical advantage, work power; efficiency
VI. STATICS (5 hrs.)
1. Concept of force and equilibrium
2. Concurrent and non-concurrent force
3. First condition for equilibrium
4. Second condition for equilibrium; torque
a. Definitions; torque, moment arm, line of motion
b. Center of gravity, determination of c.c.
c. Suggested experiments/demonstrations
1. Torque, demonstration balance
2. Parallel forces
3. Center of gravity and equilibrium
4. State equilibrium; simple crane
5. The witch
VII. ELECTROSTATIC (5 hrs.)
1. Electrification 5. Suggested experiments/demonstrations
2. Coulomb's law a. mapping equipotential lines and field
3. Electric field and potentials b. electroscope
4. Potential difference c. electrostatic generator
VIII. CURRENT ELECTRICITY (7 hrs.)
1. Definition; current, resistance and voltage
2. Sources of emf
3. Ohm's Law
4. Simple circuits
a. Ster/es, parallel and series-parallel combination
5. Suggested experiments
a. Ohm's law
b. Resistor color code
c. Voltaic cell
d. Measurements of resistance
IX.ELECTRICAL ENERGY AND POWER (5 hrs.)
1. Definition: Energy and power
2. Heating effect of electric current
3. Computation of electric bills
4. Suggested experiments/demonstration
a. Electrical Equivalent of heat
b. Reading kilowatt-hour meter
X. MAGNETISM (2 hrs.)
1. Theory of magnetism 4. Suggested experiment/demonstrations
2. Laws of magnets a. Magnetic field
3. Terrestrial magnetism b. Magnetizing iron bar
SUGGESTED REFERENCES:
1. Young & Freedman 2000, Sears and Zemansky’s UNIVERSITY PHYSICS w/ Modern Physics, 10th ed..
2. Asperilla, Jose, et al. College Physics, Manila: Alemar. Phoenia Publishing House, 1969.
2. Weber, White and Manning, et al. College Physics, New York: MacGraw-Hill. Book Co. 1974.
3. Resnick and Halliday, Physics, New York: John Wiley and Sons Inc. 1978
4. Smith and Cooper. Elements of Physics, New York: McGraw-Hill Book Co.:1972
5. Buckwalter, Gary. College Physics New York, McGraw-Hill Book Co. 1987
6. Wilson, Jerr. College Physics. Englewood Prentice Hall 1994
7. Physics: An Introduction by Bolemen (1989)
8. College Physics by Buck Walter (1987)
9. College Physics by Wilson (1994)
10. College Physics, 7th Edition by Sears, Zemansky & Young C. 1991.
11. Fundamentals of Physics, 4th Edition by Halliday, Resnick & Walker C. 1994.
12. Classical and Modern Physics Vol. I & 2 (Combined) by Kenneth Ford C. 1972.
13. Handbook of chemistry and physics (1994)
Prepared by:
MARLON F. SACEDON
Instructor 1
-----------------------
x
FBD of Box:
FBD of Box:
o
y
x
Ø
a2
a1
[pic]
S1
L
S2
h
m
m
C
B
D
A
m
Disk
Block
Penny
12 cm
[pic]
m2
m1
[pic]
x
y
[pic]
[pic]
[pic]
[pic]
O
[pic]
[pic]
x
y
370
S
36.90
B
A
36.90
C
B
A
F
Lincoln
St. Joseph
Manhattan
Clarinda
W
E
S
N
166 KM
2350
106 KM
1670
147 KM
850
KANSAS
IOWA
MISSOURI
NEBRASKA
600
370
[pic]
[pic]
[pic]
O
[pic]
[pic]
x
y
370
W
E
S
N
4.0 km
450
3.1 km
2.6 km
stop
*
*
start
L3
L2
L1
F3
F2
F
F1
L
400
B
A
B
A
Fy = T + (-W) = 0
T = W
(Fy = O
T
W
W
(
N
(
P
vx = vo
vx = vo
(
vy
R
Both body starts from 0 velocity along x-axis and vx = vo
Velocity along y-axis is zero starts from rest, but accelerated towards the ground by “g”. Hence vy = 0 + gt
voy
vo
(
vx
vy
vy = voy – gt
vx = vox
H
vy = 0
vx
vy
vx
vy
vo
Maximum value of the range will be when
sin 2 ( = 1
2 ( = 90o
( = 45o
5kg
6kg
30o
40o
A
C
(
(
N
W
vB
-vA
(v
r
r
S
B
vA
A
Ta – mg = [pic]
Tb + mg = [pic]
r = L sin (
v = [pic]
T = [pic]
Tb
mg
Ta
vb
(
F
(
Fy
R
L
Fx
W
va
W
F = [pic]
[pic]
N
(
F
W = FS
(
S
D
F
W = F cos ( S
S
D
L
P
m
F
(
h
m
m
h
F
v
S
m
S
N
N
S
N
N
S
N
N
S
S
N
N
S
N
S
N
S
Geograghic
N-pole
Magnetic
N-pole
Magnetic
S-pole
P1
P2
P1
P2
P1
P2
P4
P3
P5
P6
[pic]
P3
[pic]
[pic]
[pic]
y
x
Dx(+)
[pic]
o
[pic]
[pic]
[pic]
Dy(-)
[pic]
[pic]
(
(
x
[pic]
Ey(+)
Ex(+)
O
Ax
x
y
Ay
Bx
By
[pic]
[pic]
[pic]
Ry
Rx
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
72.4 m
57.3 m
17.8 m
360
320
(
Y (north)
x (east)
y
o
50 N
FBD of Box:
50 N
25o
o
y
x
of Box:
50 N
25o
o
y
x
50 N
25o
FBD of block B
FBD of M1:
x
y
o
floor
Wall
cable
F
A
B
FBD of strut:
β
F2
F1
x
y
o
m
M1
FBD: Roll of paper
strut
x
y
o
cable
cable
strut
β2
β1
m
x
y
o
Wall
β1
F
Roll of paper
y
x
A
B
(a)
y
B
(b)
x
A
[pic]
[pic]
Hot reservoir TH
Cold reservoir TC
[pic]
W
Schematic energy-flow diagram for heat Engine engine
[pic]
P
T2=600K
2
3
1.00 atm
T3=455K
T1=300K
1
V
O
b
a
d
C2
C1
................
................
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