Q1 Calculate the solid angle subtended by the periphery of ...



PHYSICAL WORLD AND MEASUREMENTSection AVery Short Answer Type Questions 1 mark eachWhat is a coherent system of units? Give an example.Why the fundamental quantities are called so?Define international standard of length.What is an astronomical unit?Define light year. What is its value?Define parsec? Express it in meters.Define one angstrom unit.Distinguish between A0 and A.U.What is a micron?What is a nanometer? Express it in meters.Name the unit in which the size of the nucleus is measured. Express it in meters.What is the order of magnitude of light year?Is the measure of an angle dependent upon the unit of length?Express 0.53 A0 in m.Can you increase the accuracy of a screw gauge arbitrarily by increasing the number of divisions on the circular scale?Define atomic mass unit. Express it in kg.Is light year the unit of time?How many times is millisecond larger than a microsecond?What does SONAR stand for?How many A.U. make one parsec ?Define mole.Name the two supplementary SI units.Define radian.Define steradian.The average wavelength of light from a sodium lamp is 5893 A0. Express it in nm.If X = a + bt + ct2 , where x is in metres and t is in seconds. What are the units of b and c?Write down the dimensional formula of (i) gravitational constant (ii) planck’s constant.State the number of significant figures in the following: (i) 0.20 mm (ii) 6.54 x 10-23.What is the plane angle subtended by a circle at its center ?What is the solid angle subtended by a sphere at its center ?Which of the following length measurement is most accurate and why ? ( i) 2.0 cm (ii) 2. 00 cm (iii) 2.000 cmWhat is the percentage error in volume of a sphere, when error in measurement of its radius is 2 % ?Solution (Section A)PHYSICAL WORLD AND MEASUREMENTANSWERS TO VERY SHORT ANSWER TYPE QUESTIONSCoherent System of units is a set of fundamental or basic units from which units for other physical quantities can be derived by simple multiplication or division or both without introducing numerical values. Eg: S.I. system of units.The fundamental quantities are independent of each other for measurement while other physical quantities depend on them, so these are called so.Meter is international standard of length; it is defined as the length of the path travelled by light in a vacuum in 299792458 second.An astronomical unit is the average distance of the Sun from the Earth, i.e., 1A.U.= 1.496 x 1011 m.Light year is the distance that light travels in vacuum / air with velocity of 3 x 108 m/s in one year. 1L.Y .= 9.46 x 1015 m.Parsec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second. 1 parsec = 3.08 x 1016 m. One angstrom = 10-10 m.One angstrom is microscopic unit of length One angstrom = 10-10 m while an astronomical unit is macroscopic unit. 1A.U.=1.496 x 1011 m.One micron is a unit of length equal to 10-6 m.One nanometer = 10-9 m.The size of nucleus is measured in Fermi. 1fm = 10-15 m. Since, 1 LY = 9.46 x 1015 m. So, the order of magnitude is 16.As an angle is the ratio of the length of an arc and the radius i.e., it is the ratio of two lengths, so the measure of an angle does not depend upon the unit of length.Since, 1angstrom = 10-10 m. So, 0.53 angstrom = 5.3 x 10-11 m.Yes, the accuracy of the screw gauge can be increased by increasing the number of divisions on the circular scale as the least count is pitch/no. of divisions on circular scale but not arbitrarily due to limit of resolution of eye.Atomic mass unit is defined as (1/12) of the mass of an atom of carbon 12 isotope including the mass of electrons. 1amu=1.66 x 10-27 kg.No, the light year is not the unit of time. It is the unit of distance.millisecond is 1000 times larger than a microsecond. SONAR: Sound Navigation And Ranging. 1A.U.=1.49 x 1011 m, 1 parsec=3.08 x 1016 m, so,One mole is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kg of carbon12 isotope.The two supplementary SI units are radian and steradian.Radian: one radian is defined as the angle subtended at the center of the circle by an arc whose length is equal to the radius.Steradian: one steradian is the angle subtended at the Centre of a sphere by a part of its surface having an area equal to square of its radius. Since,1 nm = 10-9 m,1angstrom = 10-10 m, so, 5893 angstrom = 5893 x10-10 m = 589.3 nm.X=a+bt+ct2, X is in metres(m), t is in seconds, so according to principle of homogeneity, the dimensions on both side of equality are same. So, unit of b = m s-1, and unit of c = m s-2. Dimensional formula of (i) gravitational constant =[ M-1L3T-2] , (ii) Planck’s constant = [ML2T-1 ] Number of significant figures (i ) 2 , (ii) 3. 2π rad 4π str2 .000 cm, because it is resolved up to third decimal place.± 6 %Section B SHORT ANSWER QUESTIONS- 2MARK With Solution 1. What are the dimensional formulae of the following physical quantities: (i) pressure (ii) power (iii) density (iv) angle ? A. (i)[ML-1T-2] (ii)[ML2T-3] (iii)[ML-3] (iv) No dimensional formula2). Suggest an indirect method for measuring the height of a tree on a sunny day.A). Using two triangles one is original height and other is shadow3). Why we treat length, mass and time as basic fundamental quantities in mechanics? Name the seven fundamental quantities in SI system.A). In mechanics, these quantities represent basic scientific notations, cannot be expressed in terms of one another and all quantities in mechanics can be expressed in terms of these . See the table from ncert4). using the principle of homogeneity of dimensions, find which of the following is correct : (i) T2= 4π2 r2 (ii) T2 = 4π2 r3 /G (iii) T2 = 4π2 r3 /GMA).(i) Incorrect (ii) Incorrect (iii)Correct5). If the unit of force is 100 N, unit of length is 10m and unit of time is 100s, what is the unit of mass in this system of unit ?A) Force=MLT-2 =100N,L=10m and T= 100sM= 100/LT-2=100T2/LM= 105Kg6). Magnitude of force F experienced by a certain object moving with speed V is given by F=KV2 Where K is constant. Find the dimension s of KA). K=[F]/[v2] substituting dimensional formulae[K] = [ML-1]7). What are significant figures. Write significant figure for 34.6700 and 0.00980210A). 4, 58).Distinguish between classical physics and quantum physicsA).Classical physics mainly deals with macroscopic phenomena which may be at laboratory, terrestrial and astronomical scales. Quantum physics deals with microscopic phenomena at the minute scales of atoms, molecules and nuclei9).Name the four fundamental forces in nature. Arrange them in the order of their increasing strengthA). There are four fundamental forces in nature which govern the diverse phenomenon of macroscopic and microscopic world Gravitational force, Electromagnetic force, Strong nuclear force and Weak nuclear force1:1025:1036:103810). Convert Joule into CGS system using dimensional analysisA). Using dimensional analysis 1J= 107erg Section C Short Answer Type Question( 3 Marks)With Solution 1). A physical quantity X is related to four measurable quantities a, b c, and d as follow : X = a2 b3 c5/2 d-2 The percentage errors in the measurement of a , b, c, and d are 1%, 2% ,3% , and 4% , respectively. What is the percentage error in quantity X ? If the value of X calculated on the basic of the above relation is 2.763, what value should you round off the result ? A). ?X/X%= 24% or 0.24So there are two significant numbers in the percentage error X=2.763=2.82). Find the value of 60 j per min on a system that has 100 g, 100 cm and 1 min as the base unitsA). Using dimensional analysis, power= [ML2T-3]a=1,b=2 and c=-3=2.16X106 new units of power3). The velocity of sound wave ‘v’ through a medium may be assumed to depend on : (i) the density of the medium ‘d’ and (ii) the modulus of elasticity ‘E’. Deduce by the method of dimensions the formula for the velocity of sound. Take dimensional constant K =1.A) v=KdaEbSubstituting dimensional formulae for solving the values of a,b,ca=-1/2 and b=1/24). A plant moves around the sum in nearly circular orbit. Its period of revolution ‘T’ depends upon: (i) radius ‘r’ of orbit (ii) mass ‘M’ of the sun and (iii) the gravitational constant G. Show dimensionally that T2 ? r3 A). T=KraMbGc substituting dimensional formulae for solving the values for a,b,c we get T2 ? r35). Find Percentage error in Z, If Z = A4 B1/3 /CD3/2.Given that the percentage errors in A,B,C and D are 4%,2%, 8% and 3%A). ?Z/Z= 4 x4 %+1/3 x 2%+1 x 8%+3/2 x 3% =16%+2/3%+8%+9/2%6). A physical quantity P is related to four Observation a, b c, and d as follow : P = a3 b2/√ c d The percentage errors in the measurement of a , b, c, and d are 1%, 3% ,4% , and 2% , respectively. What is the percentage error in quantity p? If the value of p calculated on the basic of the above relation is 3.763, what value should you round off the result ?A). ?p/p= 3X1%+2X3%+1/2X4%+1X2% =3%+6%+2%+2%= 13%7). In the expression P= El2 m-5 G-2; E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity. A). p= El2/m5G2Substituting dimensional formulaep= M0L0T0p is a dimensional constant8). The velocity ‘v’ of water waves depends on the wavelength , density of water ‘p’ and the acceleration due to gravity ‘g’ . Deduce by the method of dimensions the relationship between these quantities.A). v=Kλa pbgcSubstituting dimensional formulae of the quantities we get the expression of v9). Show that angular momentum has the same physical unit as the Planck’s constant h which is given by the relation E=h v .A). Angular momentum L= mvrAngular momentum dimensional formula [L] = ML2T-1Planck's constant h= E/ v = ML2T-2/T-1=ML2T-110). If velocity of light c, Planck’s constant h and gravitational constant G are taken fundamental quantities, then express mass, length and time in term of dimensions of these quantities.A). [C} =LT-1,[G]=M-1L3T-2, [h]= ML2T-1[h}[C]/[G]= M2[M]= h1/2C1/2G-1/2[h]/[C]= ML[L]= [h]/[C][L]=h1/2C-3/2G1/2As [C]=LT-1[T]=h1/2C-5/2G1/2 (Value Based Question),Section EFour marksQ1 Anita always confused in concepts of fundamental quantities she discussed with friends and? they decided to take Speed of light and Planck constant and gravitational constant G and expressed Mass Length and time in the form of these fundamental?a). What are value shown by her.b)Expressed the mass length in these terms.?? Q2. Anita was confused in accuracy and precision and she discuss with her teacher and teacher gave her problem to understand the problem ?Time for 20 oscillations of a pendulum is measured as t1= 39.6 s; t2= 39.9 s; t3= 39.5 s. What is the precision in the measurements? What is the accuracy of the measurement?a) what value shown by herb) find a meaningful relationQ3.Anita always tries to define new standards and tries to find new standarda)what are value shown by her?b) find the value of new standard?If the unit of force is 100N, unit of length is 10m and unit of times 100s. What is the unit of Mass in this systemQ4. Anita is fascinated by satellite and she tries to found relationships using dimensions?a). what are value shown by her.?b). Solve the problem using Dimensions?An artificial satellite is revolving around a planet of Mass M and radius R from Kepler third law about the period of satellite around a common central body square of the period of revolution T is promotional to the cube of the Radius R. Show using dimensions?Section DSolution(Value Based Question) Four marksQ1? using the concept of homogeneity? And , dimensional dependence can solve be? solved andM=√chG.?Q2. (a) Precision is given by the least count of the instrument.For 20 oscillations, precision = 0.1 sFor 1 oscillation, precision = 0.005 s.(b) Average time= 1.98Max. observed error = (1.995 –1.980)s = 0.015s.Q3Concept is principle of homogeneity of dimensions according which the dimensions of same quantity can be added or subtracted? or the can be in multiplication or division.?Q4 Use the? dependence of dimensional analysis to find relation and 105?.Q5Using the concept of arc and radius we can solve?Section E FIVE MARKS QUESTION (10) Q1 Calculate the solid angle subtended by the periphery of an area of 1cm2 at a point situatedsymmetrically at a distance of 5 cm from the area.Q2. A new system of units is proposed in which unit of mass is α kg, unit of length β m and unit of time γ s. How much will 5 J measure in this new system?Q3. In the expression P = E l 2 m–5 G–2, E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.?Q4. Force of viscosity F acting on a spherical body moving through a fluid depends upon its velocity (v), radius (r) and co-efficient of viscosity ‘η’ of the fluid. Using method of dimensions obtain an expression for ‘F’. Q5. The speed of sound v in a gas might plausibly depend on the pressure p, the density?ρ, and the volume V?of the gas. Use dimensional analysis to determine the exponents x, y, and z?in the formula?v = CpxρyVz,where C?is a dimensionless constant. Incidentally, the mks units of pressure are kilograms per meter per second squared?Q6. ?If length of pendulum is increased by 2%. The time period change by what percentage?Q7 If v stands for velocity of sound, E is elasticity and d the density, then find x in the equation v = (d/E)xQ8. 10. (a) State which of the following are dimensionally current(i) Pressure = Energy per unit volume(ii) Pressure = Momentum *volume *time(b) The density of cylindrical rod was measured by the formula = 4mπD2lThe percentage in m, D and l are 1%, 1.5% and 0.5%. Calculate the % error inthe value of density?Q9. An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Derive the relation.?Q10 The measurements recorded as m ??3.5kg ??0.1 kg ,????20m / s ?1m / s, r ?12.5m??0.5mFind the maximum possible (1) fractional error (2) % error in the measurement of force. How will you record the reading?SOLUTION( Section E)Q1.Solid angle =areasquare of radius =4 × 10– 2 steradianQ2 as energy has dimensions M1L2T-2 So 1J in new units becomes 2β2α-1J. Hence 5 J becomes 52β2α-1JQ4.Let F?∝ va, F?∝ rb?and F?∝?ηc?? ? ? ? ? ? ? ? ? ? ? ? ?…... (1)So, F = Kvarbηcwhere ‘K’ is a dimensional constant.Dimensional formula of ?F = [M1L1T-2]Dimensional formula of ? v = [L1T-1]Dimensional formula of ? r = [L1]Dimensional formula of ??η = [M1L-1T-1]Substitute for the dimensional formulae in equation (1),[M1L1T-2] = [L1T-1]a?[L1]b?[M1L-1T-1]c[M1L1T-2] = [Mc?La+b+c?T-a-c] ? ? ? ? ? ? ? ? ? ? ? ? ? …... (2)In accordance to principle of homogeneity, the dimensions of the two sides of relation (2) should be same.So, c = 1 ? ? ? ? ? ? …... (3)a+b-c = 1 ? ? ? ? ? ? …... (4)-a – c = -2 ? ? ? ? ? ?…... (5)Putting ?c = 1 in (5), we get a = 1Putting a = 1 and c = 1 in (4), we get b = 1Substituting for a, b and c in (1), F = kηrv, which is required relation.Q3 Hint placing the dimension we can find solution.Q5.A comparison of the exponents of [L], [M], and [T]?on either side of the above expression yields,1 = -x – 3y +3z,0 = x+y,-1 = -2x.The third equation immediately gives x = ??; the second equation then yields y = – ??; finally, the first equation gives z = 0. Hence,Q6 increases by 1%Q7 x=1/2Q8.use dimensions to check relation and percentage error expressionT ??g xRy??L0M0T1?=?L3/2 M0T0??L1M0T0? ?L1M0T0??X=-1/2 y=-1Q10 use relation fractional change in force = fractional change in mass + fractional change in radius nal change in velocityTWO MARKS QUESTIONHOTSQ1.If heat dissipated in a resistance can be determined from the relation: H = I2Rt joule , If the maximum error in the measurement of current, resistance and time are 2% ,1% , and 1% respectively, What would be the maximum error in the dissipated heat? Ans: % error in heat dissipated is ±6 %. Q2. Name any three physical quantities having the same dimensions and also give their dimensions. Ans : Any group of physical quantities, like work , energy and torque and their dimensions [ ML2 T-2]. Q3. In Van der Wall’s equation ( P + a/V2 )( V – b ) = RT, Determine the dimensions of a and b. Ans : [a] = [ML5 T-2] and [b] = [ M0 L 3 T 0 ]. Q4. Give the limitations of dimensional analysis. Ans …………………………… Q5. If X= a+ bt2 , where X is in meter and t is in second . find the unit of a and b? Ans : unit of a is meter and unit of b is m/sec2. HOTSTHREE MARKS QUESTIONHOTSQ6. What is meant by significant figures ? State the rules for counting the number of significant figures in a measured quantity? Ans. ………………………… Q7. Show that the maximum error in the quotient of two quantities is equal to the sum of their individual relative errors. Ans : For x = a/b , Δx/x = ± ( Δa/a + Δb/b) Q8. Deduce the dimensional formulae for the following physical quantities. A) Gravitational constant. B) Power C) coefficient of viscosity D) Surface tension. Ans: (A) gravitational constant = [M-1 L3 T-2], B) Power = [ML2 T -3] C) Coefficient of viscosity = [ ML-1 T-1] D) Surface tension = [ ML0 T -2] Q9. Name the four basic forces in nature. Arrange them in the order of their increasing strengths. Ans : (i) Gravitational force (ii) Electromagnetic force (iii) nuclear force (iv) Weak force The relative strengths of these forces are Fg :Fw:Fe:Fs=1:1025:1036:1038 . Q10. Convert 1 Newton force in to Dyne. Ans : 1N = 105 Dyne. NUMERICALS LEVEL 1(Question)Q1. Food contains chemical energy and for historical reasons, food energy is normally givenin non-SI units of Calories. One Calorie with a capital “C” equals 1000 calories, and 1 calorie is defined as 4.18 J. A typical person consumes 2000 Calories of food in a day, and converts nearly all of that directly to body heat. Compare the person’s heat production to the rate of energy consumption of a 100-watt light bulb?Q2. When a car or truck travels over a road, there is wear and tear on the road surface, which incurs a cost. Studies show that the cost C per kilometer of travel is related to the weight per axle w by C α w4. Translate this into a statement about ratios.Q3. A triangle has an area of 6.45 m2 and a base with a width of4.0138 m. Find its height with correct significant no.?Q4. The Earth's surface is about 70% water. Mars's diameter is about half the Earth's, but it has no surface water. Compare the land areas of the two planets.Q5 (a). How many cubic inches are there in a cubic foot? (b)The central portion of a CD is taken up by the hole and some surrounding clear plastic, and this area is unavailable for storing data. The radius of the central circle is about 35% of the outer radius of the data-storing area. What percentage of the CD's area is therefore lost?LEVEL 1(Solution)Q1 find full day energy and divide with time and compare with bulbQ2.C1/C2 = (w1/w2)4Q3.The area is related to the base and height by A = bh/2.So 3.21391200358762 m (calculator output)Ans 3.21 mQ5 hint: 1 inch =2.54 cm) First find an approximate answer by making a drawing, then derivethe conversion factor more accurately using the symbolic method. n1u1=n2u, ans=0.000579 cubic footLEVEL 2(Question)Q.1 ) The density of mercury is 13.6 g cm-3 in CGS system. Using dimensional analysis find its value in SI units. Using the formula n2 = n1 (M1/M2)a( L1/L2)b (T1/T2)c =13.6 [1g /1000g ] [1 cm /100 cm ]-3 = 13.6 X 103 kgm-3 Q.2 Taking velocity, time and force as the fundamental quantities, find the dimension of mass. Using Force = MASS X Acceleration = Mass X Velocity /Time MASS = Force x Time /Velocity [Mass] = [FTV-1]Q.3). The percentage errors in the measurement of mass and speed are 2% and 3% respectively. How much will be the maximum error in the estimate of kinetic energy obtain by measuring mass and speed? Given ?m/m X 100 = 2% ?V/V X 100 =3% Using K = ? mv2 ?K/K X 100 = ?m/m X 100 +2 ?V/V X 100 = 2% +2 X 3% = 8% Q.4. Deduce the dimensional formulae of the following physical quantities:(i) Specific heat (ii) Gas constant (iii) Boltzmann’s constant (iv) Coefficient of thermal conductivity (v) Mechanical equivalent of heat. (i) __ HEAT _______ = ML2T-2 = [L2T-2K-1] Mass X temperature M.K (ii) PV= NRT force x volume________ = [ML2T-2K-1 ] Mol X area x temp.(iii) K = _______HEAT_______ = [ML2 T-2 /K = [ML2 T-2-1 ] Temperature(iv) K = heat X distance ___ = [MLT-3 K -1] area x time x temperature difference (v) j = Work = [M0 L0 T 0] HEAT Q.5) If the units of force , energy and velocity are 20 N, 200 J and 5ms-1 , find the units of length, mass and time. 60 joule/ 60 s = 1 watt [power] = ML2 T -3 a = 1, b =2, c =-3 n2 = n1 [M1/M2]a [L1/L2 ]a [ T1/T2] 1 [1000/100]1 [100/100]2 [1/60]-3 = 2.16 X 106 LEVEL 3(Question) Q.6 )The frequency v of an oscillation drop may depend upon radius r of the drop, density p of the liquid and surface tension S of the liquid. Establish an expression for v dimensionally. v = K r a p b Sc [v] = T-1 [r] = L , [P] = ML-3 , [S] = MT-2 Substituting dimension Equation T-1 = [L]a [ML-3] b [MT-2] c a= -3/2 b= -1/2 C = ? v = Kr -3/2 p -1/2 S ? = K √S/Pr3 Q.7) The escape velocity v of a body depends upon (i) the acceleration due to gravity of the planet and (ii) the radius of the planet R. Establish dimensionally the relation between v, g and R.SOLUTION v =Kg2 R b LT -1 = [LT-2 ]a [ L]b = L a +b T -2a a + b =1, -2a = -1 a= ? b= ? hence K g ? R 1/2 =K √g RQ.8 If density p, acceleration due to gravity g and frequency v the basic quantities, find the dimension of force.SOLUTION p = ML-3 , g = LT -2 , v = T -1 M = Pg 3 v -6 , L = g v -2 ,T = v -1 [force] = MLT-2 = pg 3 v-6 gv-2 . v2 = [ pg4 v -6 ]Q.9 The length and breadth of a rectangle are (5.7 + 0.1) cm and (3.4 +0.2)cm. Calculate area of the rectangle with error limits.SOLUTION A= l b =5.7 X 3.4 = 19.38 cm2 Using ?A/A = (?L/L +?B/B)?A = A (?L/L +?B/B) = (0.1 /5.7 +0.2+3.4) X 19.4 = (0.1017+0.059) X 19.4 =1.47 =1.5 Area with error limit = (19.4 ± 1.5) = cm2Q.10 A physical quantity X is given by X = a2b2/c √ d . If percentage error of measurement in a, b, c, and d are 4%, 2% 3% and 1% respectively, then calculate the percentage error in X. SOLUTION X = a2b2/c √ d = 2 X 4% + 3 X 2% + 3% + ? X 1% = 17.5% *************************************************************************************8** ................
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