Measurements and - Physics



UNIT 1:

Measurements and

Mathematics

The following 4-letter words are forbidden here:

Inch Pint

Foot Acre

Yard Mile

Measurements

Physics is based on observation and measurements of the physical world. Consequently, scientists have developed tools for measurement and adopted standard conventions for describing natural phenomena (e.g. distance, displacement, speed, velocity, acceleration, momentum, and current etc.).

A unit: is a standard quantity with which other similar quantities can be compared. (e.g. mph, pounds)

DEMO – have students walk heel-to-toe to measure the length of classroom. Have them report their answers in shoe units.

What would happen if the person w/ the largest foot gave directions to the person w/ the smallest?

Standard: reference to which all other examples of the quantity are compared (meter, gram, second)

o Ex - The following shows how the meter, the second, and the kilogram were derived:

Meter: 1 meter is the distance traveled by light in a vacuum in 1/299,792,458 sec.

Second: 1 second is the time required for a specific wavelength of light emitted by a cesium-133 atom to complete 9,192,631,770 vibrations.

Kilogram: 1 kilogram is still the mass of a 39 cm tall cylinder housed in the Bureau of Weights and standards in Paris.

Why do we need units in science?

All measurements must be made with respect to some standard quantity (meter, gram), thus scientists from different countries can compare experiments and measurements. If scientists from different countries had different systems for measurement then it would be difficult to compare experiments and results.

o Ex – If car A is traveling at 38 and car B is traveling at 54, is there any conclusion that we draw about car A and car B and the corresponding values?

No, the values can be compared only if a unit is applied to each. You would need to state that car A is traveling at 38 mph and car B is traveling at 54 mph, thus car 2 is traveling faster than car 1.

SI System (metric system): provides standardized units for scientific measurements. All quantities measured by physicists can be expressed in terms of the seven fundamental units.

Seven Fundamental Units (only the first 4 are written):

Quantity Being Measured Name of Unit Symbol

(1). Length meter m

(2). Mass kilogram kg

(3). Time second s

(4). Electric Current ampere A

We are only concerned with the first 4 fundamental units!

o Ex – (a). If the mass of an object is 55 what are the appropriate units for physics class?

- (b). An elephant dropped from the top of the empire state building would take

10 ____________ to reach the ground?

(a). kg (b). seconds

The meter: the fundamental SI unit of length, used to measure the length of an object or the total length of a path an object travels; metric ruler or meter stick (1 meter = 100 cm); The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

Show students a meter stick: 1 meter = 100 cm

o Ex – Determine the length of the blue line below:

5.2 cm

Second(s): the fundamental SI unit of time, in lab we will utilize stop

watches in order to measure the length of time an object falls, travels, etc.

All problems that we analyze involving time need to be converted to seconds.

o Ex – If a rocket travels 100 miles in a time of 10 minutes to leave Earths atmosphere how many seconds did the trip take? 10min * 60 sec/min = 600 seconds

Show students how to use the stopwatch, have two students jump in the air and measure their hang time. A student will use the stopwatch to measure the hang time.

Spring Scale: A spring scale is a weighing scale used to measure

force, such as the force of gravity, exerted on a mass or the force of a

person's grip or the force exerted by a towing vehicle. This force is

commonly measured in newtons, for which one newton corresponds

to the weight of about a tenth of a kilogram of mass, or about the

weight of an apple. The spring scale can also be correctly called a

Newton Meter, because it is used to measure Newton’s.

o Ex – The spring scale in the above picture is measuring the

weight or the force of the hanging mass. If the scales

units are in Newton’s, what is the force that the mass

is exerting on the spring scale (you may need to use

your magnifying glass)?

The spring scale shows 4.5 Newton’s

Conversions: Before we begin converting in the metric system we first need to better understand how it works. The simplicity of the metric system stems from the fact that there is only one unit of measurement (or a base unit) for each type of quantity measured (length and mass in our case). The base unit for length is the meter, and the base unit for mass is the gram.

So length, for example, is always measured in meters in the metric system; regardless of whether you are measuring the length of your finger or the length of the Mississippi River, you use the meter.

The metric system is a called a decimal-based system because it is based on multiples of ten. Any measurement given in one metric unit (e.g., kilogram) can be converted to another metric unit (e.g., gram) simply by moving the decimal place. For example, let's say a friend told you that he had a mass of 72,500.0 grams (159.5 lbs), you can convert this to kilograms simply by moving the decimal three places to the left; in other words, your friend weighs 72.5 kilograms.

Let’s Think:

~ In the above example, how did we know to move the decimal place to the right three times? Look below and you will find a chart that will give you the answer. This chart shows the common metric prefixes. ~

| Common Metric Prefixes Symbol Unit Multiples Scientific Notation |

|tera – (meter) or (gram) T 1,000,000,000,000 1012 |

|giga – (meter) or (gram) G 1,000,000,000 109 |

| mega – (meter) or (gram) M 1,000,000 106 |

|kilo – (meter) or (gram) k 1,000 103 |

|Base Unit – 1 meter or 1 gram m or g 1 100 |

|deci – (meter) or (gram) d 0.1 10-1 |

|centi – (meter) or gram c 0.01 10-2 |

|milli – (meter) or (gram) m 0.001 10-3 |

|micro – (meter) or (gram) [pic] 0.000001 10-6 |

|nano – (meter) or (gram) n 0.000000001 10-9 |

|pico – (meter) or (gram) p 0.000000000001 10-12 |

*Here's a shortcut: if you are converting from a smaller unit(milligrams) to a larger unit (grams) (moving upward in the table shown above), move the decimal place to the left in the number you are converting. If you are converting from a larger unit to a smaller unit (moving down in the table), move the decimal to the right.

o Ex – (a). Convert 1.42 meters (m) into centimeters (cm)? (b). Convert 872 centimeters (cm) into meters (m)?

(a). 142 centimeters (b). 8.72 meters

o Ex – (a). If Derek Jeter’s mass is 8,262 grams, what is his mass in kilograms and megagrams?

(b). If an elephants mass in 5.201 megagrams, what is its mass in grams?

(a). 8.262 kg and .008262 M (b). 5,201,000 grams

o Ex – (a). Measure the width and length of your desk and record the value in centimeters; and then convert this value to meters. (b). Measure the height of a partner and record this value in centimeters; and then convert this value to meters.

(1). Width = 34 cm and length = 44 cm; .34 m and .44m

(2). Height = 130 cm; 1.3m

Scientific Notation:

Throughout the year in our problems, experiments, exams, and on the regents you will use measurements that have very large or very small values. For example, the mass of an electron is 0.000000000000000000000000000000911 kg, and the speed of light traveling in a vacuum is 300,000,000 m/s. Rather than writing out these values with all those 0’s, we can express them using scientific notation.

Scientific Notation consists of a number (A) that is greater than or equal to 1 and less than 10, followed by a multiplication sign (x), and the base 10 raised to some power “n”. The general form is written as: A x 10n.

This means the mass of an electron can be expressed as 9.11 x 10-31. It is important to note that there should always be at least one digit to the left of the decimal place.

In order to write the mass of an electron in scientific notation you have to move the decimal place to the right 31 times this is why “n” is equal to -31. How do you know what direction and when to move the decimal though?

Converting # to scientific notation:

When “n” is positive: 45,000.0 move decimal to the left 4.5 x 104

When “n” is negative: 0.00045 move decimal to the right 4.5 x 10-4

Converting scientific notation to #:

When n is positive: 5.8 x 106 move the decimal to the right 5,800,000.0

When n is negative 5.8 x 10-6 move the decimal to the left 0.0000058

o Ex – Express the diameter of a nickel, 0.0221m, in scientific notation.

2.21 x 10-2

o Ex – The chance of being struck by lightening is 576,000 to 1, express this probability in scientific notation.

5.76 x 105

o Ex – The speed of a rifle bullet is 7 x 102 m/s and the speed of a snail is 1 x 10-3 m/s. How many times faster than the snail does the bullet travel?

700 m/s / .001 m/s = 700,000 times faster

o Ex - The power of sunlight striking Earth is 1.7 x 1017 watts. How many 100-watt light bulbs would produce this amount of power? This is why solar energy is a viable alternative energy source.

1.7 x 1015 or 1,700,000,000,000,000

Estimation:

We can estimate the length, height, width, diameter, or mass of various objects (e.g. mass of baseball, length of a baseball bat). On every single regent’s exam (including yours this June) the first question in part B-1 will be an “estimation” question. You need to be able to do 2 things to get this questions right: (1). Have a reasonable understanding of the size of various objects, and (2). Be able to convert scientific notation into usable values or numbers that we can better understand (e.g. 102m = 100m).

The only way to understand and work with these questions is through practice, so let’s have some fun:

o Ex – (a). What is the approximate height of this soda bottle? Record your answer in centimeters and meters.

.3 meters or 30 centimeters

- (b). What is the approximate height of Sachem East? Record your answer in centimeters and meters.

10 m or 1,000 centimeters

- (c). What is the approximate mass of a Lotus Elise (in case you did not know it is a fancy sports car with a very low center of gravity and short length allowing it to make extremely sharp turns)? Record your answer in grams and kilograms.

1,000 kg or 1,000,000g

o Ex – (a). The height of a doorknob above the floor is approximately:

(1). 1 x 102 m (2). 1 x 101 m (3). 1 x 100 m (4). 1 x 10-1 m

(3). 1 m

- (b). The length of our classroom is probably closest to

(1). 10-2 m (2). 10-1 m (3). 101 m (4). 104 m

(3). 10m

- (c). The thickness of a dollar bill is closest to

(1). 1 x 10-4 m (2). 1 x 10-2 m (3). 1 x 10-1 m (4). 1 x 101 m

(1). .0001m or 1 centimeter; choice (2). is 1 centimeter

- (d). The mass of physics textbook is closest to

(1). 103 kg (2). 101 kg (3). 100 kg (4). 10-2 kg

(3). 1 kg

Converting Units

Due to such a variety in units (non-SI, SI, and English), conversions are often necessary. Some conversions require little calculation, where as others require much more. In order to carry out more complex conversions, knowledge of some basic conversions is required.

• Ex - 1 day = 24 hours (hr)

|1 foot (ft) |= |12 inches (in) |1 hour (hr) |= |60 minutes (min) |

|1 inch (in) |= | 2.54 centimeters (cm) |1 minute (min) |= |60 seconds (s) |

|1 meter (m) |= |1.1 yards (yd) |1 pound (lb) |= |454 grams (g) |

|1 mile (mi) |= |1609 meters (m) | | | |

1. Convert one year (yr) into seconds (s).

[pic]

2. Convert 45 miles per hour (mi/hr) into meters per second (m/s).

[pic]

3. Convert one foot (ft) into centimeters (cm).

[pic]

4. Convert 17,520 hours (hr) into years (yr).

[pic]

5. Convert 5 meters per second (m/s) into miles per hour (mi/hr).

[pic]

Mathematics

Much of what we will do this year in physics will require some type of math skills. You will be solving equations for single variables, graphing lines and curves, and constructing best fit lines. These are all concepts that you have already covered in math class and should be familiar with. However, we will be brushing up on these skills and focusing on the ones that are the most appropriate for regent’s physics.

Solving Equations:

As 11th and 12th graders you have completed at least 5 math classes, and if you were lucky enough you started solving equations in sixth grade. So let’s practice those skills and put them to good use.

Step 1: Write down the givens

Step 2: Rearrange the equation to isolate the variable

Step 3: Substitute in the appropriate values (with units)

Step 4: Compute your answer

Step 5: Go back to the original and check your work

o Ex – Solve the following equations for v.

(a). KE = 1/2mv2 (b). p = mv (c). n = [pic]

Graphing Data:

The data collected in a physics experiment are often represented in graphical form. A graph makes it easier to determine whether there is a trend or pattern in the data.

Equation of a Line: y = mx + b

m = slope = [pic]= [pic]= [pic]

b = y-intercept

The Line of Best Fit is a straight or curved line which approximates the relationship among a set of data points. This line usually does not pass through all measured points.

o Ex – The position of a moving truck was measured at one second intervals and recorded in the following table. Graph the data on the grid provided and draw the line of best fit. Describe the slope of the line as positive, negative, or zero.

|Time (s) |Position (m) |

| 0.0 | 0 |

| 1.0 | 18 |

| 2.0 | 40 |

| 3.0 | 62 |

| 4.0 | 80 |

| 5.0 | 100 |

Slopes of common curves:

Positive Slope: Negative Slope: Zero Slope:

Undefined Slope:

Mathematical Relationships:

Two quantities are directly proportional if an increase in one causes an increase in the other. The direct portion y = 2x is shown below:

Two quantities are inversely/indirectly proportional if an increase in one causes a decrease in the other. The equation y = [pic] or xy = 12 below exemplifies an inverse proportion:

Two quantities have a constant proportion if an increase in one causes no change in the other. The equation y = 6, shown below is an example of a constant proportion:

Two quantities have a direct squared proportion if an increase in one causes a squared increase in the other. The direct squared proportion y = x2 is shown below:

Two quantities have an indirect squared proportion if an increase in one causes a squared decrease in the other. The equation y = [pic] below demonstrates an indirect squared proportion:

Measuring Angles:

A common unit for measuring angles is the ______________________ (°), which is 1/90th of a right angle. The protractor is an instrument used for measuring angles in degrees.

The figure below shows a protractor being used to measure angle AOB. Point O always has to be aligned with the vertex of the angle, and the diameter of the protractor (think of it as a semicircle) lies on OA, one side of the angle. The other side of the angle intersects the protractor at some angle.

o Ex – What is the measure of angle AOB, __________°.

o Ex - How do you think we can construct an angle of 25°? Using the line below, OA, and your protractor see what you can come up with.

O A

Before I do lab I still need to write in this packet: (1) metric (SI) conversions grams to kg and meter to cm, (2). Estimating the size of objects (3). Show prefixes on the reference table (4). Math practice

*Do the math review after the lab!

Conversions (of all SI units)

Estimation

Now we will complete the first lab of the year, it is titled “Linear Measurements & Thumb Wrestling”

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