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Physics 201
Lab Manual
PCC Rock Creek
Fellman
2
Table of Contents
General Instructions for Lab Reports…………………………………………………………………… 3
Uncertainty Analysis and Propagation………………………………………………………………… 4
Micrometer Caliper Instructions…………………………………………………………………………….6
Vernier Caliper Instructions……………………………………………………………………………………7
Lab 1 Conversions and Measurement.………………………………………………………………. 8
Lab 2 Position, Velocity and Acceleration………………. ……………..………………………… 10
Lab 3 Projectiles………………………………………….……………………………………………….…….. 12
Lab 4 Tension and Friction………………………………………………………………………………… . 13
Lab 5 Circular Motion and Centripetal Force…………………………………………………….. 14
Lab 6 Conservation of Energy………..…………………………………………………………………. 17
Lab 7 Momentum and Collisons………………………………………………………………………… 18
Lab 8 Center of Mass…………………………………………………………………………………………. 20
Lab 9 Angular Momentum…………………………………………………………………………………. 21
Make-Up Lab Conservation of Energy in Rolling Motion………….………………………. 22
Sample Lab Report A………………………………………………………………………………………….. 23
Sample Lab Report B………………………………………………………………………………………….. 27
3
General Instructions for Lab Reports
For every lab completed, each person will be responsible for handing in a lab
report. These reports will be due one week from the day the exercise is completed,
at the beginning of class. Reports may be typed or handwritten as long as they are
neat, easy to follow, and contain the following elements:
Purpose - What you hope to accomplish through the exercise, in your own words.
Procedure - Brief procedural notes in your own words. It is not acceptable to
simply write “We followed the instructions in the lab handout.” Instead, your report
should outline the steps you followed in enough detail so that it would make sense to
a person who has never seen the lab instructions.
Data - All original data in a well organized format.
Uncertainties - The range of possible values associated with every measurement
you take. (Ex: 3.5 cm  .1 cm) These uncertainties are usually due to precision
limitations of measuring equipment and will also propagate through calculations.
(See uncertainty handout.) Remember: Every measurement we take in the lab has
an uncertainty associated with it.
Calculations - Include formulas, sample calculations, and results of all calculations.
Graphs - (When appropriate) With clearly labeled axes.
Diagrams - (When appropriate) Free body diagrams, electric circuit diagrams, etc.
Sketches of Equipment - (Always appropriate) Also include a basic diagram of
the experimental setup if applicable.
Comments - Your thoughts and observations throughout the process.
Results - Clear statement of results.
Conclusion - The most important part!!! Almost anything will do as long as it shows
that you thought about it. Examples: What did you learn? Did the results make
sense? If not, what are the possible reasons? Relate the concept to an everyday
experience. Relate it to something in the text. (Get the idea?) The conclusion
should be at least a paragraph.
** Note that with the exception of the purpose (at the beginning of the
report) and the conclusion (at the end), these elements should not each be
contained in a separate section of the report, but instead will flow naturally
throughout the report. Please see the sample lab reports at the end of this
booklet **
Once again, neatness is VERY important. A well organized, easy to read report
with moderately good results will receive a higher grade than a report with excellent
results that are hidden in scribble.
4
Uncertainty Analysis and Propagation
Every measurement has a range of uncertainty associated with it. This
uncertainty is usually a result of precision limitations of the instrument used to make
the measurement. Any calculations done using a measurement will also have a
degree of uncertainty. This is a measure of how confident you are in the result of
your calculation. The propagation of the uncertainties through various calculations
has to be carefully considered.
One way to proceed with the concept of uncertainty propagation is called the
“Worst Case Calculation”, and is shown in the following two examples.
Example #1
Length = 93.10 cm  .05 cm
Width = 3.540 cm  .003 cm
Area = L  W = (93.10 cm)(3.540 cm) = 329.6 cm 2
Worst cases:
L & W both largest possible value:
Area = (93.15 cm)(3.543 cm) = 330.0 cm2
L & W both smallest possible value:
Area = (93.05 cm)(3.537 cm) = 329.1 cm2
(+.4 cm2)
(-.5 cm2)
Report final result as: 329.6 cm2  .5 cm2
Example #2
Mass = 32.32 g  .01 g
Volume = 18.8 cm3  .1 cm3
Density = M / V = 32.32 g / 18.8 cm3 = 1.72 g/cm3
Worst cases:
Largest M / Smallest V:
Density = (32.33 g) / (18.7 cm3) = 1.73 g/cm3
Smallest M / Largest V:
Density = (32.31 g) / (18.9 cm3) = 1.71 g/cm3
Report final result as: = 1.72 g/cm3  .01 g/cm3
(+.01 g/cm3 )
(-.01 g/cm3 )
5
On the other hand, when your data consists of a number of measurements to be
averaged for a final result, the uncertainty can be reported as the standard
deviation.
Example #3
You are measuring the range of a projectile. The experiment has been repeated
5 times, yielding 5 different distances: 3.41 m, 3.69 m, 3.33 m, 3.57 m, and 3.50
m.
First find the average:
_
x =  xi = (3.41+3.69+3.33+3.57+3.50)m = 3.50 m
n
5
Now find the standard deviation:
_
st. dev. =
 (xi - x )2
=
(n-1)
_
x
3.50 m
3.50 m
3.50 m
3.50 m
3.50 m
xi
3.41
3.69
3.33
3.57
3.50
m
m
m
m
m
_
(xi - x )
- .09 m
+.19 m
- .17 m
+.07 m
.00 m
.078 m2
4
=
.14 m
(xi - x )2
.0081 m2
.0361 m2
.0289 m2
.0049 m2
.0000 m2
.078 m2
(The standard deviation may also be obtained by using your calculator’s standard
deviation function.)
Report final result as: 3.50 m  .14 m
*******************************************************************
Notice the appropriate use of significant figures and decimal places in the three
examples above:
*Each final result has no more significant figures than any measurement involved in
the calculation.
*The smallest decimal place of the uncertainty matches the smallest decimal place of
the measurement or result.
6
Micrometer Caliper





In order to measure an object, the object is placed between the jaws and the
thimble is rotated until the object is lightly secured.
Make sure to secure the object with no more pressure than was required to
close the jaws at zero when empty. You may use the small thimble at the end
of the barrel for the final tightening (see small figure above, bottom right
corner). This thimble has a clutch in it that will ensure that you don’t over
tighten the caliper (you will hear clicking when the jaws tighten).
The first significant figure is taken from the last graduation showing on the
sleeve directly to the left of the revolving thimble.
Note that an additional half scale division (0.5 mm) must be included if the
mark below the main scale is visible between the thimble and the main scale
division on the sleeve.
The remaining two significant figures (hundredths of a millimeter) are taken
directly from the thimble opposite the main scale. See examples below:
The reading is 7.38 mm
The reading is 7.72 mm
7
Vernier Caliper
The Vernier Calipers we will be using have a smallest measurable value of 0.02mm
or 0.002 cm.
Using the calipers to measure the size of objects will allow for a smaller uncertainty
than if we used a ruler. In addition, you will shortly see that the design of the
calipers is more appropriate than a ruler for making certain measurements.
The diagram below shows how to read correct value on a Vernier caliper. As shown
there is a main scale with major divisions marked in centimeters and subdivisions in
millimeters. Notice that there are also marks (shown in white here) on the shaded
area in the diagram below. These marks allow you to determine the size of the
object to the nearest tenth of a millimeter.

How do we read this scale?
This example is for a caliper that has a smallest measurable value of 0.1 mm but the
concept is the same for all Vernier style calipers.
 Find where the index mark in the shaded area intersects with the main scale
(4.3 cm in the diagram shown). This gives you the length to the nearest
millimeter.
 Now find the line in the shaded area that exactly lines up with the millimeter
marks on the main scale. In the diagram below this would correspond to the
second mark. This gives a reading to the nearest 0.1 mm.
So the reading shown in the diagram below would be:
4.32 cm or 43.2 mm
8
Lab 1
Conversions & Measurement
The purpose of this exercise is to familiarize the beginning physics student with unit
conversions, measurement techniques, and measuring devices.
Part A
Conversions
1. Measure the height of one of your lab partners in inches. Convert this value to
meters, centimeters, millimeters, and kilometers. (Don’t forget to report and
convert the uncertainty in the measurement as well.)
2. Convert the speed limit of 55 mph to km/h. (No, you can’t run out to your car to
check your speedometer.)
3. We will learn that the speed of light in a vacuum is 3.0 x 108 m/s. How many
miles per hour is this?
4. Calculate the number of seconds that you have been alive.
5. A physics textbook lists the age of the universe as 5.0 x 1017 s. How many
years is this?
Part B
Measurement
Use the most accurate method possible for each step. Don’t forget about the
uncertainty of the measurements in each case.
6. Measure the thickness of a page in the textbook. How certain are you of this
measurement?
7. Measure the thickness of 10 pages. Is this measurement ten times as great as
the thickness of one page? Does the ink have a measurable thickness?
8. Measure the length and diameter of a cylinder. What is the volume of the
cylinder?
9. Measure the mass of the cylinder. Density is the amount of mass per unit
volume. What is the density of the cylinder?
10. Measure the mass of one navy bean.
11. Measure the mass of 10 navy beans. Do 10 navy beans have 10 times the mass
of one navy bean? Do the measurements agree within the range of uncertainty?
12. Measure the length of your table.
13. Measure the time it takes for one of your lab partners to walk the length of the
table. Speed is the distance traveled divided by the time it took to travel that
distance. What was the average speed of your lab partner in meters per second?
14. Convert this speed to miles per hour. Is this a reasonable value?
In this section you will gain experience with the Data Studio program and various
data taking devices.
15. Set up the air track and air cart, and observe the motion of the cart when the air
is turned on. What advantage do you think this apparatus will provide in our
studies of motion?
16. Set up the photogate to measure the velocity of the air cart. Make sure that
you enter the correct length of the object passing through the photogate.
Experiment with the range of velocities you can achieve. Record three velocities:
one slow, one medium, and one fast.
9
17. Why must you enter the length of the object that passes through the photogate?
What other information does the photogate utilize to calculate velocity?
18. Now hook up the smart pulley. By spinning it with your finger, observe how it
can record position and/or velocity (or even acceleration which we will get to in a
few days). Comment on how position and velocity values change as you spin it
at different rates.
19. Observe and record (with drawings) all of the ways in which data can be
displayed using this program.
20. Relying upon the vast experience you have gained using the Data Studio
program today, write an outline of the general procedure that must be followed
to take data using this program. Begin with turning the computer on, and list the
steps that would help a person doing this for the first time.
10
Lab 2
Part A
Position, Velocity & Acceleration
Position, Velocity, and Acceleration Graphs
When using the motion sensor, begin by familiarizing yourself with the graphing
capabilities of the Data Studio software. In particular, learn how to display only the
desired portion of your graph on the optimal scale.
1. Set up the motion sensor to graph position. When it is recording position, it acts
as the origin, with all distances in front of it being positive. Before you start
recording any data, sketch four position vs. time graphs that you would expect to
obtain for the following situations:
a) standing still
b) walking at a constant slow speed away from the detector
c) walking at a constant fast speed away from the detector
d) walking at a constant slow speed toward the detector
2. Now create position graphs using the motion sensor for each of the four cases.
Do they match your predictions? If not, describe how you would move to make a
graph that looks like your prediction.
3. Now set up the motion sensor to graph velocity. Before you start recording any
data, sketch four velocity vs. time graphs that you would expect to obtain for the
previous situations, in addition to four new ones:
e) walking at an increasing speed away from the detector
f) walking at an increasing speed toward the detector
g) walking at a decreasing speed away from the detector
h) walking at a decreasing speed toward the detector
4. Now create velocity graphs using the motion sensor for each of the eight cases.
Do they match your predictions? If not, do you now understand why?
5. Sketch constant acceleration graphs for all eight situations (most importantly,
decide for which situations the acceleration would be positive and for which it
would be negative). Verify the sign of the acceleration in each case using the
motion sensor.
Note: please include your predictions in your report as well as representations of the
computer graphs. I may be a little skeptical if anyone managed to predict
everything correctly.
Part B Reaction Time
6. Have one of your lab partners hold a meter stick at one end, letting it hang
vertically downward. Now have another person in your group hold their thumb
and index finger approximately 2 inches apart at about the midpoint of the meter
stick. When the person holding the meter stick unexpectedly drops it, record the
distance the meter stick falls before the other person catches it.
7. Use this distance along with the accepted value for gravitational acceleration (g)
to calculate the reaction time.
8. Repeat the process so that everyone in your group gets to measure their reaction
time. If you want to, write your name and reaction time on the board so that we
can see who is the fastest!
11
Part C Acceleration Due To Gravity
In the previous section, we used the accepted value for gravitational acceleration at
the earth’s surface. In this section, we will pretend that we don’t know its value, and
so we are going to do an experiment to determine it.
9. Set up the photogate to measure velocity. As you drop the picket fence through
the photogate, the computer will record the velocity of each solid panel as it
passes the sensor. To arrive at a value for the acceleration, obtain a graph of
velocity vs. time, and read the average slope of the graph.
10. Calculate the percentage error between this value and the accepted value in your
textbook.
11. Would dropping a solid colored panel have worked? What information from the
experiment did the software utilize to create the graph?
12. Give a detailed description of how you would calculate the acceleration by hand
using this information.
12
Lab 3
Projectiles
The purpose of this lab is to explore an actual example of two-dimensional motion.
CAUTION!!!! These projectiles can be dangerous! Make sure that the range is clear
before loading the apparatus. Also, please use only the plastic balls unless
specifically asked to do otherwise. There have been complaints from our downstairs
neighbors in the past.
1. Use a piece of paper and carbon paper to measure the average range of the
ballistic launcher aimed in the horizontal direction. There are three different
launch positions, make sure that you are using the first launch position (1
click) for this step. Get at least five data points and find the average range.
2. What other information do you need to calculate the initial velocity? Record
this data as well.
3. In treating this as a textbook problem, what factor(s) do we ignore in our
calculation?
4. Calculate the initial velocity of the projectile. Don’t forget about
uncertainties. Verify your result with me before proceeding.
5. Now calculate the initial velocities (and associated uncertainties) for the 2nd
and 3rd launch positions in the same manner as above. Again, verify your
results for both of these velocities with me before proceeding.
6. Predict which one will go farther, the steel ball or the plastic ball. Test your
prediction (only once please!). Was once enough to tell the difference?
Speculate on the reason for the difference in the range of the two balls.
7. Now, with the launcher set to a 60˚ launch angle, find the spot on the floor
where the ball lands, and place a Styrofoam cup at this point. Fire the
launcher 10 times and record how many times the ball goes into the cup. This
will give you some idea of how consistent the launcher is, plus it is fun 
13
Lab 4
Part A
Tension and Friction
Tension and Weight
1. First make sure the air track is level. Use the air track, smart-pulley, and
computer interface to measure the acceleration of the air cart pulled by the
weight of three different masses:
m = 20 g, 30 g, and 40 g .
2. Sketch a generalized free body diagram of the hanging mass. Remember: free
body diagrams are sketches of ONE body and ALL forces acting on THAT body.
3. Apply Newton’s 2nd law to the hanging mass. Use the resulting equation to solve
for the tension in each of the three trials.
4. Now sketch a generalized free body diagram of the cart. Where is the normal
force coming from in this case?
5. Apply Newton’s 2nd law to the horizontal motion of the cart. Use the resulting
equation to solve for the mass (M) of the cart for each of the three trials.
Average these results and find the standard deviation.
6. Now measure the actual mass of the cart. How do the measured and calculated
values compare?
7. Now, without actually setting it up, calculate the acceleration you would expect
if the air cart were pulled by a mass of 25 g. Need a hint? Consider the
equations from steps 3 and 5. These were results of applying Newton’s second
law to the motion of both objects. Combine these two equations and two
unknowns to arrive at a formula for the acceleration.
8. Now, verify your answer by actually trying it out.
Part B Tilted Air Track
9. Tilt the air track using wood blocks so that the pulley end of the track is highest.
10. Hang just enough mass from the string to hold the cart near the center of the
inclined plane without motion (in equilibrium) while the air is turned on. What is
the sum of the forces on the cart at this point?
11. Give the cart an initial velocity by gently pushing it down the track, and measure
its acceleration using the computer interface. Before the brief push, what was
the sum of the forces on the cart? After the brief push, what is the sum of the
forces on the cart? What should the acceleration be? (Hint: This question
requires no calculation.)
12. Sketch a free body diagram of the cart.
Part C
Friction
13. Place the friction block flat on your table, and increase the tension by slowly
adding mass to the end of the string (running over a pulley) until the block just
begins to slide. At this point, the tension in the string is equal to fs,max, the
maximum static frictional force. Use this fact to calculate the static coefficient of
friction, s. Draw a free body diagram of the block.
14. Now, using the same setup, carefully find the tension in the string that causes
the block to slide along the table at constant velocity after a brief push. At this
point, the tension in the string is equal to fk, the kinetic frictional force. Use this
fact to calculate the kinetic coefficient of friction, k. Draw a free body diagram of
the block.
Be sure to include ALL free body diagrams in your report.
14
Lab 5
Circular motion and Centripetal Force
When an object of mass m, attached to a string of length r, is rotated in a horizontal
circle, the centripetal force on the mass is given by:
mv 2
F
r
(Equation 1)
Today we will be verifying this equation, and exploring the relationships between the
centripetal force, mass, velocity and radius of a rotating object.
1. First you must level the rotational apparatus. This experiment requires the
apparatus to be extremely level. If the track is not level, the experimental
results will be wildly different from the theoretical results. Carry out the following
steps:
 Purposely make the apparatus unbalanced by attaching the black 300g square
mass onto the end of the aluminum track that is nearest the brass object with
3 hooks.
 Adjust the leveling screw on one of the legs of the base until the end of the
track with the square mass will remain aligned over the leveling screw on the
other leg of the base. (see figure 1)
 Now rotate the track 90 degrees so it is parallel to one side of the “A” (see
figure 2) and adjust the other leveling screw until the track will stay in this
position.
 The track is now level and it should remain at rest regardless of its
orientation. ONCE LEVEL, DO NOT MOVE THE BASE OF YOUR APPARATUS!!
Figure 1
Figure 2
2. Now try to spin the rotating platform at a nearly constant rate, and measure the
time it takes to make ten complete rotations. Divide this time by ten to obtain
the period of motion.
3. In order to calculate the linear speed of a point on the edge of the platform, what
other measurement do you need? Take the appropriate data, and calculate this
speed. You will be following this same general procedure in the following steps to
calculate the speed of a rotating object.
15
4. Now remove the square black mass, and attach a clamp-on pulley to the end of
the track nearer to the 3-hooked-object. Attach a string to the open hook of the
3-hooked-object, and hang a known mass over the pulley. (see figure 3)
Figure 3
5. Calculate the weight of the hanging mass. This will be the amount of centripetal
force we will be using later in our calculation. Enter this force into a table like the
one below:
Table 1
Centripetal Force
Radius
Speed
Mass
6. Now select a radius by aligning the line on the side post with any desired position
on the measuring tape. While pressing down on the side post to assure that it is
vertical, tighten the thumb screw on the side post to secure its position. Record
this radius in the table.
7. The 3-hooked-object must hang perfectly vertically: on the center post, adjust
the spring bracket vertically until the string from which the 3-hooked object
hangs is aligned with the vertical line on the post. If there is too much slack in
the string, wind the extra string securely around the hook it is fastened to.
8. Align the indicator bracket on the center post with the orange indicator. This is
done by sliding the bracket vertically and tightening the thumb screw when it is
at the correct level.
9. Once these adjustments have been made, remove the mass that is hanging over
the pulley, and remove the clamp-on pulley.
10. Rotate the apparatus, increasing the speed until the orange indicator is centered
in the indicator bracket on the center post. This indicates that the string
supporting the hanging object is once again vertical and thus the 3-hookedobject is at the desired radius.
11. Maintaining this speed, use a stopwatch to time ten revolutions. Use this
information to calculate the speed of the 3-hooked-object, and enter this speed
into the table.
12. Remove the 3-hooked-object from the apparatus and measure its mass, entering
this information into the table.
13. Now that the table is complete, use these values to verify equation 1.
16
14. If the mass of the 3-hooked-object were changed to half of its current value,
what would happen to the value of the speed? Discuss this with your group, and
formulate a hypothesis. Now carry out the experiment by removing the two side
masses from the 3-hooked object, effectively reducing its mass by half. Make a
new table, enter the new values, and make sure that equation 1 is obeyed once
again.
15. Repeat the experiment using a different radius.
16. Repeat the experiment using a different amount of centripetal force.
17. Discuss the interrelationships between centripetal force, mass, speed and radius
in relation to the behavior of this particular system (i.e. what happens when you
increase or decrease their values?)
17
Lab 6
Part A
Conservation of Energy
Gravitational Potential Energy
1. Drop a racquetball from the height of one of the counters, and measure the
height of the bounce. Do this at least five times to get an average of the bounce
height.
2. What is the initial potential energy of the ball? What is the potential energy at
the height of the first bounce?
3. How much mechanical energy was lost? Identify two ways in which mechanical
energy could have been dissipated.
Part B
Conservation of Energy in Vertical Motion
In this section, we will be using the ballistic launcher again. However, this time it
will be set up to launch vertically, and we will be using superballs to cut down on the
noise.
4. Set up the photogate so that it measures the velocity of the ball just as it leaves
the launcher. Using the first launch position, record the vertical launch velocity
of the superball. (Do it several times to get an average.) How much kinetic
energy does the ball have when it is launched?
5. How much kinetic energy does the ball have as it reaches the top of its
trajectory?
6. What must be the increase in potential energy between these two points?
7. Now measure the height difference between these two points (average several
trials), and calculate the corresponding change in the potential energy of the
superball. Does this value agree with your answer for the previous step?
8. Repeat this process for the second and third launch positions. This time,
calculate the expected height before you actually measure it.
9. Do you need to know the mass of the ball to calculate the height?
10. How did the calculated and measured heights compare in each case? Was there
a trend in your results? Give a possible explanation for any discrepancy.
Part C
Conservation of Energy on an Incline
11. Set up the air track with one end tilted up. Compress the bumper of the air cart
against the bottom end of the track, and then release it quickly so that it travels
a good distance up the track before starting back down.
12. Use the photogate to measure the velocity of the cart shortly after it is released,
and also record the highest position reached by the cart.
13. How much kinetic energy does the cart have as it is passing through the
photogate?
14. How much kinetic energy does the cart have at its highest position on the track?
15. What must be the increase in potential energy between these two points?
16. Think of a way to measure the difference in potential energy between these two
points. Do the values agree within the range of uncertainty?
17. Try the process one more time for a different initial velocity.
18
Lab 7
Part A
Momentum and Collisions
Estimation and Calculation of Momentum
An essential part of science is the investigator’s ability to estimate quantities without
precise measurement. One important aspect of this ability is used when an
experimenter designs an experiment to investigate some physical phenomenon. In
everyday life this skill is also quite valuable, so we will be practicing the art of
estimation while determining the momentum of various objects that are in motion at
different speeds. There is no right or wrong answer, but the use of logic and
reasonable assumptions in the estimation process is important for your result to be
close to the answer you would obtain from an exact measurement. For the
following, estimate both the mass and speed, and report the resulting momentum in
scientific notation.
a) a bug crawling across the sidewalk
b) a butterfly flying past you
c) a bowling ball rolling down the lane
d) a fastball thrown by a major league pitcher
e) an average bicyclist at medium speed
f) a car at highway speed
g) a 747 at cruising speed
h) a supertanker cruising from the Middle East full of oil
Part B
Momentum and Kinetic Energy in Collisions
The first objective of this section is to qualitatively observe several types of
collisions. We will then verify quantitatively the equations for velocities resulting
from various types of collisions. We will be considering both elastic and inelastic
collisions between aircarts having various initial velocities and masses.
B1. Qualitative Observation
For each of the following collisions between roughly equal masses, predict the
outcome by drawing a diagram of the carts both before and after the collision, using
vectors to illustrate their velocities in each case. Then test your predictions by
observing each actual collision. Make sure the air track is level.
Elastic collisions
a) velocities:
b) velocities:
c) velocities:
mass 1
-----
-----
-----
mass 2
0
-----
Inelastic collisions
d) velocities:
e) velocities:
f) velocities:
-----
-----
-----
0
-----
Now repeat several of the above collisions, this time using two carts of different
mass. Make sure you indicate the mass differences on your diagrams. Don’t forget
to draw your predictions first, then test your predictions by observation.
19
B2. Quantitative Measurement
You will be setting up each of the following four collisions. For each one, use the
photogates to measure the velocities of each cart both before and after the collision.
Remember that the photogates do not recognize direction, so you must include the
appropriate signs for velocity. Then, for each collision, use the appropriate equations
from your text to calculate the expected final velocities, and compare these to your
experimental values.
a) A moving aircart collides elastically with a second aircart that is initially at rest
and has twice as much mass.
b) A moving aircart collides elastically with a second aircart that is initially at rest
and has half as much mass.
c) The same situation as part a), but this time a completely inelastic collision.
d) The same situation as part b), but this time a completely inelastic collision.
For which of these four collisions would we expect kinetic energy to be conserved?
For which of these four collisions would we expect momentum to be conserved?
20
Lab 8
Center of Mass
Part A Meter Stick
1. Place the meter stick on the balancing stand in order to locate its center of mass.
Record this value. Recall that the center of mass is that point of a rigid body
where all of its mass can be considered to act.
2. Without actually setting it up, suppose that a 50 g mass were suspended from
the meter stick at the 95 cm mark, and calculate the new center of mass of the
system.
3. Now test your calculation by placing the stand at the center of mass position you
just calculated. Does it balance? If not, how far from the true balance point was
it? Your results should definitely fall within the range of experimental
uncertainty. If they don’t, find out why and correct your setup.
4. Add an additional hanging weight of your choice to the system at a position of
your choice. Find the new center of mass by calculation and experiment, and
compare them. Once again, the results should agree.
5. Until this point, you have most likely been calculating the center of mass by
measuring all distances from the 0 cm end of the meter stick. Interestingly,
distances can be measured from any position on the meter stick. Try doing the
calculation from part 4, but this time measuring all distances from some arbitrary
position on the meter stick. (Note that some distances will now be negative.)
6. Do the calculation one more time, measuring all distances from the original
center of mass of the meter stick found in part 1. Why is this the easiest way to
do the calculation?
Part B People
7. Stand with your heels and back against a wall and try to bend over and touch
your toes. Explain why this doesn’t work.
8. Measure the minimum distance of your heels from the wall for which you can
touch your toes. Compare this distance with others in the classroom. On
average, do men or women need to stand farther from the wall? What does this
imply about our centers of mass?
Part C Balancing Act (Take-Home Portion To Be Done Online)
9. Go to https://phet.colorado.edu/en/simulation/balancing-act. Download and open
the simulation. Investigate the “Intro” section by moving the tanks and trash
cans around and removing the supports to try to balance the seesaw. While you
play with this simulation, make observations and comments about when the
beam balances and when it doesn’t. Use the tools on the right side (mass labels,
forces from objects, level and position markers) to help you make your
observations. Describe what you discovered about balancing the seesaw.
10. Now move on to the “Balance Lab” section of the simulation. Scroll through the
objects (bricks, people, etc.) until you reach the 8 Mystery Objects (A, B, C, D,
E, F, G & H). Determine the mass of each Mystery Object by balancing it with
bricks and/or people of known mass. Show all of your calculations including
formulas and units. In addition, clearly summarize your results for the masses of
the 8 mystery objects neatly in a table.
21
Lab 9
Angular Momentum & Rotational Inertia
1. The purpose of this first part is to experience the effects of a changing moment of
inertia on angular velocity. Hold one weight in each hand, and sit on a rotating
stool. Start to spin with your arms extended, them bring your arms in toward
your chest and notice the difference in your angular velocity. Describe the effect,
and explain the reasoning behind it.
2. This part is fun - everyone should try it! Sitting on a rotating stool again, take
the bicycle wheel and get it spinning as fast as you can, holding it so that the
momentum vector is pointing straight up. The stool should stay still when you lift
your feet up. Now, keeping your feet up off the ground, turn the wheel (as
quickly as you can) so that the momentum vector is pointing straight down.
Explain in words and pictures (complete with momentum vectors) what happens
in terms of conservation of angular momentum.
3. For the rest of this exercise, we will be using a rotational apparatus. Make sure
the platter is level. Using the rotating platter alone, apply a torque to it by
wrapping string around one of the center spools, then running the string over the
smart pulley and hanging a 50 g mass from the loose end. Measure the
tangential acceleration using the computer interface. Use this tangential
acceleration and the radius of the spool to calculate the angular acceleration of
the system.
4. Calculate the torque that is applied to the disk by the force of the hanging
weights: torque = (force) x (lever arm), where the lever arm is the radius of the
spool around which the string was wound.
5. Using the angular acceleration found in part 3 and the torque found in part 4,
calculate the moment of inertia of the platter using Newton’s 2 nd law for
rotational motion.
6. Now, remove the metal ring from the box. Using the appropriate formula for an
object of this shape (see the screen), calculate the moment of inertia of the ring.
7. Recognizing the fact that moments of inertia are additive, calculate the total
moment of inertia that would result from adding the ring to your system.
8. Using Newton’s 2nd law for rotational motion, calculate the angular acceleration
that should result from adding the ring to your system. Convert this into a
tangential acceleration that can be measured with the smart pulley.
9. Now add the ring to your system, measure the resulting acceleration, and
compare it to the value that you just calculated.
22
Make-Up Lab: Rolling Motion
Part A
Conservation of Energy
1. In order to observe rolling motion we need an inclined surface. Raise one end of
your table by putting three stacking blocks under each leg at that end. Now
measure the time it takes for the solid ball to roll the entire length of the table,
starting from rest. This timing can be tricky, but a good method will yield
surprisingly accurate results. Make sure that the ball rolls nearly straight down
the table, and that no bumping or slipping occurs. Most importantly, average
many different trials with various people operating the stopwatch.
2. We are interested in the translational speed of the ball at the bottom of the table.
Using the fact that it experienced constant linear acceleration as it rolled, and
taking into consideration the length of your table, calculate its speed at the
bottom. (Hint: think back a few chapters.)
3. Now we will verify that mechanical energy was conserved. The two points at
which we will consider the energy of the ball are the top of the table and the
bottom.
At the top:
What is its gravitational potential energy?
What is its translational kinetic energy?
What is its rotational kinetic energy?
What is the sum of the mechanical energy at the top?
At the bottom:
What is its gravitational potential energy?
What is its translational kinetic energy?
What is its rotational kinetic energy?
What is the sum of the mechanical energy at the bottom?
Your results should reflect that fact that mechanical energy was conserved (within
the range of uncertainty).
Part B
Rotational Inertia
4. Obtain one of each of the following four different objects: solid sphere, hollow
sphere, solid cylinder, and hollow cylinder. Recall that in a race, the order these
objects finish is determined by their shape, and not by their mass or radius.
Look at the table in your textbook of the moments of inertia for the various
shapes. Rank the objects from smallest to largest rotational inertia.
5. Now roll all four objects down the table, releasing them simultaneously. (Hint:
race two at a time if you are having trouble racing all 4 at the same time.)
6. How does the order in your rotational inertia list compare to the finishing order
that you obtained in your experiment? Discuss the results you obtained in terms
of the relationship between rotational inertia and rotational acceleration.
23
Sample Lab Report A:
Lab 4: Tension and Newton’s 2nd Law
Purpose: The purpose of this lab is to study the effect of tension in different
situations. First we will consider the tension caused by a hanging mass that is
connected to another mass resting on a flat surface. Since the two masses are
connected using a string of negligible mass running over a frictionless pulley, and
one of the masses is resting on a frictionless surface, we will be able to neglect
friction. During the second part, we will again neglect friction, but this time the
hanging mass is connected to a mass resting on an inclined plane. We will attempt
to find the sum of the forces acting on the cart which is on the inclined plane.
Procedure, Data, Calculations, Diagrams:
m1
Part A: Tension and Weight
m2
1. Using the computer interface, we measured the acceleration of the air cart
pulled by three different hanging masses. The air cart represented in the
drawing by m1, is resting on a frictionless surface and is connected to the
hanging mass m2, by means of a frictionless pulley. The pulley is also
interfaced with the computer and will be used to measure the acceleration of
m1. We measured the acceleration for three different hanging masses (m 2) of
20, 40, and 60 grams. The following table is the result of these tests:
Hanging mass m2
Acceleration of m1 (in m/s2)
20g
40g
60g
.795
1.447
2.006
2. Here is a free body diagram for our hanging mass, m2:
aa forces acting on the hanging mass
The only
are gravity, which is pulling the hanging
mass toward the ground, and the tension of
the string which is pulling the mass upward.
The tension in the string is present because
of the mass it is attached to on the flat,
frictionless plane.
T
m2
m2g
24
3. Newton’s second law tells us that the sum of the forces acting on an object in
any one direction is equal to its mass multiplied by its acceleration in that
direction. Since both forces acting on the hanging block are along the vertical
axis, we can apply Newton’s second law to it as follows:
(sum of forces down) = (mass)(acceleration down)
(m2g-T) = (m2)(a)
Then, we can rearrange it to solve for the tension:
T = m2g – m2a = m2(g-a)
Since we used three different masses which resulted in three different
accelerations, we have to solve for three different values of T:
T(20g) = (.020 kg)[(9.8m/s2) – (.795 m/s2)] = .180N
T(40g) = (.040 kg)[(9.8m/s2) – (1.447m/s2)] = .334N
T(60g) = (.060 kg)[(9.8m/s2) – (2.006m/s2)] = .468 N
4. Next we will draw a free body diagram for the air cart:
In this case, the cart is resting on a cushion of air,
Thus the air cushion is where the normal force
is coming from. Since there is no motion in the
vertical direction, it is safe for us to assume that
T
the normal force, FN, is equal to and in the
opposite direction as gravity. Therefore these
two forces cancel each other out. At the same time,
since there is no friction, any amount of force acting
in the horizontal direction will produce an acceleration
for the cart.
FN
m1
m1g
5. Applying Newton’s second law to the cart tells us that the sum of the forces
acting in the horizontal direction will be equal to the cart’s mass times the
cart’s acceleration in the horizontal direction. Since the only force acting in
the horizontal direction is the tension, T, of the string, we can set up a
relationship between T, m1 and a:
(sum of the forces in the x direction) = T = (m1)(a)
 m1 = T/a
Now we are ready to predict a value for m1 for each of our three trial cases:
m2
T(m2)
a(m2)
m 1=T/a
20 g
.180 N
.795 m/s2
226 g
40 g
.334 N
1.447 m/s2
231 g
60 g
.468 N
2.006 m/s2
233 g
Averaging these results for m1, we get a value of 230 g.
25
6. We measured the actual mass of the cart on the digital scale to be
221 g, so our average was only off by about 4%.
Part B: Tilted Air Track
7. Now we will tilt the air track so that the pulley end of the track is highest:
m1
m2
8. Since the cart is no longer on a level plane, gravity will now have some
effect in pulling the air cart towards the ground, or the lower end of the
air track. Therefore, to balance the cart we must apply a force in the
opposite direction, which can be accomplished by hanging a mass from
the other end. We had to hang 15 g on the string for the cart to remain in
equilibrium. At this point, since there is no motion in any direction, the
sum of the forces acting on the cart is zero.
9. Now we will give the cart an initial velocity by gently pushing it down the
track. Since the sum of the forces is zero before the push, it must also be
zero after the push, and therefore the acceleration should be zero also.
Using the computer interface, we measured an acceleration for the cart of
.004 m/s2, which is very close to zero. We would also expect the cart to
have zero acceleration (constant velocity) after the push in consideration
of Newton’s first law, which states that an object in motion will continue in
straight line motion if no force acts.
10. Now we will sketch a free body diagram for the cart. I have included the
original direction of the force due to gravity, but only with a dotted line as
it is resolved into the directions of the tilted x and y axes for our
calculations.
FN
T
m1
m1gsin
m2gsin
m1g
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Conclusion:
Looking at the two separate parts of this lab, I find a common link: Newton’s second
law of motion, which tells us that the force acting on an object in any direction is
equal to the object’s mass times the object’s acceleration in that direction. Sure, it
sounds pretty straightforward, but as this lab has shown, we run into different kinds
of obstacles when applying this law. So what I have learned by doing this lab is that
in applying Newton’s second law, we must also consider the angle at which the
object is moving, as well as any kinds of frictional forces that may be resisting the
object’s motion, not to mention all the uncertainties involved in the motion of
everyday objects that we have not even considered in this lab.
27
Sample Lab Report B:
Introduction to Circuits
Purpose: To explore the relationship between voltage, resistance and current in a
circuit.
Procedures & Comments:
1. Using a multimeter, measure and record the voltage of two different batteries.
Voltages
Battery 1
Battery 2
1.502 V
1.513 V
Battery Pak (1&2
combined)
3.010 V
If the leads are reversed when measuring voltage, the multimeter will
display the same magnitude but will indicate that the voltage is negative.
2. A. From their color bands, determine and record the resistance and the
uncertainty rating for five different resistors.
B. Measure and record the resistance with the multimeter.
Resistor Color Guide
Band Color
Black
Brown
Red
Orange
Yellow
0
1
2
3
4
Band Color
Green
Blue
Purple
Gray
White
5
6
7
8
9
Uncertainty: Silver = 10% ; Gold =  5.0%
Resistor:
Resistor
1
2
3
1. Red, Purple, Black, Silver
2. Red, Red, Yellow, Silver
3. Red, Purple, Red, Silver
4. Orange, Orange, Red, Gold
5. Brown, Green, Yellow, Silver
Determined
Resistance, 
27X10^0
22X10^4
27X10^2
Determined
Uncertainty, %
10
10
10
Measured
Resistance, 
30.0X10^0
23.2X10^4
29.0X10^2
Actual
Difference, %
10.0
5.17
6.90
28
4
5
33X10^2
15X10^4
5
10
33.2X10^2
15.2X10^4
0.60
1.32
All of the measured values fall within the range given by the color
bands.
3. Using two resistors of the same order or magnitude (the third band of the same
color), connect them in series and measure and record the resistance of the
combination. Repeat twice again with different resistors.
Equivalent Resistance = R1 + R2
Resistors
3&4
2&5
4&5
Measured Resistance, 
6.23X10^3
3.83X10^5
1.55X10^5
Calculated Resistance, 
6.22X10^3
3.84X10^5
1.55X10^5
4. Using two resistors of the same order or magnitude (the third band of the same
color), connect them in parallel and measure and record the resistance of the
combination. Repeat twice again with different resistors.
1 / Equivalent Resistance = 1 / R1 + 1 / R2
Resistors
3&4
2&5
4&5
Measured Resistance, 
1.55X10^3
9.18X10^4
3.25X10^3
Calculated Resistance, 
1.55X10^3
9.18X10^4
3.25X10^3
5. Rules for adding Equivalent Resistances and Capacitances in Series and Parallel
Circuits:
Series
Resistance:
R=R1+R2+R3+…
Capacitance:1/C=1/C1+1/C2+1/C3+…
Parallel
1/R=1/R1+1/R2+1/R3+…
C=C1+C2+C3+…
6. A series circuit acts like a one-lane highway and therefore all of
the traffic must pass along the same path, which resists and slows
down flow.
29
A parallel circuit acts like a freeway and therefore the traffic
can always take the path of least resistance, which allows it
to flow freely.
7. Construct a circuit consisting of a battery and a resistor and
measure the current.
Resistor 3
Measured
Calculated
Voltage, V
Resistance, 
3.00 V
2.90X10^3
V/R=I
Current, A
1.03X10^-3
1.03X10^-3
Measured and calculated answers agree.
8. Construct a circuit consisting of a battery and two resistors, with
roughly the same order of magnitude, in series and measure the
current.
Resistors 3 & 4
Measured
Across
Current, A
Voltage, V
Battery
Resistor 3
Resistor 4
4.80X10^-4
3.00
4.80X10^-4
1.40
4.80X10^-4
1.60
30
In series the current must be the same through all elements.
The voltage measured across resistor 3 and 4 equaled 1.40 V and
1.60 V respectively. The sum of these two voltages equals 3.00 V,
which is the same as the voltage of the battery. In series the sum of
the voltages through all of the elements must equal the power
source.
9. Construct a circuit consisting of a battery and the two resistors
used in procedure 8 in parallel and measure the current through the
battery, the first resistor and the second resistor. Then measure the
voltage across each resistor.
Resistors 3 & 4
Measured
Across
Current, A
Voltage, V
Battery
Resistor 1
Resistor 2
1.93X10^-3
2.99
1.03X10^-3
2.99
9.00X10^-4
2.99
In parallel the voltage must be the same through all elements.
The current measured across resistor 3 and 4 equaled 1.03X10^-3 A
and 9.00X10^-4 A respectively. The sum of these two currents equals
1.93X10^-3 A, which is the same as the current through the battery.
In parallel the sum of the currents through all of the elements
must equal to the current through the power source.
Conclusion: The resistance of a resistor can be determined easily by deciphering its
color bands. These determined resistances, although not perfect, are quite accurate.
Using the multimeter to measure voltage, resistance and current is straight forward,
but one must take special consideration when measuring current not to damage to
meter. For series circuits, the equivalent resistance is equal to the sum of the
resistances of the resistors; and the reciprocal of the equivalent capacitance is equal
to the sums of the reciprocals of the capacitances of the capacitors. And for parallel
circuits just the reverse is true. The reciprocal of the equivalent resistance is equal
to the sum of the reciprocals of the resistances of the resistors; and the equivalent
capacitance is equal to the sum of the capacitances of the capcitors. In a series
31
circuit the current is the same through all of the elements in the circuit; and the sum
of the voltages through the elements in the circuit are equal to the voltage of the
power source.
And for parallel circuits, again just the reverse is true. The voltage is the same
through all of the elements in a circuit; and the sum of the currents through the
elements in the circuit are equal to the current through the power source. Whenever
possible all circuit lab work should be done on the breadboard. It is easier to handle
and is a much more efficient way to do experimentation.