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Course  Project  
Physics I with Lab
COURSE OBJECTIVES
1.
2.
3.
4.
Explain the fundamental laws of physics in both written and equation form
Describe the principles of motion, force, and energy
Predict the motion and behavior of objects based on physical laws
Demonstrate the principles of motion, force, and energy in simulated labs
BACKGROUND INFORMATION
•
Create an experiment that demonstrates and allows exploration of a physical principle
•
Design the experiment and measurement techniques to minimize measurement
uncertainty
•
Explain the experiment’s development and solutions to technical issues
•
Analyze the experimental data in terms of known physical concepts and generalize the
results to derive a functional law
•
Describe the function of the experiment, the independent quantities under investigation,
the measurement technique, and the results
PROJECT INSTRUCTIONS
Complete your project by following the steps below. Please also refer the Physics with Lab Final
Project Lab Manual for specific instructions on completing the final steps.
1. Choose a physical relationship to examine experimentally.
You may choose one of the examples from the “Suggested Experiments” section of lab manual
provided below, or you may design a similar experiment to suit your own tastes and available
equipment, but the experiment should relate to one or more concepts studied during the course
so far.
2. Design and build an experiment that demonstrates the physical property you have
chosen.
You will need to perform multiple measurements under similar or identical conditions, so your
experiment should be designed to allow you to reproduce the positions, velocities, and
orientations of various elements consistently many times.
Perform a series of measurements where you vary one quantity to be investigated and record
the outcomes.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
3. Analyze the recorded data.
First convert your raw measurements, usually distance and/or time, into the physical quantities
relevant to your property. Then generalize the series of measurements to extract the underlying
physical law.
4. Create your lab report.
To complete your project you will need to document all of your findings in a lab report. To
understand how to set up your lab report you will need to follow the directions in the Physics
with Lab Final Project Lab Manual (see below). There, you will find items that should be
addressed in your lab report.
LAB MANUAL INSTRUCTIONS
For your final project, you will perform a laboratory experiment to illustrate a physical law or
relationship. This manual will help you through the process of designing, operating, and
documenting your experiment.
At the end of the project, you will hand in a lab report. This report should document
everything you’ve done in the project, present all of your recorded data, and describe
your analysis and your findings.
A typical lab report will have the following sections:
Introduction
Describe the chosen problem
Experiment
Discuss how you designed the experiment including what choices you made to control
uncertainties
One or more figures or photographs clearly showing the experimental setup and the
layout, with relevant parts, distances, etc. labeled
Data collection
Describe how a typical experimental trial was conducted
Discuss how external factors may affect your measurements
A record, usually in table form, of each experimental trial, including the independent
variable and all relevant measurements at each step
If any experimental trials are to be excluded from the final analysis, discuss the
reasoning
Data analysis
Show how the measured quantities (length, time, etc.) are converted into the relevant
physical quantities (energy, momentum, force, etc.). This may require diagrams and
several steps of calculation.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Show how the uncertainties are calculated
One or more graphs showing the main physical effect being studied
Graphical analysis deriving the underlying physical law
Results
Summarize the experiment and the main findings
SUGGESTED EXPERIMENTS
Here are some suggested experiments to consider.
Moments of inertia
How does the moment of inertia of a cylinder depend on the mass of the contents? This
problem is discussed in Section 10.4 of the textbook (see pages 334-335).
By rolling a can down an inclined plane, you can measure its moment of inertia. If you fill the
can with different materials with different masses, you can study how the moment of inertia
varies as a function of mass. For an ideal solid cylinder with radius r, the correct formula is
I=
mr 2
2
How do your results compare to expectations?
Variations
The equation above is valid for solid cylinders. Do you expect the results to be the same if you
fill the can with liquids instead?
You can perform the same experiment with multiple cans to vary the radius instead. Or use a
different shape instead of a cylinder. Can you think of another way to measure the moment of
inertia instead of rolling a can down a slope? What about applying a known torque and
measuring the angular acceleration instead?
Or using the gravitational potential energy of as separate mass to somehow start the can
spinning and then measure the resulting angular velocity?
Other possibilities
Magnets exert forces on other magnets or on certain metals.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Can you measure the strength of a magnetic force as a function of the distance between the two
objects?
You’ll need to find a way to measure rather small forces without letting the objects fly out of
control.
Or come up with your own…
Anything that interests or excites you can be made into a good experiment. Just make sure
that you choose something that can be repeated and have one good independent variable to
control. If you want to demonstrate conservation of momentum, you can’t just collide two
objects together once, show that the equations balance, and claim success. You need to show
a systematic trend. Repeat the collision many times with varying speed of one of the objects
and show that, no matter what the initial momentum, the final momentum is the same.
EXAMPLES AND GUIDANCE FOR EXPERIMENTAL UNCERTAINTY
When designing and operating your experiment, you’ll need to pay close attention to
uncertainty. We want the experiment to test real physical effects, and not be swayed by data
that is mis-measured.
Two main classes of uncertainty need to be considered: external factors and measurement
uncertainty.
1. External factors
The outcome of a good experimental trial should be determined by only one element: the
independent variable you are studying. If there are any extraneous factors that can affect the
result, your data will be difficult to interpret accurately.
When designing and operating your experiment, be sure to keep external factors in mind. For
example, if your experiment uses a ball rolling down a plane, pay close attention that the ball
doesn’t slide instead. Sliding and rolling are different types of motion and that will give you
different results.
If the ball does slide during one trial, make a note and consider excluding this data point from
your final analysis. If the ball consistently slides during many trials, you may need to redesign
the experiment to avoid this, perhaps by using a different angle of incline for the plane or by
using different materials.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Try also to reproduce the starting condition of the apparatus identically for each trial. Release
the ball at the same place each time, perhaps by marking a “start” line on the plane. Make sure
that the plane hasn’t moved or tilted between trials.
Anything that might affect the rolling motion of the ball has to be controlled. Also, mention in
your lab report how your experiment controls such factors. Discuss whether any external
influences might still exist and how significantly they might affect your data.
2. Measurement uncertainty
Measurements made in the real world are imperfect and subject to uncertainty. The smaller the
uncertainty on a given measurement, the more precise the measurement is. This will be more
useful in distinguishing between alternative physical theories.
When quoting any measurement, it is essential to give both the measured quantity and its
uncertainty, so that the reader can easily see the precision of the measurement. A crude
balance might give the mass of a projectile as (50±5) g, while a more sophisticated scale might
measure (49.3±0.1) g for the same object. The precision of a measurement also dictates how
many significant figures to show. It doesn’t make sense to quote any digits smaller than the
measurement uncertainty.
A measuring device often limits uncertainties. A typical ruler has markings for centimeters and
millimeters. Under ideal circumstances, you might be able to estimate the length of an object to
within one-fifth of one division on the ruler, or 0.2 mm. If the object is irregularly shaped,
however, your precision might be limited to 1.0 mm, or even worse, and it is up to you to
estimate this.
One trick, which may help estimate uncertainty in difficult cases, is to repeat the measurement
several times. In this case, the mean value of the set is a good estimate of the true value, while
the standard deviation of the set is an estimate of the uncertainty. Recall the definition of
standard deviation:
s=
√
1
∑ ( x i − ̄ x )2
N− 1
3. Dealing with measurement uncertainty
An important part of designing an experiment is trying to minimize uncertainty. One way to
minimize uncertainty is to compare the physical quantities that need to be measured to the
measuring apparatus.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Imagine that you are asked to measure the thickness of a sheet of paper and are given a
standard ruler, with millimeter markings, so that the uncertainty might be 0.2 mm.
On the left, measuring one sheet is nearly impossible with the markings on a standard ruler. On
the right, measuring 500 sheets is much easier with the same ruler, since the quantity being
measured is now much larger than the uncertainty in the markings.
If you actually try this measurement, you’d probably measure the thickness to be (0.2±0.2) mm.
This is not a particularly precise value; it has 100% uncertainty! The trick is to measure the
thickness of many sheets of paper together, say 500, which have a total thickness of (48.7±0.2)
mm.
Dividing by 500 sheets gives the thickness of one sheet as (0.0974±0.0004) mm, a much better
result. Trying to measure a large object relative to the intrinsic uncertainty in the ruler is always
easier. Note that these measurements all rely on the assumption that all sheets of paper are
essentially the same thickness; if they are not, combining them together will spoil the
measurement. Be aware of this when designing your experiments.
Similar tricks work for other measurements as well. A stopwatch is not very precise when trying
to measure an event lasting a fraction of a second, but if that event repeats periodically and you
can measure 10-to-20 repeats over several seconds, the measurement becomes much better.
4. Data analysis
Once you have recorded raw data, you need to convert these measurements into the physically
important variables. Doing this should be straightforward using the techniques you’ve learned
during the course. With your experimental apparatus, you can easily measure masses,
distances, and time intervals, which you can then combine to calculate energies, momenta, and
forces as needed.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Imagine that we wish to test conservation of energy by rolling a toy car down a hill from several
different heights. We can measure the velocity of the car at the bottom of the hill by timing its
passage between two marked points a measured distance apart. By measuring time and
distance, we calculate velocity:
v=
x 2− x 1
t 2− t 1
then kinetic energy:
KE=
1
mv 2
2
4. Propagation of uncertainties
When converting these quantities, we need to consider how the uncertainties on the original
measurements affect the calculations. Here are the most common rules for propagating
uncertainties. Imagine that we have independently measured two quantities, x and y, with their
respective uncertainties denoted Δx and Δy.
•
Addition or subtraction: uncertainties add in quadrature. This means that, if z=x+y, the
uncertainty on z is given as:
δ z= √(δ x) 2+ (δ y )2
ñ Multiplication or division: the fractional uncertainties add in quadrature. If z=x/y, the
uncertainty on z is given as:
√
δz
δx 2 δ y 2
= (
)+(
)
z
x
y
3
ñ Exponentiation: the fractional uncertainty is multiplied by the exponent. If z=x , then:
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
δz
δx
=3
z
x
With these rules in mind, we can propagate uncertainties through nearly every calculation.
The textbook gives a slightly different rule for propagating uncertainties through multiplication:
that the fractional uncertainties add normally, not in quadrature. The two formulas are equivalent
when the uncertainties are small, but, if the uncertainties are significant, you should use the one
given here.
Consider again the example given above of determining the kinetic energy of a car by timing its
passage between two points. Assume that we first measure the mass of the car to be (81±1) g
and that we measure the distance between our starting and ending points to be (50.0±0.5) cm.
Before starting the experiment, we perform several test measurements with the stopwatch to
estimate our timing uncertainty as ±0.08 s. Here are the raw data for several trials:
Trial #
Measured time
(seconds)
1
0.60±0.08
2
0.54±0.08
3
0.44±0.08
4
0.35±0.08
5
0.30±0.08
For the first trial, we measured (0.60±0.08) s to cover (50.0±0.5) cm, so the calculated velocity
is 83.3 cm/s. To get the uncertainty, we use the rule for division that fractional uncertainties add
in quadrature:
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
√
√
δv
δ x 2 δt 2
0.5 2 0.08 2
= (
)+( )= (
)+(
) = 0.134
v
x
t
50
0.60
Multiplying 0.134 by 83.3 cm/s, we have v = (83±11) cm/s.
Converting velocity to energy follows a similar strategy. First we square the velocity that,
according to the rule of exponents, doubles the fractional uncertainty:
δv
= 0.134
v
⇒
δ (v 2 )
= 0.268
v2
So for the first trial we have v2=(6900±1800) cm2/s2. Multiplying this now by the mass of the car
requires the same rule as for division:
√
√
2 2
δ(mv 2)
δ m 2 δ( v )
1 2
=
(
)
+
(
)
=
(
) + 0.2682= 0.268
m
81
mv 2
v2
Now we have mv2=(560000±150000) g cm2/s2.
Multiplying by the constant one-half applies to both the value and the uncertainty, so that
KE = (280000±75000) g cm2/s2 = (0.0280±0.0075) J
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Repeating this calculation for the other four trials gives:
KE (g cm2/s2)
Trial #
Measured
time (s)
v (cm/s)
1
0.60±0.08
83±11
2
0.54±0.08
93±14
350,000±105,000
3
0.44±0.08
114±21
530,000±200,000
4
0.35±0.08
143±33
830,000±380,000
5
0.30±0.08
167±45
1,100,000±590,000
280,000±75,000
Finally, note how the uncertainty on the measured time is much larger than the uncertainty on
the measured length or mass. If this is the case, you can safely neglect these terms in your
calculation and just consider the uncertainty on the time. Or, if possible, redesign the
experiment slightly to reduce the uncertainty on the measured time.
Note also how the uncertainty on time measurement affects the uncertainty on the energy of
trial #5 so much more than for trial #1. This can make analysis of the data much more
complicated. If you discover such an effect when recording your data, think about how you could
redesign the experiment to reduce the uncertainty. In our example, one good idea might be to
measure over a longer distance. At first, we measured velocity over a distance of 50 cm, but,
once we see such large uncertainties, we could repeat trials #4 and #5 measuring over 75 or
100 cm instead.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
5. Graphical analysis
The simplest way to analyze a trend in the data is to plot the calculated quantity against the
independent variable. In the example above, assume that the car was released at heights of 3,
5, 7, 10, and 15 cm during the five trials. In that case, the plot looks like this, with the
uncertainties shown as error bars:
We can clearly see the trend of increasing kinetic energy with increasing height.
If we apply conservation of energy and recall that gravitational potential energy is defined as PE
= mgh, we can see that our data is perfectly consistent with expectations, which are shown as a
blue line overlaid on the same data:
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Imagine now that we wanted to use this experiment to actually measure something rather than
just confirming what we already know about gravity. Assume that we know PE = mgh but do not
know the value of the gravitational constant g. We can use our data to measure g instead. Our
data should lie along a line with a slope of mg when plotted against h. We therefore attempt to
draw a line through all of our data points.
Since the data has uncertainties, more than one line will meet this criteria:
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
Note that any line between the two dashed lines here will pass through all of our data points
within their error bars. Taking the slopes of these two outermost lines and dividing by the mass
of the car, we find that the gravitational constant g is between 880 and 1150 cm/s2, or
g = (1020±150) cm/s2
This calculation would be a primary result of our experiment. It is also worth noting that the
derived value is consistent with the “true” value of 980 cm/s2. If we had not obtained something
close to the correct value, we would need to investigate further and explain what had gone
wrong with the data.
While this type of graphical analysis can be done by hand, many popular software packages
and even some pocket calculators automate the process of finding the best line or curve to fit a
set of data. This process is usually called “linear regression.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.
Course  Project  
Physics I with Lab
PROJECT SUBMISSION
1. A title page is not required for project submissions. Because evaluators do not see student
names when reviewing student work, it is important that students not include any personal
identifiers in their project submissions.
2. Save your document as a PDF file. In Microsoft Word, you can use the Save As option to
select PDF as your file format.
3. If your project requires a video, you should post the video to a free video hosting site like
www.youtube.com, www.photobucket.com, or one of the other free webhosting websites.
The following website maintains a list of video hosting sites; http://www.videohostings.com/.
In the written materials that you submit as part of the assignment, you should include the
title of the video and a link for the faculty member to use to grade your submission.
4. Upload the PDF file in your course.
5. Your assignment will not be returned to you so keep a copy for your files.
© 2013. Any unauthorized reproduction or distribution of this material is prohibited without express
written permission. Students are expected to maintain the integrity of the assignment by refraining from
reproducing or posting the assignment or their completed work where it can be viewed by current or
future students.

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