Experiment 2 Rotational Dynamics
Background
You may find it useful to review chapters from your introductory physics textbook
dealing with rotational motion, torque and angular momentum.
The average angular velocity of a rotating object can be calculated using
Δθ
(1)
ω=
Δt
where Δθ is the angle through which the object rotates during a time Δt. We usually
express Δθ in radians, and in fact for many of the following equations, angles MUST be
expressed in radians. If the angular velocity is constant, then equation (1) also represents
the instantaneous angular velocity.
For a rotating object experiencing constant angular acceleration, α,
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θ = θ o + ω ot + α t 2 .
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If it starts at rest, this simplifies to
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Δθ = α t 2 . (2)
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Equation (2) can be used to calculate the angular acceleration of a rotating object.
Angular acceleration is determined by the net torque acting on the object. First, torque is
defined through the equation
  
τ =r ×F
(3)
In the case of a force applied at right angles to the lever arm, which is the case when a
string is wound around a pulley, the magnitude of the torque is given by
τ=RF, (4)
where R is the radius of the pulley and F is the tension in the string.
Moment of inertia is a measure of the resistance of an object to changes in its rotational
motion. The value of the moment of inertia depends on the axis of rotation and on the
distribution of mass in the object. For some simple situations we have
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I=mr2
A point mass m a distance r from the axis of
rotation
A hoop or ring of mass m, inner radius R1, and
outer radius R2, rotated about an axis
perpendicular to it and passing through its
center
A uniform solid disk of mass m and radius R
rotated about an axis perpendicular to it and
passing through its center
A thin rod of mass m and length L rotated
about an axis perpendicular to it and passing
through its center
I=(m/2)(R12 +R22)
I=(1/2)mR2
I=(1/12)mL2
Moments of inertia of more complex objects can sometimes be found by breaking the
object down into components and adding up the individual contributions to the moment
of inertia.
The moment of inertia relates torque to angular acceleration:
(5)
τ net = Iα
The moment of inertia also shows up when calculating angular momentum:
L=Iω
(6)
In the absence of external torques on a system, the angular momentum is conserved. This
means that in a collision between rotating objects, to a good approximation the total
angular momentum of the objects just prior to the collision is equal to the total angular
momentum of the objects just after the collision. During such a collision, friction acting
between the objects will allow the objects to achieve a common final angular velocity.
While this friction does result in a decrease in the kinetic energy of the system (that is,
the collision is inelastic), the angular momentum is NOT affected by the friction between
the two objects since internal torques do not change the total angular momentum of the
system. However, if there is friction external to the system, then the total angular
momentum will not be constant over time.
Apparatus
Pasco Rotational Dynamics Platform
As configured for the first part of this experiment, this device consists of a heavy black
metal base with a freely rotating vertical metal rod. Attached to the rotating rod is a
horizontal metal platform (it looks like a fancy ruler). You will also need a stopwatch
and a mass set.
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Procedure
Part 1: Varying torque
1. There are two adjustable weights mounted on the top of the horizontal rotating
platform. One may have coins glued to it. Do not remove the coins since they are
present to balance the weights. The weights should be centered at 15 cm on either side
of the middle of the rod. A hole through one edge of the block allows you to see through
to the scale below. Important: DO NOT OVERTIGHTEN THE SCREWS (the tops
break off easily). Tighten the screws just enough to keep the blocks from sliding.
SAFETY GOGGLES MUST BE WORN WHILE THE PLATFORM IS ROTATING.
These are provided for you in the lab.
2. Directly underneath the rotating platform, mounted on the vertical rotating rod, you
will see three pulleys (“platform pulleys”). There should be a black thread tied to that
system of pulleys. Wind the thread on the uppermost platform pulley . Drape the free
end of the thread over the other pulley (the “guide pulley”) that hangs over the edge of
the table. Hook a 50 g mass onto the free end of the string. When released, the hanging
weight pulls on the string that in turn applies a torque to the pulley, causing the rotating
platform to undergo angular acceleration. You will determine this acceleration with the
help of equation (2).
3. Line the rotating platform up with the horizontal rod on which the guide pulley is
mounted. Release the platform from rest at the same time you start the stopwatch. Stop
the watch when four full revolutions have been completed. Record this time and repeat
until you have three complete measurements. In between each measurement, rewind the
thread, being careful that it stays on the uppermost platform pulley. Each time you need
to wind the hanging mass back up to a high enough point that the thread will not get
unwound completely (and start winding back up in the other direction) while you are
timing the four revolutions.
4. Change the hanging mass to 70 g by hooking a 20g mass to the bottom of the 50 g
mass, and repeat step 3.
5. Change the hanging mass to 100 g and repeat step 3.
6. Now repeat steps 2-5 for the middle platform pulley and then the bottom platform
pulley. In the end, you should have taken 27 measurements (three measurements for each
of three hanging masses for each of three pulleys). It is not necessary to re-tie the thread
to a different pulley, but you do need to be careful each time you rewind it that it is going
on the correct pulley. Also, particularly for the bottom pulley, you will need to be careful
to start the mass at a high enough position so that it does not reach the bottom before the
four revolutions are complete. Also watch carefully to make sure the string does not
come off the bottom pulley during the timed interval.
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7. Detach the rotating platform by loosening the black set screw underneath the platform.
Measure the length of the platform, and then measure the mass of the rotating platform
and the two sliding masses separately. The black base at the center of the rotating
platform has a mass of 70.9g. It contributes a negligible amount to the moment of inertia
of the platform since that mass is concentrated near the center of the platform. When
you go to calculate the moment of inertia of the platform, subtract this 70.9g mass
from the total platform mass you measured on the scale. Use calipers to measure the
diameter of the three platform pulleys and remember to convert these numbers to radii for
your calculations. Make sure you are measuring the radius of the pulley out to the point
at which the thread makes contact with the pulley (not past that to the lip of the pulley).
Part 2: Varying moment of inertia
1. Reattach the rotating platform to the shaft, being careful to line up the set screw with
the flat side of the shaft. Wind the string around the middle platform pulley. Adjust the
sliding masses so that they are each centered 3cm from the middle of the rotating
platform. DO NOT OVERTIGHTEN THE SCREWS. Hang a 50 g mass from the thread
and time the platform through 4 revolutions starting from rest. Repeat to obtain three
measurements.
2. Repeat step 1, moving both masses successively through the following positions : 8
cm, 13 cm, 18 cm, 23 cm. In each case, these positions refer to the center of each mass.
In the end, you will have taken 15 measurements (three measurements for each of five
mass positions).
Part 3: Conservation of angular momentum
1. Take the hanging mass off of the string and then loosen the set screw underneath the
rotating platform . Be careful not to allow a knot to form in the thread—it has a tendency
to twist up. Slide the platform off, setting it aside in a safe location. Wind the thread
completely around the pulley and use a small piece of tape to secure it to the pulley to
keep it from getting tangled. You will not need the thread for this part of the experiment.
2. You should see a nearly solid black disk with a circular groove in it. Mount it where
the rotating platform was. If this disk does not slide onto the shaft smoothly, you do not
have the flat side of the hole lined up properly with the flat side of the shaft. You should
also see a black hollow cylinder that is made to fit in the groove. Spin the solid disk.
Hold the hollow cylinder just above the groove on the disk and practice dropping the
cylinder onto the disk while the disk is rotating, so that the hollow cylinder lands in the
groove. Make sure you let go of the cylinder completely before it makes contact with the
rotating disk.
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3. Remove the hollow cylinder from the disk. Give the disk a spin so that it is taking
roughly 0.5s to 1.0s to complete a revolution. Time the disk for ten consecutive
revolutions. As soon as possible, drop the cylinder onto the groove and then immediately
time another 10 revolutions. Repeat this process for a total of five pairs of
measurements. With the help of equation (1), these times will allow you to calculate the
angular velocity before and after the collision. It is important to allow as little time as
possible to elapse between the time measurements and the collision in order to minimize
the effect of friction.
4. Remove both the hollow cylinder and the solid black disk. Remove the tape that was
securing the thread and remount the rotating platform. This is a “lab courtesy” step
which gets the apparatus ready for the next group.
5. Measure the mass of the hollow cylinder and the solid disk.
6. Measure the diameter of the solid disk and of the hollow cylinder. For the solid disk,
one way to do this accurately is to first measure the circumference by measuring the
length of a thread that is wrapped around the outside. The radius is this length divided by
2π. Alternatively, you may be able to get an accurate reading by using the meter stick to
measure the diameter—on one side of the disk there are two small screw holes that can be
used to help locate a diameter. Either way, make sure you estimate the uncertainty
generated by the experimental procedure (which is likely to exceed the instrument
uncertainty). For the hollow cylinder, measure both the inner and the outer diameter (or
one of those two and the thickness) using the calipers. Be sure to return the solid disk
and the hollow cylinder to the table with the rest of the rotational dynamics apparatus
when you are done.
Data Analysis and Writing the Results Section
This section will help guide you through writing up the Results section of your report. A
complete Results section should show all the raw data (in tabular form when you have a
lot), calculated results with uncertainties, sample calculations of each type, an
explanation of the data and calculation, and a discussion of the results. Make sure that
this section includes sufficient text to be readable. That is, do not just show a data table
but introduce it with a sentence or two indicating what data you are showing (e.g., “The
following data was acquired to test the principle of conservation of angular
momentum.”). Put another way, the Results section is not just a collection of numbers
and calculations; it is TEXT interspersed with data and calculations. As always, consult
the sample lab report online for further information and keep in mind the comment on
significant digits in the Experiment 1 write-up.
In this and subsequent experiments, when you need a value for g, you should use the
accepted value for this course, including the provided uncertainty. This value will be
posted in the lab, and will likely be a value determined by student results from
Experiment 1.
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Part 1
For each of the pulley and mass combinations, calculate the angular acceleration using
equation (2). Use the variation in measured times to estimate an uncertainty in each
angular acceleration, as discussed in the Uncertainty Analysis presentation. A fair
question to ask is, should the uncertainty in Δθ be included too? If you assume that there
is only random error (as opposed to systematic error) in this value, then the uncertainty
in Δθ will show up as an uncertainty in Δt, since it will only affect when you stop the
watch. This uncertainty has already been accounted for. If this is the approach you plan
to take, then you should state so explicitly in your Results section. In general, I will be
looking at each input into your calculation for an uncertainty. If you have not included
the uncertainty, then you need to justify why it is not needed.
Use equation (4) to calculate the applied torque, taking the tension in the string to be
approximately mg. Don’t forget to determine the uncertainty in the calculated value of
the torque! Using a spreadsheet program, plot the applied torque (vertical axis) vs. the
angular acceleration (horizontal axis) for all nine mass/pulley combinations. If you have
correctly analyzed the data and equation (5) is correct, the result ideally would be a
straight line passing through the origin. Add a trend line showing a linear fit. Under
options, do not specify that it passes through the origin but do have it display the
equation. The best-fit equation will likely not pass through the origin. As you discuss
your results in this section, discuss the physics behind this non-zero intercept and discuss
what the slope of the trend line represents. (This shoud serve as a reminder that
discussion DOES belong in the results section!) Remembering that both the calculated
torque and the calculated acceleration have uncertainties, sketch those in as error bars on
your printed plot or figure out how to get the spreadsheet to do that (this is possible in
Excel—highlight the data series and then pull up the format data series menu). Make
sure the final printed plot has all appropriate labeling.
Part 2
Knowing the mass and length of the rotating platform, you can estimate its moment of
inertia if you treat it as a thin rod. Add to this the contributions from the two adjustable
masses, treating them like point masses. You should express everything in kilograms and
meters. Calculate the approximate applied torque as above and then for each mass
position you can predict an angular acceleration from equation (5):
τ
α pred = appl
I
Compare this value to the measured angular acceleration, calculated the same way you
did in Part 1. Each of these accelerations (both predicted and measured) should have an
uncertainty calculated. If the uncertainty intervals overlap (e.g., if αpred=2.3+/-0.4 and
αmeas=2.1+/-0.3), then you have verified the theoretical prediction to within the
uncertainties in your experiment. If the uncertainty intervals do not overlap (e.g., 2.3+/6
0.4 and 1.7+/-0.1), then you have not verified the theoretical prediction. Among the
possible reasons are (1) you underestimated the uncertainties; (2) the theory has been
incorrectly applied because a significant effect was overlooked (such as friction); and (3)
the theory is incorrect. Your discussion of the results should include your analysis about
whether the uncertainty intervals overlap or not. You can postpone conclusions from
these observations to the Conclusions section (which, fortunately for you, is not required
for this lab).
Part 3
Treating the solid disk as uniform and the hollow cylinder as a hoop (see the moment of
inertia table on page 2), calculate their moments of inertial. We expect
Li = L f
I diskω i = ( I disk + I cylinder )ω f
ωf
I disk
=
ωi I disk + I cylinder
Δθ
Δθ
Δt f
Δti
=
I disk
I disk + I cylinder
Δti
I disk
=
Δt f I disk + I cylinder
The ratio on the right hand side can now be calculated and should be the same for each
trial. The ratio on the left hand side can be calculated separately for each trial. Note that
is does not make sense to average the Δt’s first since you do not expect them to be the
same for each trial. Furthermore, since angular momentum should be conserved
separately for each trial, we expect agreement in each trial to within expected
uncertainties in this last equation. Your analysis should include a table showing the
calculated values for each trial of the left hand side of the last equation compared to the
predicted value (the right hand side of the above equation), and an indication for each
about whether they are within expected uncertainties. In Parts 1 and 2 you developed a
good feel for expected uncertainties when timing rotational motion. You can use an
estimate from those numbers for your uncertainty in Δt here. You should also estimate
uncertainties in the mass and the radius for your moment of inertia calculations. This will
allow you to determine the uncertainty in the predicted ratio (the right hand side of the
equation above).
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What is due?
For this lab, you should turn in a title page and the Results section. The title page should
include the name of your lab partner, the date you turned the report in, and an Abstract.
The Abstract should describe the nature of the experiment in a few sentences and briefly
summarize the results. This is not a mystery novel you are writing—it is expected that
you reveal the essential results in the Abstract. See the introductory material posted
on the web site for a sample lab report (showing all sections, not just those required
for this lab). Check this introductory material on the web site every time you are
learning how to write a new section. About half the points that are lost by students on
these first few reports typically arise because they did not read the introductory material
outlining expectations for the report, and they did not consult the sample report.
A word on including numerical results in abstracts: In your previous experiment, your
goal was to measure the local value of g. This result has some general interest to others
who might be doing experiments in Ypsilanti and to those interested in how g varies
geographically. For that reason, an abstract for that experiment could usefully include the
numerical value of g, along with its uncertainty. In this experiment, however, you are
doing things like calculating the moment of inertia for a particular configuration of a
particular apparatus. That value will not have any general use outside the context of your
experiment so it should not be quoted in the abstract. Instead, you might include
statements like, “We found that we were able to calculate angular accelerations to within
an uncertainty of x%. We then compared them to the predicted values (known to within
y%) determined by dividing the torque by the moment of inertia. We found these two
calculations did/did not agree.” That is, rather than quoting any actually values of
acceleration or moments of inertia, which are not important, you should focus on whether
you verified the key equation, τ=Iα.
Grading comment
Pay careful attention to the way sections of lab reports are graded (see the introductory
material posted on the web site). In particular, the “uncertainty analysis multiplier” is a
grading device designed to ensure that you take uncertainty analysis seriously. Your
score on all sections related to the results is multiplied by a number ranging from zero to
one. The multiplier will be 1.0 if you do a credible (not necessarily perfect) job of
estimating uncertainties and other sources of error, and of carrying these estimates
through numerically in all of your calculations. If you skip this, your uncertainty analysis
multiplier will be 0, meaning you will get no credit for any of the relevant sections.
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