Lab 7 – Moment of Inertia and Rotational Motion
Purpose
The guiding questions for this lab are “what is the experimental value for the moment of inertia
of a pair of ‘point’ masses?” and “does this value agree with the theoretical prediction?”
Overview
We will measure the angular acceleration of an object with a computerized data acquisition
system. That value and the applied torque will be used to determine the moment of inertia. We
will make two measurements, one for the apparatus alone, and one with two “point” masses
attached to the apparatus. The difference of these two results will give us the moment of inertia
of the “point” masses alone, which we will then compare to the theoretical result. Your cover
memo or abstract should clearly and prominently summarize these two key results, which should
also be featured in your summary of the experiment.
Prelab and Lab Procedure
The pre-lab questions on LON-CAPA are based on the theory section for “Lab 17” in the lab
manual (pp. 179-181), but the procedure for our experiment is completely different because
we will use a computerized data acquisition system. The steps for doing this experiment and
analyzing the data are on a separate handout, and we will also give you a different data sheet
and a different set of post-lab questions that must be answered in your report.
The procedure and data sheet refer to the numbered equations on this handout.
Theory
2
The moment of inertia of a point mass is given by MR , where M is the mass and R is the
distance from the mass to the axis of rotation. We will use two masses, each equally distant
from the center, in this experiment, so the theoretical moment of inertia will be
I point = M total R 2
[1]
where Mtotal = M1 + M2 is the total mass of the two point masses and R is their distance from the
rotation axis. These are attached by a rod, pulley, and axle to the Rotary Motion Sensor, which
also contribute an unknown amount to the total rotational inertia of the system. We will measure
I experiment = I apparatus + I point
[2]
and make a separate measurement of the moment of inertia of the apparatus, so we can subtract it
and determine the rotational inertia of the “point” masses alone.
The experimental moment of inertia is found by applying a known torque to the object and
measuring the resulting angular acceleration. Since τ = I α,
I experiment =
τ
α
[3]
where α is the angular acceleration and
τ =rT
[4]
is the torque due to the tension T in the thread wrapped around the 3-step pulley of radius r. This
is a two-body problem, so the tension is not equal to mg. The mass m hanging on the thread is
also accelerated. We must solve Newton’s Second Law for the hanging mass
mg − T = ma
[5]
to solve for the tension
T = m( g −a )
[6]
where the linear acceleration a = α r with r = the radius of the pulley. Once the angular
acceleration of the rotating system is measured, we can calculate the linear acceleration and the
tension, and use the tension to find the torque and moment of inertia.
Equipment required
!
!
PASCO Science WorkshopTM 700 interface connected via a SCSI interface to a ...
computer running the PASCO DataStudio software. The computer must be booted up
with the interface turned on for the operating system to identify it properly.
!
Rotary Motion Sensor with the Mini-Rotational Assembly attached.
!
Table clamp and support rod.
!
Mass hanger (about 10 g) and set of small (5 to 20 g) masses.
!
Lightweight string or thread.
!
Triple beam balance.
!
Vernier caliper and 30 cm ruler.
Lab 7 – Moment of Inertia and Rotational Motion
Set up the apparatus
1.
The Rotary Motion Sensor, Super Pulley,
3-step pulley, and rod should already be
assembled as shown in the drawing. (A
photo is on the web.) The Rotary Motion
Sensor should already be plugged into the
computer interface. If the interface is
turned off, the computer will usually need
to be rebooted before it will “see” the
interface and sensor.
2.
Remove the two masses and weigh them
together. Enter their mass in Table 1.1.
3.
Attach a mass to each end of the rod, equidistant from the center of the rod. You may
choose any radius you wish. The masses do not have to be at the end as shown in the
picture, but do not put them too close to the center of rotation.
4.
Measure the distance from the axis of rotation to the center of each mass. Be sure the
distance is the same to both and record this radius in Table 1.1.
5.
Attach one end of the thread to the 3-step pulley and the other end to the mass hanger.
(Important: Do not use too much thread. The effective radius of the pulley changes as
you wind the thread around it, and excess thread will complicate your measurement.)
The thread will be wound around the middle pulley, so you can either (a) tie the thread to
the small hole in the pulley or (b) tie it loosely around
the top pulley and then run it through the notch in the
pulley to secure it.
6.
Drape the thread over the Super Pulley as indicated in
the drawing. The clamp-on Super Pulley must be set at
an angle so the thread goes straight over the Super
Pulley and is tangent to the point where it leaves the 3step pulley. Be sure the mass hangar can fall freely
without hitting the table.
7.
Measure the diameter of the middle pulley with a vernier caliper and enter in Table 1.1.
8.
Calculate the radius of the pulley and enter it in Table 1.1.
Set up the “DataStudio” program
1.
Double-click on the PasPortal icon on the bottom of the
screen. It is a funny looking blue and green thing.
2.
The “How would you like to use it” window will appear.
Click on the “Launch DataStudio” icon.
3.
4.
A picture of the SW700 interface will appear. The
rotary motion sensor is already connected to the
left-most connectors, so you need to click on the
one on the far left as shown in the picture here.
The “Choose Sensor” window will appear. There is an
alphabetical list of sensors in the left-hand column. Double
click on the “Rotary Motion Sensor”. After you do this, an
icon of the sensor will appear, showing how it is supposed to
be plugged into the SW700 interface. See picture below.
The yellow connector should already be plugged
into the left-most socket like you see here.
We select “Angular Velocity” and “rad/s” units.
You don’t have to change any other settings
but click the sensor tab to confirm that it is
collecting data at 10 Hz.
5.
Click on the “Measurement” tab and verify that the box for “Angular Velocity” for
Channels 1&2 with units of (rad/s) is checked and that none of the other boxes is
checked. It should look like the example at the bottom of the previous page.
6.
There is a list of display options at the bottom of the left
column of the Setup window. Double-click on the one that
says “Graph”, and a display called “Graph 1” will open up.
This is where our data will appear.
7.
Resize the “Graph 1” window so it fills most of the screen. We are now ready to take
data, which should be angular velocity (y axis) as a function of time (x axis).
Collect and analyze the data
1.
Select a mass to put on the mass hanger (30 g to 50 g works well) and weigh it along with
the mass hanger on your balance. Enter the value in the first column of Table 1.2.
2.
Carefully wind the string onto the middle pulley, checking that the Super Pulley turns
freely and that the string goes straight over the Super Pulley. One person should hold
onto the rod until you are ready to start.
3.
Release the rod and click on the “Start” button. Data
analysis is simplified if you start collecting data just after
the rod is released, like we did in the air table experiments.
4.
Click the “Stop” button to end the collection of data.
Again, data analysis will be simpler if you stop collecting
data before you run out of string and/or the mass hits the
floor, but there is an easy work-around if you don’t.
5.
You will now have a set of data displayed in your “Graph 1” window. The buttons
across the top of this window (see below) are used to analyze the data. If you position the
mouse cursor over a button, a window will tell you what it does. We will mainly use the
leftmost button (to select part of the data to view or fit), the menu under the “Fit” button,
and the menu under the “Data” button.
6.
A sample data set is shown at right. If your data look like
this bogus set, you need to start over and try again, but
these fake data make it easy to see how the options work.
7.
Here we have data both before and after the acceleration
took place. We can click on the left-most button, which is
a selection tool, and use the cursor to click-and-drag a box
around the portion of the data we want to fit as shown here.
8.
Use the pull-down menu under “Fit” to select a “Linear Fit”.
9.
The fitted line will be shown along with a box containing
the fit parameters. The fit parameters are the same ones
we get from “Graphical Analysis”. We get uncertainties
as well as the correlation coefficient “r”. Notice that the
data being fit are highlighted in yellow.
10.
Record the slope ± its standard deviation in Table 1.2
along with the correlation coefficient “r”.
11.
Remove the two masses (M1+M2) from the rod and repeat steps 1 through 10. Because
the moment of inertia of the rod and the rest of the apparatus is so small, you must use
less mass this time. Usually the mass hangar alone (about 10 g) is enough. Record
those data in the second column of Table 1.2.
12.
You can either delete the previous run or have the new data appear
over the old ones. The “Data” menu controls which set is active
and being fit.
Calculations
For both the “point mass” and “apparatus alone” cases:
1.
Copy the value for the slope into Table 1.3 for α. Use the pulley radius in Table 1.1 to
calculate the linear acceleration, a, then use Equation [6] to find the string tension, T.
2.
Use Equation [4] to calculate the torque, and then use Equation [3] to calculate the
moment of inertia. Enter the results in Table 1.3. Remember to include units.
3.
Subtract the moment of inertia of the apparatus alone from the combined experimental
moment of inertia of the “point” masses plus the apparatus. (See Equation [2].) Record
this experimental estimate of the rotational inertia of the “point” masses in Table 1.4.
4.
Calculate the theoretical value of the “point” masses from Equation [1] and enter the
result in Table 1.4.
5.
Calculate the percent difference and record it in Table 1.4.
Other details
!
The DataStudio program can produce printouts of your fits. Try one and see, but you
don’t need to include it at the end of your lab report unless your instructor tells you to.
!
Shut down the computer program when you are done.
Answer “No” to the dialog box that shows up.
Remember to shut down the program and log off of the computer!
Lab report
Answer the questions on the page attached to the data sheet and include it in your report along
with the other things required by your lab instructor.
Be sure you have answered the two guiding questions listed on the lab introduction sheet:
1) What is the experimental value for the moment of inertia of a pair of ‘point’ masses?
and
2) Does this experimental value agree with the theoretical prediction for two point masses?
This information should be in both your cover memo abstract and final summary, as required by
your syllabus and lab instructor’s directions for your lab report.