Revised Rotational Inertia Experiment - Physics 4A

Rotational Inertia Experiment
Physics 4A – Chabot College
Scott Hildreth
Part A: Measuring Rotational Inertia
and Using the Parallel Axis Theorem.
Using the rotating support and some
accessory masses, we’ll investigate the
rotational inertia of selected geometric
shapes, and explore the relationship
between torque, angular acceleration, and
linear acceleration.
1. The rotational inertia experiment
uses a drum wrapped with string,
which will be pulled by a
descending or “driving” mass to
create a torque that turns the
drum. Measure the diameter of
the drum of the rotating support
and record (along with your
uncertainty!) You’ll need this measurement to determine the radius of the drum (R* in the picture), and
with that, the torque that rotates the apparatus.
2. Connect the computer to the Labpro, and connect the photogate and smart pulley to the Lapro DIG/SONIC
1 port. Open up the “Atwoods Machine” file in “Physics with Computers”; note that a graph of velocity vs.
time will be displayed. The velocity here will be the *linear* velocity of the descending drive mass, m, as
it pulls the string around the pulley, and off of the rotating drum.
3. Determine the masses of the accessories: (a) flat disk; and (b) the two small solid cylinders. Record these
with your uncertainties. Note that they may be larger in mass than the scale’s capability to measure, so
you’ll have to use additional masses on the opposite scale pan to achieve a balance. How will this process
affect your uncertainty in measurement of the masses? How many significant figures are reasonable?
What are the percentage uncertainties in your measurements?
4. Using the pulley arrangement, add a small amount of mass to the descending side so that the drum begins
to turn at a constant rate. You might assess whether the drum is turning by looking at the descending mass
– it should move at constant speed linearly if the drum support is rotating at constant angular speed.
Record this mass mo– it establishes the frictional resistance of the drum and pulley, and will have to be
subtracted from other measured descending masses in the experiment to determine actual torques.
5. Attach a 50-g weight hanger to the string, and add the frictional mass mo to the hanger. The descending
mass should now be m = (50+ mo) grams, and you can let it go after clicking the COLLECT button on the
Computer to begin data collection. Once the drive mass hits the ground, stop the collection, and identify
the portion on the velocity vs. time graph where the velocity was constantly and smoothly increasing.
Analyze that portion, and determine the slope –the linear acceleration.
6. From the linear acceleration, find the rotational acceleration of the system
f = a/R*
Note that you use the drum radius here – not the radius of the plate or pulley. The string is providing a
torque around the drum, and it is that radius we need to use to determine how fast the system is increasing
its rotational velocity.
7. Calculate the experimental rotational inertia of the rotating support, based on the known values of r (drum),
the descending mass, and the angular acceleration. Since the support will be used in the rest of the
experiments, you’ll have to subtract this value from the measured rotational inertias you get for the other
accessories. Since the descending mass is pulled downwards by gravity, and upwards by the tension T in
the cable, its equation for motion (in the vertical “y” direction, with DOWN being positive) is:
mg - T = ma
(note the signs of these terms! Do they make sense??)
T= mg-ma = m (g-a)
The tension on the cable pulls on the drum, creating an applied torque that turns the drum. That applied
torque is equal to the force time the distance from the axle (and since the cable pulls at right angles away
from the drum, the angle involved between the force and radius vector is 90 degrees, and the lever arm is
equal to the radius):
a = lever arm x Force = R* T
The friction in the rotating support system creates another torque, f , directed backwards (retarding the
drum’s acceleration). If you measured the mass mo correctly, and the descending mass was moving at a
constant rate, it wasn’t accelerating, and Tfriction = mog so:
f = R* mog
But torques produce angular accelerations, so the NET torque results in the angular acceleration of the
rotating support.
 net = Ia - f
I (experimental) =  net/= (a - f )/  = [ (R* T) – (R* mo g) ] / 
= 
Here we know = a/R* from the measurements above, so the value for I can be decomposed into known
and measured quantities – the mass of the descending hanger, the frictional mass, the radius of the drum,
and the acceleration of the descending mass.
As usual with an algebraic derivation, you should check to see if the units match. Rotational inertia should
have units of mass x distance2 (kg m2). The right side of this equation has units of
Distance x Force/ angular acceleration = meters x Netwons / rad/sec2 = m (kg m/sec2) sec2 = kg m2 (Correct)
8. Experiment (a) Now place the flat disk accessory on the rotating support, add more mass to the hanger so
that it descends with constant speed (record this value as moa ). (Question! Why would this be more than
the previous value of mo from the support alone?) Now add extra mass so that the hanger descends slowly
to the floor. Record the distance fallen and time taken for (3) trials.
9. Experiment (b): Repeat with the disk plus two solid cylinders. For this experiment, place the two solid
cylinders on top of the disk at the edges, but not too close so that they hit the pulleys or supports. Measure
the distance “d” between the center of the support wheel and the center of the cylinders, needed for the
parallel axis theorem! Again, first determine the small mass mob needed to make the system descend at a
constant rate.
10. Calculate the measured rotational inertia of the support plus disk, and support plus disk + cylinders, and
then subtract the rotational inertia of the support alone found earlier. Record the experimental inertia of the
accessories alone.
11. Calculate their theoretical inertia using the masses and radii, according to the formulas in the textbook for
similar geometric solids rotating about central axes. Compare and determine your percentage errors.
Account for the difference(s) between your results and the theoretical values with a brief paragraph of
For the experiment using two cylinders on top of the rotating disk, the rotational inertia will be the sum of
the inertia of the disk they rest on, the rotation inertia of each cylinder, AND their extra inertia because they
are rotating a distance “d” from the axle (= Md2):
I (disk + cylinders) = I disk + 2 (I cylinder) + 2(Mcyl d2)
You can read more about this use of the “Parallel Axis Theorem” in the textbook.
Part B: Estimating Rotational Inertia
You’ll use a similar apparatus to investigate the rotational inertia of a vertical flywheel. Your goal here will be
to first estimate the rotational inertia of the flywheel geometrically, publish your estimate by providing that
with your team members’ names submitted to the professor, and then to the experiment, comparing that
estimate to the experimental value you find.
The team with the closest published match between their estimate and the actual measured results
from the experiment for their flywheel will not have to submit a lab report for this lab and will
receive full marks for the lab report itself, IF you can complete the experiment and analysis within
the class period.
That team need only submit their data tables from the experiment; their completeness and clarity will be
factored into the overall score for the experiment. All other teams will be expected to submit a complete
write-up, including graphs and data analysis along with the data tables, in one week.
You should create your own data table for this Part of the Lab – clearly labeling all measured quantities. The
clarity and completeness of your data will be factored into your overall team lab score if you are not fortunate
to have been the team with the closest match between experiment and theory.
B1. First estimate the theoretical rotational inertia of the flywheel by measuring (and recording, with
uncertainties) all appropriate variables. Be sure to record the descending mass m, which will be supplied
by the professor.
Draw and clearly label two free-body diagrams of the problem – including one for the descending
mass (pulled down by gravity, and up by the tension in the paper tape), and one for the flywheel (rotating
from the torque created by the tape’s tension pulling down on the edge of the wheel).
Just as with the earlier Atwood’s machine problems with two masses hung from a pulley, or the two-mass
system with one sliding up a slope while connected by a string over a pulley to another mass falling down,
there are TWO variables that are in common with both the rotating flywheel and the descending mass.
What are they? What two equations allow you to solve for for I?
From the value of the linear acceleration of the descending mass, establish the rotational acceleration of
the wheel, from that, the inertia of the flywheel. Determine the % uncertainty in your estimate from the
uncertainties in the measured quantities.
Make a clear sketch of the flywheel, with its dimensions, and clearly show how you arrived at the
estimated value of its rotational inertia. Note that there are two flywheels – BLUE and GREEN. Be sure
you identify which one you are using to measure and analyze, because you’ll have to use the same
flywheel for the experiment later.
Be careful with significant figures here!
B2. Submit your team’s estimate for I(flywheel), with uncertainty, including the color of the wheel (BLUE or
GREEN), and the names of your team members to me:
Flywheel Color:
Theoretical I = _____________ +/- ______________ (what units??)
B3. Now do the experiment! To determine the experimental value of I, you’ll again use a measured
descending mass “m” pulling on a paper tape that unrolls from the flywheel, and a spark timer (set to
create a spark every 1/10th of a second) and thermal strip paper record which will give you dots on the
paper as a record if its motion.
Establish the acceleration of the descending mass by measuring how far the paper had moved in time, and
fitting that data to a spreadsheet program with a quadratic equation to determine the linear acceleration a
of the descending mass. (If possible, we’ll try to use the same Vernier apparatus to use the computer and
logger pro software to collect this data as well. But you need to be able to use the supplied tape.)
B4. Determine the percentage difference between your theoretical estimate and your experimental value.
B5. Defend your results – if your experimental value was different from the theory, what happened to account
for the difference? How close are the two values given the uncertainties you have in your measurements?
Data Table for Part A
Diameter of Support Drum:
Radius of Support Drum (R*):
Mass of Accessories:
a) disk
c) individual cylinder
Dimensions of Accessories:
a) disk radius
b) cylinder radius
(Don’t forget units!)
Distance from center of support to the center of the cylinders
Rotating Support Alone
mo (frictional mass)
a) Support with Disk
moa =
Descending mass m
Linear Acceleration
From Vernier
LoggerPro Graph
Angular Acceleration
Calculated I of
Net I (Measured) of accessories alone
Subtract I support from the calculated values in a & b
Theoretical I
of Disk, or Disk + Cylinders
Percentage Difference
d =
b) Support with two
mob =