PHY101 Experiment
10
Rotational inertia, also called moment of inertia, is defined by I = ∫ 𝑟 2 𝑑𝑚 [unit: kg m
2
]. It is a measure of the distribution of mass away from the object’s axis of rotation. In essence, when an object moves linearly and rotationally (a rolling ball; the tire of a car), it is not enough to know its mass: we need information about the distribution of mass across the body ( I ) to fully understand its motion.
Rotational inertia I can be determined from the geometry of the object, when density is uniform (often found in tables). In the case of a disk, the solution to the integral becomes
𝐼 𝑑𝑖𝑠𝑘
= 1
2
𝑀𝑅 2
[ I from geometry]
Rotational inertia I can also be determined from motion, applying Newton’s 2 nd
Law in its linear and rotational forms:
𝐹⃗ 𝑛𝑒𝑡
= 𝑚𝑎⃗ [linear] 𝜏⃗ 𝑛𝑒𝑡
= 𝐼𝛼⃗ where 𝛼 = 𝑎 𝑅 [rotational]
Let’s imagine a yoyo, attached to a string, spinning and moving down. In this case, both rotational and linear motions are combined. Ignoring friction, the torque on the yoyo will be provided by the tension in the string
(otherwise it would not spin). On the other hand, if we pay attention only to the center of mass (CM) of the yoyo, its motion is purely linear downward motion. The net force on the CM is gravity minus tension. The equations become:
𝑀𝑔 − 𝑇 = 𝑀𝑎 [linear: CM]
𝑇𝑅 = 𝐼(𝑎 𝑅) [rotational: rim of yoyo]
We want to find I . Isolating it from the system of equations above, and eliminating the unknown T, we get:
𝐼 =
𝑀(𝑔 − 𝑎)𝑅 2 𝑎
Our experiment today is about comparing different methods to determine I .
[ I from motion]
PROCEDURE
Part 1 Simple disk motion
1.
In LoggerPro, insert the movie file <YoyoDisk.mov>.
2.
Enlarge the movie window.
3.
Calibrate the scale of the movie.
4.
Collect points for the center of mass of the falling disk (red dot).
5.
Click on the graph window to analyze the physics behind the motion.

6.
Determine I from the motion of the disk and compare it with the determination of I from geometry. If you keep the mass in grams (do not convert to kg), then you can keep three decimal places in all calculations in order to obtain a good comparison of values. Apply this throughout the entire lab.
7.
Make a Data Table and an Analysis Table. Keep track of initial uncertainties. Error propagation will not be required for this lab.
Part 2 Disk plus weight
A small weight is connected through a light string to a compound wheel system (refer to model in classroom).
The weight is allowed to fall and spin the wheels in the process. A photogate collected data of the rotation of the wheels, which is displayed in a graph angular velocity vs. time . The angular velocity of the wheels was recorded during the fall of the weight, and for some time after the weight had landed on the floor.
Density of nylon:
 ny
= 1300 kg/m
3
Density of aluminum:
 al
= 2700 kg/m
3
4.0 cm
1.0 cm
1.5 cm
NYLON
ALUMINUM
12.5 cm mass of weight: m = 50.0 g
The linear and rotational forms of Newton’s 2 nd
Law will look the same as before, except that m = mass of hanging weight; r is the radius of the nylon wheel connected to the string; and I sys
is the combined moment of inertia of both wheels. In terms of angular acceleration

, instead of linear acceleration a, the equation for I sys becomes:
𝐼 𝑠𝑦𝑠
= 𝑚(𝑔 − 𝛼𝑟)𝑟 𝛼
[no friction]
What if there was significant friction in the wheels’ axle of rotation? Then friction would produce a torque (
 f
) opposing the torque from the tension in the string. The rotational form of Newton’s 2 nd
Law is modified to account for friction:
𝑇𝑅 − 𝜏 𝑓
The torque done by friction can be written as: 𝜏 𝑓
= 𝐼
= 𝐼 𝑠𝑦𝑠 𝛼 𝑓 𝑠𝑦𝑠
(𝑎 𝑅)
In that case, the adjusted determination of moment of inertia becomes:
𝐼 𝑠𝑦𝑠
= 𝑚(𝑔 − 𝛼𝑟)𝑟 𝛼 + 𝛼 𝑓
In this part of the lab, we will compare several different ways to determine I sys
.
[with friction]

1.
Make your own Data and Analysis Tables for this part of the lab. Always show all calculations in detail, and record your values with labels (subscripts) that will distinguish one determination from another.
2.
Calculate the mass for the aluminum and nylon wheels (M al
, M ny
) using the definition of density 𝜌 =
.
3.
Using the geometric formula for moment of inertia I of a disk, calculate a value for the combined wheel system I sys
= I al
+ I ny
. This will be our control value for future comparison.
4.
Open the file <CompWheels.cmbl>.
5.
Remember that the data was collected while the weight was falling, and for some time after that. When the weight hit the ground, the wheels kept spinning. There was a clear difference between the two regimes. Identify the portion of the graph that corresponds to the time when the weight was still in the air, falling down.
6.
Determine a value for I sys based on the part of the graph where the weight is falling.
7.
Once the weight landed on the floor, what was the only torque acting on the wheels? Did the wheels’ angular speed increase or decrease in that interval of time?
8.
Measure the angular acceleration provided by friction,
 f
.
9.
Now calculate a new value for I sys
taking friction into account.
10.
Another adjustment we can do in this problem is to add the mass of the light string, which has been ignored so far. Assume m string
= 5 g, and add that to the mass of the falling weight, in order to give us an upper limit determination on the moment of inertia. Calculate I sys
again, with string and friction taken into account. Record it in your table.
11.
Compare all values and discuss which determination of I sys
is closest to the control value according to your calculations. Give an explanation.
Submit a lab REPORT. There is no need to propagate errors this time, but you need to discuss the sources of errors and estimate them in the conclusions.