Goals:
- Use torque and angular acceleration to find the moment of inertia for a rotating object.
- Use rotational energy to find the moment of inertia for a rotating object.
- Use geometric formulas to compute the moment of inertia of a rotating object.
Angular Position, Velocity and Acceleration
For a rotating object, we describe the position using an angle, angular velocity,


, (measured in radians) from some starting position. We define
(radians per second) to be the rate of change in angle, so we can write
 avg

 
 t
and the small time limit

(radians per second per second) to be gives the instantaneous angular velocity. Finally, we define the angular acceleration, the rate of change in angular velocity, so we can write
 avg



 t and the small time limit gives the instantaneous angular acceleration.
All these quantities are related by rates of change, just like velocity is a rate of change in position and acceleration is a rate of change in velocity. It is not surprising that the formulas describing constant angular acceleration are the same as those we learned for constant linear acceleration earlier in the course:
Formulas for constant angular acceleration:

  
0 0 t


0
 t


2  
2
0

2
 
0

Torque, Moment of Inertia and Angular Acceleration
 avg
1
2
 t
2
1
2
  
0

Earlier in the course Newton’s second law allowed us to relate force to acceleration, and the new formula
 
I
 allows us to relate torque to angular acceleration (
 is the torque on a rotating object, I is the moment of inertia and

is the angular acceleration).
Torque can be thought of as “twisting force” and is given by the formula
  
F sin
     where r is the lever arm from the contact point to the rotation axis.

Computing Moments of Inertia
Moment of inertia can be viewed as “the resistance to angular acceleration”. We have to rely on the formulas in the book in order to compute moments of inertia (these can be found “from scratch” by using integral calculus). One qualitative thing you should always remember is that the moment of inertia gets larger as more mass moves farther from the rotation axis.
Therefore, the moment of inertia of a ring is greater than the moment of inertia for a disk (if the mass is the same). The formulas for a ring and a disk are given below.
Ring: I

MR
2 Disk: I

1
2
MR 2
Note that the disk has a smaller MOI because more mass is located closer to the rotation axis on average!
Rotational Energy
The total kinetic energy contained in a rotating object (from all the little pieces moving at all their different speeds) is given by the formula:
KE rotational

1
2
I

2 , in perfect analogy to the old formula

1
KE mv
2
2 .

Set up your rotating assembly (bicycle hub and support plate) so that it is reasonably level. Place the heavy disk on top and center it as closely as possible (you can tell how centered it is by rotating the assembly). Now wrap 1 meter of string around the small cylinder at the bottom of the assembly (it helps to start by taping the end). It is probably easiest to give yourself about 110 cm. of string and mark the string with a dot at the 1 meter mark. You will unwind the string by pulling on a spring scale, so you need a loop in the end of the string as well. NOTE: the moment of inertia of the hub and support plate assembly is negligible compared to the disk within typical experimental error!
heavy disk bicycle hub spring scale
1. Keep the spring scale tensioned near 500 g as you pull out 1 m of string. Measure the time it takes to pull the whole string out, and measure the final angular velocity of the disk (use 20 rotations to take the angular velocity measurement).
Time to unwind the string: ________ s
Angular velocity of the disk at the end of the experiment: ________ rad/s
2. Now we want to figure out the moment of inertia for the disk by using angular kinematics.
First, compute the torque that was applied to the hub during the experiment:
Torque applied to the hub: ________ Nm
Now, compute the angular acceleration of the assembly by using the appropriate angular kinematics formula(s):
Angular acceleration: ___________ / 2
Finally, use the known torque and angular acceleration to compute the moment of inertia of the disk:
Moment of inertia for the disk: ___________ kg m
2

3. We can repeat the calculation of moment of inertia by using energy instead of torque and angular acceleration.
First, compute the work done as the string was pulled out:
Work done by pulling string: __________ J
The work is entirely pumped into the rotational kinetic energy of the disk. Now you can use the known rotational kinetic energy and the angular velocity to compute the moment of inertia. Show your calculations below.
Moment of inertia for the disk: __________ kg m
2
4. Now compute the moment of inertia for the disk by using geometry and a scale. Show your work below.
Moment of inertia for the disk: ___________
 2
5. Make sure that all three computations of the disk MOI are within about 10% of each other. If one of the measurements is significantly different, there is likely an error in your experimental method!

Now we want to repeat the experiment with a different moment of inertia. Replace your disk with a heavy RING – centered and secured thoroughly with tape. This time, we will use the MOI and the total work to predict final angular velocity.
1. Compute the moment of inertia of the ring by using geometry and a scale (show calculations below):
Moment of inertia for ring: __________

2
2. Now use work/energy to predict the final angular velocity of the assembly when you repeat the experiment with the ring instead of the disk:
 predicted

___________ rad/s
3. Run the experiment, and measure the angular velocity after the string has been pulled out:
 measured

____________ rad/s
4. Compute the percent difference between the predicted and measured angular velocities:
Percent difference: ___________