Experiment 9: Moments of Inertia

Figure 9.1: Beck’s Inertia Thing with masses

EQUIPMENT

Beck’s Inertia ThingVernier Caliper30cm RulerPaper ClipsMass Hanger50g MassMeter StickStopwatch

49

50 Experiment 9: Moments of Inertia

Advance Reading

Text: Torque, Rotational Motion, Moment of Inertia.

Objective

To determine the moment of inertia of a rotating sys-tem, alter the system, and accurately predict the newmoment of inertia .

Theory

Moment of Inertia (I) can be understood as the ro-tational analog of mass. Torque (τ) and angular ac-

celeration (α) are the rotational analogs of force andacceleration, respectively. Thus, in rotational motion,Newton’s Second Law:

F = ma (9.1)

becomes:

τ = Iα. (9.2)

An object experiencing constant angular accelerationmust be under the influence of a constant torque(much like constant linear acceleration implies con-stant force). By applying a known torque to a rigidbody, measuring the angular acceleration, and usingthe relationship τ = Iα, the moment of inertia can bedetermined.

In this experiment, a torque is applied to the rota-tional apparatus by a string which is wrapped aroundthe axle of the apparatus. The tension T is suppliedby a hanging mass and found using Newton’s secondlaw.

Figure 9.2: String wrapped around axle.

If we take the downward direction as positive, and ap-ply Newton’s second law, we have:

ΣF = mg − T = ma (9.3)

so the tension is

T = m(g − a) (9.4)

The rotational apparatus has an original moment of in-ertia I0 with no additional masses added. Whenadditional masses are added, it has a new moment ofinertia Inew. The added masses effectively behave aspoint masses. The Moment of Inertia for a point massis Ip = MR2, where M is the mass and R is the radiusfrom the point about which the mass rotates. Thus,the relationship between I0 and Inew is given by

Inew = I0 + Ip1 + Ip2 + ... = I0 +M1R2

1+M2R

2

2+ ...

(9.5)where M is an added mass and R is the distance ofthis mass from the center of the wheel (i.e. from theaxis of rotation). So, if multiple masses are added atthe same radius, we have

Inew = I0 +ΣIp = I0 + (ΣM)R2 (9.6)

In comparing this to Eq. 9.1, we consider that allmasses, along with the disk, experience the same an-

gular acceleration. If we were looking for the Force

on a system of connected masses all experiencing thesame acceleration, we would simply sum the massesand multiply by acceleration (i.e. a stack of boxes be-ing pushed from the bottom). Similarly, when lookingfor the Torque on a system, we must sum the momentsof inertia and multiply by angular acceleration.

52 Experiment 9: Moments of Inertia

PROCEDURE

PART 1: Moment of Inertia of apparatus with

no additional masses.

1. Using the vernier caliper, measure the diameter ofthe axle around which the string wraps. Calculatethe radius of the axle.

2. Wrap the string around the axle and attach enoughmass to the string to cause the apparatus to rotatevery slowly. The angular acceleration of the diskshould be nearly zero. Record this mass and use itto calculate the frictional torque.

3. Holding the disk, place an additional 50 grams(mass hangers are 50 grams) on the string. Measurethe distance from the bottom of the mass hanger tothe floor.

4. Release the disk, be sure not to impart an initial

angular velocity. Using the stopwatch, measure thetime until the mass hanger reaches the floor.

5. Repeat Step 3 five times. Record the times in atable and calculate the average time.

6. Using the average time, calculate the linear acceler-ation of the masses.

7. Calculate the angular acceleration of the disk usingα = a

r. Refer to Step 1 for r.

8. Calculate the tension on the string, Eq. 9.4. Be sureto use the total hanging mass.

9. The applied torque on the spinning disk is providedby the tension of the string. Use the values fromStep 1 and Step 7 to calculate the net torque,which is applied torque minus friction torque.

10. Repeat Step 2 through Step 8 for 100 grams on themass hanger in addition to the mass from Step 1.

11. Using Graphical Analysis, plot the net torque vs.angular acceleration. Be sure to enter the ori-

gin as a data point. Determine the moment ofinertia I0.

PART 2: Moment of Inertia of apparatus

with additional masses.

12. Measure the distance from the center of the disk tothe outer set of tapped holes (Where you will attachthe three large masses).

13. Attach the three masses to the disk. Using the massstamped on the top/side of the masses, calculate thenew moment of inertia, Inew, for the system.

14. Repeat Step 1 through Step 10 for the altered sys-tem. Calculate the percent difference between theexperimental value and the calculated value.

Questions

1. What are the units for Torque, Moment of Iner-

tia, and Angular Acceleration? Show all work.

2. If the Torque applied to a rigid body is doubled,what happens to the Moment of Inertia?

3. Compare friction compensation in this experimentto friction compensation in Newton’s Second Law.