Experiment 10: Moments of Inertia

Figure 10.1: Beck’s Inertia Thing with masses

EQUIPMENT

Beck’s Inertia ThingVernier Caliper30cm RulerPaper ClipsMass Hanger50g MassMeter StickStopwatch

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2 Experiment 10: 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 (10.1)

becomes:

τ = Iα. (10.2)

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 10.2: 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 (10.3)

so the tension is

T = m(g − a) (10.4)

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 +M1R21 +M2R

22 + ...(10.5)

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 (10.6)

In comparing this to Eq. 10.1, we consider that allmasses, along with the disk, experience the same an-gular acceleration. If we were looking for the Forceon 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.

Prelab 10: Moments of Inertia 3

Name:

1. What is Moment of Inertia? Explain using words only. (20 pts)

2. Draw two force diagrams. One for the mass hanging from the sting, and a second for the disk as viewed fromabove. (20 pts)

3. The rotational apparatus pictured below begins at rest. Upon release, the hanging mass falls 75cm in 10 seconds.The apparatus experiences no friction torque. Calculate the following: (50 pts)

Figure 10.3

a) The acceleration of the falling mass.

b) Angular acceleration of the rotational apparatus.

c) The tension in the string.

d) The torque applied by the falling mass.

e) The net torque applied.

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Name: Section: Date:

Datasheet - Exp 10: Moments of Inertia

Objective

To determine the moment of inertia of a rotating system, alter the system, and accurately predict the new momentof inertia .

Theory

Moment of Inertia (I) can be understood as the rotational analog of mass. Torque (τ) and angular acceleration(α) are the rotational analogs of force and acceleration, respectively.

Thus, in rotational motion, Newton’s Second Law: F = ma becomes: τ = Iα.

PROCEDURE

Part 1: Moment of Inertia of apparatus with noadditional masses.

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

2. Holding the disk, place 50 grams (mass hangers are50 grams) on the string. Measure the distance fromthe bottom of the mass hanger to the floor.

3. Release the disk, be sure not to impart an initialangular velocity. Using the stopwatch, measure thetime until the mass hanger reaches the floor.

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

5. Using the average time, calculate the linear ac-celeration (a) of the masses with the kinematicequation.

6. Calculate the angular acceleration (α) of thedisk using α = a

r . Refer to Step 1 for r.

7. Calculate the tension (T ) on the string, Eq. 10.4.(The total force (ma) is equal to the force of gravityminus tension).

8. The applied torque on the spinning disk is providedby the tension of the string. Use the values fromStep 7 and the radius of the axle to calculate thetorque (τ).

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

10. Using Graphical Analysis, plot the net torque vs.angular acceleration for both situations. Be sureto enter the origin as a data point. Determinethe moment of inertia of the disk, I0.

Part 2: Moment of Inertia of apparatus withadditional masses.

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

12. Attach the three masses to the disk. These massesare 1.35kg each. Calculate the new moment of iner-tia, Inew, for the system (you may treat the addedmasses as point masses, use I0 from part 1).

13. Repeat Step 2 through Step 10 for the altered sys-tem. Calculate the percent difference between theexperimental value and the theoretical (calculated)value.

You should have 2 graphs to include in yourlab report.

When determining independent and de-pendent and variables, consider what youchanged between Part 1 and Part 2.

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Daxle:

raxle:

yfloor:

Rmasses:

Raw data:

Trial t-Part 1(a) (s) t-Part 1(b) (s) t-Part 2(a) (s) t-Part 2(b) (s)

1

2

3

4

5

Average

Calculated Data:

Part 1(a) Part 1(b) Part 2(a) Part 2(b)

a

α

T

τ

(Be sure to include units in your lab report!)

Experimental I0:

(Use this to calculate Theoretical Inew)

Theoretical Inew:

Experimental Inew:

Percent Difference Inew:

Questions

1. What are the units for Torque, Moment of Inertia, 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. Why did you need to calculate acceleration to determine I0? Could you have calculated a theoretical I0 withoutrunning any trials?

4. Were any torques ignored in this experiment? What are they? Do you believe they may have significantly alteredyour results?