Experiment 9: Moments of Inertia

Home , Acceleration, Angular acceleration, Moment of inertia, Torque

Figure 9.1: Beck’s Inertia Thing with masses

EQUIPMENT

Beck’s Inertia Thing Vernier Caliper 30cm Ruler Paper Clips Mass Hanger 50g Mass Meter Stick Stopwatch

49 50 Experiment 9: Moments of Inertia

Advance Reading so the tension is Text: Torque, Rotational Motion, Moment of Inertia. T = m(g − a) (9.4) Objective The rotational apparatus has an original moment of inertia I0 with no additional masses added. When To determine the moment of inertia of a rotating sys- additional masses are added, it has a new moment of tem, alter the system, and accurately predict the new inertia Inew. The added masses eﬀectively behave as moment of inertia . point masses. The Moment of Inertia for a point mass 2 is Ip = MR , where M is the mass and R is the radius Theory from the point about which the mass rotates. Thus, Moment of Inertia (I) can be understood as the ro- the relationship between I0 and Inew is given by tational analog of mass. Torque (τ) and angular acceleration (α) are the rotational analogs of force and 2 2 I = I0 + I 1 + I 2 + ... = I0 + M1R + M2R + ... acceleration, respectively. Thus, in rotational motion, new p p 1 2 (9.5) Newton’s Second Law: where M is an added mass and R is the distance of this mass from the center of the wheel (i.e. from the F = ma (9.1) axis of rotation). So, if multiple masses are added at becomes: the same radius, we have

τ = Iα. (9.2) 2 Inew = I0 +ΣIp = I0 + (ΣM)R (9.6) An object experiencing constant angular acceleration In comparing this to Eq. 9.1, we consider that all must be under the inﬂuence of a constant torque masses, along with the disk, experience the same an- (much like constant linear acceleration implies con- gular acceleration. If we were looking for the Force stant force). By applying a known torque to a rigid on a system of connected masses all experiencing the body, measuring the angular acceleration, and using same acceleration, we would simply sum the masses the relationship τ = Iα, the moment of inertia can be and multiply by acceleration (i.e. a stack of boxes be- determined. ing pushed from the bottom). Similarly, when looking In this experiment, a torque is applied to the rota- for the Torque on a system, we must sum the moments tional apparatus by a string which is wrapped around of inertia and multiply by angular acceleration. the axle of the apparatus. The tension T is supplied by a hanging mass and found using Newton’s second law.

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)

52 Experiment 9: Moments of Inertia

PROCEDURE PART 2: Moment of Inertia of apparatus with additional masses. PART 1: Moment of Inertia of apparatus with no additional masses. 12. Measure the distance from the center of the disk to the outer set of tapped holes (Where you will attach 1. Using the vernier caliper, measure the diameter of the three large masses). the axle around which the string wraps. Calculate the radius of the axle. 13. Attach the three masses to the disk. Using the mass 2. Wrap the string around the axle and attach enough stamped on the top/side of the masses, calculate the mass to the string to cause the apparatus to rotate new moment of inertia, Inew, for the system. very slowly. The angular acceleration of the disk 14. Repeat Step 1 through Step 10 for the altered sys- should be nearly zero. Record this mass and use it tem. Calculate the percent difference between the to calculate the frictional torque. experimental value and the calculated value. 3. Holding the disk, place an additional 50 grams (mass hangers are 50 grams) on the string. Measure Questions the distance from the bottom of the mass hanger to 1. What are the units for Torque, Moment of Iner- the floor. tia, and Angular Acceleration? Show all work. 4. Release the disk, be sure not to impart an initial angular velocity. Using the stopwatch, measure the 2. If the Torque applied to a rigid body is doubled, time until the mass hanger reaches the floor. what happens to the Moment of Inertia? 5. Repeat Step 3 five times. Record the times in a 3. Compare friction compensation in this experiment table and calculate the average time. to friction compensation in Newton’s Second Law. 6. Using the average time, calculate the linear acceleration of the masses. 7. Calculate the angular acceleration of the disk using a α = r . Refer to Step 1 for r. 8. Calculate the tension on the string, Eq. 9.4. Be sure to use the total hanging mass. 9. The applied torque on the spinning disk is provided by the tension of the string. Use the values from Step 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 the mass 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 of inertia I0.