Experiment 111-6Moment of Inertia Introduction The rotational equivalent of Newton’s Second Law F = ma is τ = I α in which τ is the torque, I is the moment of inertia, and α is the angular acceleration. In the same way that mass is a measure of inertia (or resistance to acceleration) the moment of inertia is a measure of resistance to rotation. However, mass is an intrinsic property — you can change the shape and/or orientation of an object without changing its mass. In contrast, the moment of inertia depends on both the mass and the geometry of the object as well as the position of its rotation axis. For example, it’s easier to spin a metal rod like an axle than it is to twirl it like a baton. In the apparatus illustrated in Figure 6.1, a disc of mass M and radius R is mounted on an axle that spins in a bearing. The cord wrapped around the axle exerts a torque τ = Tr (6.1) due to the tension T acting at the radius r. When the load mass m is released, it accelerates downward with a = 2∆ s t 2 , (6.2) where t is the time for the load mass to fall from rest through a distance ∆s. This causes the axle/disc system to rotate with an angular acceleration given by α=a r. (6.3) The moment of inertia of the axle/disc system is given by I =τ α = Tr 2 a or, since T = m ( g − a ) , g I = mr 2 − 1 . a (6.4) Figure 6.1 Moment of Inertia Apparatus 23 Then find the average time as well as the estimate of error in the mean. Record the data in a table like Table 6. Calculate the moment of inertia I via equation (6. Record the time t and the distance ∆s through which the mass has fallen. The error calculations can be simplified by assuming that the uncertainty in the load mass is negligible. 6. Calculate the downward acceleration of the load mass using equation (6. 4.Moment of Inertia Experiment 111-6 Purpose The primary objective of this experiment is to find a relationship between the moment of inertia and the radius for discs of the same mass. Axle diameter: Use a vernier caliper to measure the diameter of the axle when it is bare db . Then you only have to consider the uncertainties in r and a. Mount the disc on the end of the axle. The total accelerating mass. Wind the cord around the axle in a single layer enough to raise the accelerating mass to the top of the half-metre stick. A secondary objective is to develop a general equation relating the moment of inertia and radius for discs of any mass. should be m = 150 g.1). • Place the apparatus on the protective mat to avoid damaging the bench top.1.4). (see Figure 6. You’ll need five tables. 3. including the mass holder. Calcu- For each disc: Starting with the largest (disc 1) follow the steps outlined below. 24 . which is large enough to minimize the effects of friction but not so large as to make the times too short to measure accurately. D (in m) with a metre stick. Procedure • All the measurements are to be entered directly into your lab record book. Use a stopwatch to measure the time (in s) it takes for the mass to fall to the bottom end of the half-metre stick. 1. 2. Uncertainties are to be estimated for all measurements and errors found for all calculated quantities. Measure the mass M (in kg) with a digital balance and the diameter. Assume that the major source of error is in the timing and that the uncertainty in ∆s is negligible. Do a total of six timing trials for each disc. late the effective radius r = ( db + d w ) 4 (in m) at which the tension in the cord acts.2). 5. Record the total accelerating mass m. Wind the cord around it in a single layer and then measure the “wrapped” diameter d w . and the power n. what should n be? Once you have established what the power ought to be. Now write the equation that relates the moment of inertia to the disc radius in terms of the slope of the line k. M = ____________________ Dimensional Analysis: Work out the units for the moment of inertia. Given these considerations. plot a graph of I versus R n and determine the slope and intercept of the best-fit line. to what power should the disc radius be raised in the relationship between disc radius and moment of inertia? That is. it should be apparent that k is somehow related to the mass M of the discs. I (units?) mr 2 = ____________________ Average Disc Mass. R = ________ m Acceleration.2 and enter the relevant data and calculated results. if I ∝ R n . Is k a simple fraction. Table 6. such as 1/2.Moment of Inertia Table 6. From the units of the slope in the equation you’ve obtained. of the 25 . or 1/4. 1/3. In the introduction to the experiment it was pointed out that the moment of inertia depends on the mass and the geometry of the object. It should have the form I = kR n + I 0 . the intercept. a = ________ m/s2 Time trial 1 2 3 Moment of inertia. I = ___________________ 4 5 6 Average Standard deviation Error estimate in the mean Experiment 111-6 Falling time t (s) t σ δt = α After all the data have been acquired: Construct a summary table like the one illustrated in Table 6. M = ________ kg Disc Radius. 2 (m ) R? ? Acceleration a (m/s2) Disc Moment of Inertia.1 Data & Results for Disc#___ Disc Mass.2 Summary of Results Disc radius R (m) Accelerating mass × (axle radius) . When a figure skater does a spin and then pulls in his or her arms. Why is this? 26 . How might it have affected the results you obtained? The mass of the axle and bearing is about 225 g. Explain why. the spin rate increases. Some Points for Discussion Friction in the axle bearing plays a role in the experiment. but its moment of inertia is very small. which is roughly half that of the discs.Moment of Inertia Experiment 111-6 average disc mass? What is the physical significance of the intercept I 0 — what does it represent in the experiment? Once you have determined the physical interpretation of the slope and the intercept. write a more general equation in terms of the disc mass and the intercept I 0 .