Lab 6. Rotational Motion Experiment Type: Open-Ended

Overview In this experiment, students will determine the moment of inertia (or called rotational inertia) of an object about an axis of rotation using two methods: (1) calculate the moment of inertia from the mass and geometric dimensions, (2) determine the moment of inertia from the equation, net "r = I#r , by measuring the torque and the

angular acceleration.

Objectives On completion of this experim!e nt, students should have learned how to: 1) measure the angular acceleration of a rotating object 2) determine the moment of inertia of a rotating object

Review of concepts:

Moment of Inertia (or rotational inertia)

r Analogous to net F = mar for linear motion, rotational motion is governed by r net r = Ir , where r = r F = torque, I = rotational inertia, r = angular acceleration.

Mass (or linear inertia) is the object’s “resistance” to change in its linear motion.

The moment of inertia is the object's resistance to change in its rotational motion.

Unlike mass, the moment of inertia is not a single number for a particular object. The moment of inertia of an object depends on the axis of rotation and the mass distribution of the object. Which do you think has the greater moment of inertia, a ruler rotated about an axis going down its length or a ruler rotated about an axis which goes perpendicular through the face?

In our experiment, we would like to measure the moment of inertia of an object that is composed of three cylinders of metal.

Figure 0-1 We wish to find the moment of inertia of this sandwich of cylinders

Method 1: You can find the moment of inertia directly by measuring the mass and radius of each cylinder, since you know that the moment of inertia is Eq. (7-1).

I = ! r 2dm (7-1)

In this equation, r is the (radial) distance (in cylindrical coordinates) from the axis of rotation and dm is an infinitesimally small piece of mass at that distance. Solving equation (7-1) for the sandwich of cylinders, the moment of inertia is

1 1 1 I = m r 2 + m r 2 + m r 2 (7-2) 2 1 1 2 2 2 2 3 3

In Eq. (7-2) the subscripts 1, 2, and 3 indicate the individual cylinders as labeled in Fig. 7-1. The r1, r2, and r3 variables are the radii of the cylinders, and the m1, m2, and m3 variables are the masses of the individual cylinders. Unfortunately, the cylinders are stuck together, and there’s no way to measure each individual mass. However, we know that the cylinders are made out of the same material and we measure the total mass. Can we use these information to determine the mass of the individual cylinders? What assumptions are made in your technique for determining the masses?

Method 2: “Newton’s Second Law” for rotational motion:

net "r = I#r (7-4)

relates the net torque, "r , to the product of the moment of inertia, I, and the angular r acceleration, " . We can measure the angular acceleration and we can measure the applied force and the “moment ar!m ” to determine the torque applied, so we can solve for the moment of inertia. The torque , τ , will be provided by hanging a mass, ! applied mh, on the cylinder sandwich, as shown in Fig. 7-2. !

Figure 0-2 A string will be wrapped around one of the cylinders. A hanging mass will be attached to the string to apply a known force to the cylinder sandwich

Of course, friction is also present, so we need to count the torque due to friction, τfriction. which acts in the opposite direction of the applied torque. Thus, the net torque is:

" applied # " friction = I$ (7-5)

Since we do not know how large is the frictional force, if we only do one measurement of the applied torque and the angular acceleration, we still cannot solve for the moment of inertia. H!o wever, if we assume that frictional torque is constant (independent of the magnitude of applied torque), then Eq. (7-5) is a linear equation when plotting applied torque vs, angular acceleration. By varying the applied torque (either changing the applied force or the “moment arm”) and measuring the corresponding angular acceleration, we should find a line whose slope is the moment of inertia and whose “y- intercept” is the torque due to friction. What if " friction is not a constant but linear proportional to the applied torque, now what would be the “y- intercept” of the applied torque vs. angular acceleration graph? Suppose the frictional torque occurs mainly at the contact points of the rotating axle with the end plates. How can you vary the applied torque without varying t!he frictional torque? (Remember that the applied torque can be varied by either changing the applied force or the “moment arm”. Which method of changing the applied torque will NOT change the amount of friction at the contact point?)

This method requires you to measure the angular acceleration for different values of applied torque. You’ll have to figure that out on your own, but here’s a hint: is there a relation between the angular acceleration of the cylinders and the linear acceleration of the hanging mass (assuming the string is not slipping).

In the first week of this lab, concentrate on developing a procedure that will allow you to determine the moment of inertia for the system using these two methods. You should try to complete the data collection this week, but it’s okay if that doesn’t happen. You should at least have preliminary data to test if your ideas work. If they do not work, modify your approach.

In the second week you should refine your experiment. How can you get more accurate timing? Did you select the hanging masses that worked well in the experiment? What assumptions did you make that can be eliminated? You should think about all these questions (and more!) in the week between the two lab periods.