The moment of inertia I of a body is a measure of how hard it is to get it rotating about some axis. The moment I is to rotation as mass m is to translation. The larger the I, the more work required to get the object spinning, just as the larger the mass m, the more work required to get it moving in a straight line. Alloy rim wheels on bicycles have a lower I than steel rim wheels and so are easier to get spinning, making fast bicycle acceleration easier.

The moment of inertia is always defined with reference to a particular axis of rotation — often a symmetry axis, but it can be any axis, even one that is outside the body. The moment of inertia of a body about a particular axis is defined as:

where the sum is over all parts of the body (labeled with an index i), m_i is the mass of part i, and ri is the distance from part i to the axis of rotation. Performing this sum is easy if the body consists of discrete point masses. But if the body is a continuous object of some arbitrary shape, then performing the sum requires the techniques of integral calculus. In this course, we simply tell you the answer for various shapes. For a disk with an axis through the center of symmetry, the moment of inertia is

Notice that the thickness of the disk does not enter into the expression for I_disk, which depends only on the radius and the total mass.


In this experiment you will measure I for a disk mounted on an axle. The axle can be thought of as a very thick disk and you can use the same expression to compute Idisk and Iaxle. The total I of the disk + axle is the sum of these two.

(2)

In this experiment, you will determine I in two ways. First, you will measure the masses and radii of disk and axle and then compute I from the formula above. Then you will compute I by timing the wheel as it rolls down inclined rails and using the principle of conservation of energy.

Consider the wheel, consisting of disk and axle, rolling down an inclined set of rails after starting from rest at the top, like so:

The total energy at any time is the sum of the translational kinetic energy, the rotational kinetic energy, and the gravitational potential energy.

Here, M is the total mass of disk+axle, v is its translational speed , w is its angular velocity, and h is the height of the center of mass. Initially, the wheel is at rest at height ho, so its initial kinetic energy (both translational and rotational) is zero and its total energy is all potential.

When the wheel reaches the bottom of the rails, h=0, and the energy is all kinetic:

where v_f is the final translational speed and w_f is the final angular speed.

Because the rolling friction is very small, we can assume that the total energy is constant as the disk rolls down the rails, and so the initial energy is equal to the final energy.

Note that here and Eqn (8) below, r is the radius of the axle, NOT the radius of the big disk!

Using equations (6) and (7), one can find I in terms of M, r, g, v_f, and h_o.

(You are asked to derive Eqn (8) in the questions.) You will use this expression to determine I using two different choices of the initial height h_o, thus yielding two new values of I that you can compare with I calculated from Eqn (2).

Part 1. Measurement of I from dimensions and masses of disk and axle

In this experiment, it is a good idea to use centimeters and grams, rather than meters and kilograms for all your measurements. This is because the moment of inertia of our disks turns out to be a very small number in MKS units (roughly 10^-3 kg× m²) and it is a little awkward to work with small numbers. If you use cgs (centimeter-gram-second) units, you must be consistent and always use cgs units, so use g = 979.6 cm/s² (not 9.796m/s²)

Gently slide the axle out of the disk and weigh both separately to find their masses. Measure their diameters to find their radii, r for the axle and R for the disk. At this stage you do not need to know r very precisely, but you will in part 2, so measure the diameter of the axle very carefully three or four times with the calipers. Use the average of your measurements and estimate the uncertainty in r. (If you don't know how to use the calipers, ask your instructor.)

Using Eqn (1), find I_disk and I_axle separately and then compute . (Is I_axle significant, compared to I_disk, or can it be ignored?) In using Eqn (1) to compute I_disk, we are making a small error by ignoring the hole in the center of the disk. Compare the "missing mass" of the hole to the mass of the axle to determine whether this omission is significant.
 

Part 2. Measurement of I using energy conservation.

One end of the rails can be raised and lowered to one of three positions. Place the rails in the lowest position, at which they are approximately level, and then using the adjustable screws in the base, make the rails exactly level. Use the bubble level to get a rough level and then place the wheel on the rails to get a precise level. (If the rails are exactly level, the wheel will not start rolling.)

Raise the movable end of the rails to one of the two upper positions and then fix the two starting blocks, one on each rail, at some convenient position near the top of the track. Make sure that the starting blocks are level with each other, so that the axle can be started resting against both blocks and will roll straight down the track when released. Using the meter stick attached to one rail, record the positions of the sharp tip of the axle in the starting and stopping positions and compute the distance d through which the wheel rolls. Leave the starting blocks fixed from now on, so that the value of d is the same for all timings.

To determine the heights h₁ and h₂ through which the wheel descends, begin by measuring the height changes H₁ and H₂ of the end of the rail when it is raised from the level position to the two upper positions. H₁ and H₂ can be measured quite accurately by measuring the separations of the "notches" that hold up the end of the rail.

Unfortunately, H₁ and H₂ are not the actual heights through which the wheel descends, since it does not roll the whole length of the rail. Instead, the situation is as shown below, where d is the distance traveled by the wheel, while D is the total length of the track (D is measured from the center of the pivot at the bottom to the center of the support at the top.) The two triangles shown are similar triangles, therefore . Use this relation to calculate h₁ and h₂ from measured values of d, D, H₁, and H₂.

Now use the stopwatch to measure the time t₁ for the wheel to roll down the rail when it is in position 1 (height H₁). This is best done with the same person operating the stopwatch and releasing the wheel. Make a few trial runs to determine the best procedure. Have each member of your team measure t₁ a few times and record all values. From your measurements, determine an average value of t₁ and estimate its uncertainty. Repeat this whole procedure for t₂, measured when the rail is in position 2 (height H₂).

Now calculate the wheel's final speed v at the bottom of its travel for each of the two positions. Be careful! The quantity d/t is the average speed of the wheel. In the case of constant acceleration, the average speed is related to the final speed by

Label the two final speeds v₁ and v₂.

Finally, with all your measurements, using Eqn (8), compute I for each of the two positions of the rail. Be careful to display the correct number of significant figures in your final answers. Display your three final values for I: your value from part 1 and your two values from part 2. If there is a large discrepancy among the values, comment on possible sources of experimental error.

Question:

Equation (8) involves g, the acceleration of gravity. This seems to suggest that you would get a different value for I if you conducted the same experiment on the moon, where g is different. But the definition of I (Eqn (8)) does not depend on location. Like the mass m, the moment I of an object is the same on the moon as on the Earth or anywhere else. So how do you explain the presence of g in Eqn (8)?