Shown in the photograph at the left below is a physical pendulum, with a bicycle wheel mounted at the end of the pendulum arm. The bicycle wheel can be free to rotate about its axis, as seen by clicking the photograph at left below, or it can be tied down, as seen in the photograph at the right, so that it cannot rotate at all.
If the wheel is tied down, so that it cannot rotate about its axis, it oscillates with some period, as seen in a video by clicking your mouse on the photograph at the right above. The pendulum is allowed to make ten oscillations while the clock runs, so the period is the final measurement on the clock divided by ten.
Now suppose that the cord tying the wheel is released, allowing the wheel to rotate about its axis, if it wants to. The pendulum will be pulled to the side as in the case above and again released from rest, but with the wheel free to rotate about its axis.
With the wheel free to rotate, which of the following statements are correct?
- (a) The period of oscillation will be greater with the wheel free.
- (b) The period of oscillation will be less with the wheel free.
- (c) The oscillation will immediately damp out because the wheel will begin to rotate.
- (d) A coupling resonance will occur between pendulum oscillations and wheel rotations.
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The answer is (b): The period of oscillation will be less with the wheel free, as seen in this video (click here) .
This experiment illustrates the "Parallel Axis Theorem," which states that the moment of inertia of a rigid body about an arbitrary axis I is equal to the moment of inertia of its center of mass about the arbitrary axis added to the moment of inertia of the object rotating about its center of mass.
Suppose that the mass of the wheel m is much greater than that of the pendulum, the length of the pendulum is L and the bicycle wheel radius is r.
When the wheel is tied down so that it cannot rotate, according to the parallel axis theorem its moment of inertia is equal to mL2 + mr2.
When the wheel is free to rotate, its moment of inertia, is equal to mL2. Because the wheel does not rotate, this pendulum is equivalent to a pendulum consisting of the mass of the wheel at the radius L. The moment of inertia is therefore mL2.
The former is greater than the latter, so the pendulum will oscillate more slowly when the wheel is tied down and cannot rotate.