Parallel-Axis Theorem

What it shows:

One can show that the period of oscillation of an object doesn't change for different suspension points, as long as they're the same distance from the COM. This is consistent with what the parallel-axis theorem tells us about the moment of inertia of the object.

How it works:

The parallel-axis theorm states that if Icm is the moment-of-inertia of an object about an axis through its center-of-mass, then I, the moment of inertia about any axis parallel to that first one is given by I = Icm + md2. Here m is the object's mass and d is the perpendicular distance between the two axes.

The objects in this demonstration are two 12½-cm×52-cm wooden boards with a 10-cm diameter hole in the middle. The COM of the board is located at the center of the hole. The board can be suspended from any point around the edge of the hole as shown in the photograph.

wooden boards

Since all points around the edge of the hole are the same distance from the COM, the parallel-axis theorem tells us that the moments of inertia will all be the same, regardless of the suspension point. Thus the period of oscillation, T = 2∏√(I/mgd), should be the same because I and d are the same, regardless of the suspension point. To demonstrate that that is so, different suspension points are chosen for the two boards, which are then displaced from their respective equilibrium positions and released at the same time. Indeed, the two boards oscillate in sync.

Setting it up:

A length of aluminum angle, C-clamped to the edge of the lecture bench, provides a simple suspension mechanism for the boards. Since we're only comparing the period of oscillations of the two boards, no timer is required unless you wish to quantitatively predict the period.

Comments:

That the period of oscillation will be the same, regardless of how the board is oriented, is not at all obvious making this is a good demonstration and prediction of the parallel-axis theorem.