Moment of Inertia and parallel axis theorem

Consider Newton’s second law. It can be thought of as a way to define mass: mass is the proportionality constant that relates the net force acting on an object and the acceleration of that object. Suppose you hit two different masses with the same net force, which one will move the most, the one with the larger mass or the one with the smaller mass?

Tape 2-100 gram masses to a meter stick. Then hold it at one “axis” and tap it. Now hold it at the other “axis”. Try to tap it the same distance from the axis with the same force. How far does it swing now compared to the first case? Everyone should try this, and everyone should watch while their partners try it. Did it swing farther when the mass was far from the axis or close to the axis?

The equivalent of Newton’s second law for rotation is:
    Net Torque = Moment of Inertia * Angular Acceleration.
This can be thought of as a way to define moment of inertia: moment of inertia is the proportionality constant that relates the

    _________________________ acting on an object and the _________________________
of that object. Fill in the blanks above.

If the moment of inertia is the proportionality constant relating torque and angular acceleration, what must its units be in terms of kg, meter, sec?

In which case was the moment of inertia larger? when the masses were close to the axis or when they were far from the axis of rotation?

If you attached more mass to the meter stick, would you expect its moment of inertia, I, to increase or decrease? If you doubled the mass, how would I change?

If you moved the masses farther from the axis of rotation, would you expect its moment of inertia, I, to increase or decrease? If you doubled the distance, how would I change?

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Diagrams of 5 objects (a solid square, a solid cylinder, a hoop, a square outline, and a rod) are shown below. The center is the axis of rotation (into and out of the page) around which they rotate. The objects all have the same mass, and the vertical dimension is the same for all of them.

Without looking at any formulas from the table in the book, rank the objects from largest [ = 1] to smallest [= 5] moment of inertia. Explain how you made your ranking.

Finally, we can find the moment of inertia about points other than the center of mass of an object. In the book, they use the integral definition to derive the Parallel Axis Theorem:
I = I_com + Mh² where h is the distance from the center of mass to the axis of rotation

Find I for the meterstick about one end.

Examples:

1. Consider holding a meter stick horizontally. It is pivoted at one end and released from rest. What will be the angular speed of the meter stick when is vertical.

2. A pulley is made from a solid disk of radius 15 cm and mass 400 grams, and rotates with no friction. A string passes over the pulley and is attached to masses of m1= 200 grams and m2= 300 grams. The system is released from rest. (a) Find the speed of m2 after it falls s = 25 cm. (b) Find the acceleration of m2. (c) Find the tension in each part of the string. N.B. Since the pulley mass is not equal to 0, it is no longer ideal and the tensions in the 2 strings will be different.

This page maintained by Anne G. Young. Last modified 08-Mar-2004 .