10/26/09 - Moment of Inertia and Combined Rotation and Translation

Since torque is the rotational analog of force, that means that just as an unbalanced force produces a linear acceleration, an unbalanced torque produces a rotational acceleration.

For an unbalanced force, the linear acceleration is inversely proportional to the mass:

For an unbalanced torque, the rotational acceleration is inversely proportional to the moment of inertia:

The moment of inertia I of an object is a combination of the mass of an object and the way it's distributed. Experimentally it's been found that mass close to the point of rotation is easier to accelerate than mass farther away from the point of rotation. For example, rotating a baseball bat from the small end (because most of the mass is far from the small end) is harder than rotating a baseball bat from the large end (where most of the mass is concentrated). The moment of inertia is actually the sum (or integral) of a bunch of slices of a mass times the distance squared they are from the point of rotation: I = Σ r2∆m.

			Example:

			Our textbook has a table of moments of inertia for different common distributions of mass on p. 262.

			Most motions are a combination of translations (movement from one position to another without rotation) and rotations.