what is the parallel axis theorem?

Answer 1

moment of inertia of a body abt an axis parallel to center of mass axis= moment of inertia abt center of mass axis+ mass of the body * (distance of the axis from center of mass axis)^2

Answer 2

The parallel axis theorem has to do with the Moments of Inertia (the M.I.'s) of a mass distribution of total mass M around axes NOT passing through a centroid [C, at (x_c, y_c, z_c)] of the mass distribution.

If the new axis is parallel to the z-axis and passes through the point (X, Y, z_c), then

M.I._z (new) = M.I._z,C + M [(X - x_c)^2 + (Y - y_c)^2].

This result is of course most readily visualized and appreciated if the centroid is itself the origin of coordinates. The M.I. about the displaced axis is then simply the M.I. around a parallel axis through the centroid plus M times the square of the distance between those two axes.

The theorem follows from the fact that, in general, the M.I. around the z-axis is given by Sigma (delta M) [x^2 + y^2], where delta M is an element of mass at the point (x, y, z), and around any other axis parallel to the z-axis through (x_1, y_1, z_1) it is Sigma (delta M) [(x - x_1)^2 + (y - y_1)^2]. When you examine the resulting cross-terms in these sums, the definition of the centroid's position itself makes that cross-product in the sum disappear, resulting in the parallel axis theorem.

This and the x-axis or y-axis equivalents are the results of main interest in elementary questions. If, however, one is interested in the Moment of Inertia TENSOR, which is involved in the Euler Equations for a rotating body, etc., there are corresponding results for the other, "non-diagonal" terms such as Sigma (delta M) [(x - x_c)(y - y_c)].

Live long and prosper.