Routh's rule is a mnemonic formula for finding the moment of inertia of a symmetrical solid. The rule works for a circular or elliptical cylinder rotated about the cylinder axis, or for a circular or elliptical disk about any of the axes of symmetry.
Routh's rule states that the moment of inertia I of a body of mass M about an axis is given by the formula in the picture.
where the denominator is 3 for a rectangular body, 4 for a elliptical body, or 5 for an ellipsoidal body.
The moment of inertia of a solid uniform right circular cylinder of radius R rotated about the axis of the cylinder. Then the numerator in Routh's rule is R
2 + R
2 = 2R
2, while the denominator is 4 (a circle is a special case of ellipse), so,
I = M(2R
2)/4 = MR
2/2
