From a disc of radius R , a concentric circular portion of the radius r is cut out so as to leave an annular disc of mass M . The moment of inertia of this annular disc about the axis perpendicular to its plane and passing through its centre of gravity is

Step-by-step solution

Mass per unit area = M π R 2 - r 2 ∴ mass of the whole disc = M · π R 2 π R 2 - r 2 = MR 2 R 2 - r 2 Moment of inertia of the disc of radius r = 1 2 M · π R 2 2 π R 2 - r 2 · r 2 = 1 2 Mr 4 R 2 - r 2 Moment of inertia of the whole disc = 1 2 Mr 2 · r 2 R 2 - r 2 = Mr 4 2 R 2 - r 2 ∴ Moment of inertia of the annular disc = M 2 R 4 - r 4 R 2 - r 2 = M R 2 + r 2 2