Problem 1.280

A stationary platform P which can rotate freely about a vertical axis (Fig. 1.72) supports a motor \(M\) and a balance weight \(N\). The moment of inertia of the platform with the motor and the balance weight relative to this axis is equal to I. A light frame is fixed to the motor's shaft with a uniform sphere \(A\) rotating freely with an angular velocity \(\omega_{0}\) about a shaft \(B B^{\prime}\) coinciding with the axis \(O O^{\prime} .\) The moment of inertia of the sphere relative to the rotation axis is equal to \(I_{0} .\) Find: (a) the work performed by the motor in turning the shaft \(B B^{\prime}\) through \(90^{\circ}\); through \(180^{\circ}\); (b) the moment of external forces which maintains the axis of the arrangement in the vertical position after the motor turns the shaft \(B B^{\prime}\) through \(90^{\circ}\)