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 $90^{\circ}$