Problem

A crane has a boom A of mass mA and length ℓ that can rotate in the horizontal plane about...

A crane has a boom A of mass mA and length ℓ that can rotate in the horizontal plane about a fixed point O. A trolley B of mass mB is mounted on one side of A such that the mass center of B is always at a distance e from the longitudinal axis of A. The position of B is controlled via a cable and a system of pulleys. Both A and B are initially at rest in the position shown, where d is the initial distance of B from O measured along the longitudinal axis of A. The boom A is free to rotate about O and, for a short time interval 0 ≤ t ≤ tf, B moves with constant acceleration a0 without reaching the end of A. Letting IO be the mass moment of inertia of A, modeling B as a particle, and accounting only for the inertia of A and B, determine the direction of rotation of A and the angle θ swept by A from t = 0 to t = tf. Neglect the mass of the cable and of the pulleys.