A lightweight rigid cross consists of two perpendicular massless rods, each of length 0.40 m. Identical point masses of 0.30 kg are fixed at all four ends, forming a symmetric “plus” shape that can rotate in a horizontal plane about a vertical axle through its center. The cross is spun to an initial angular speed of 15 rad/s and then released so that no external torques act on it.
A small solid disk of mass 0.50 kg and radius 0.10 m is smoothly dropped coaxially onto the rotating cross. The disk sticks to the cross without slipping, and the combined system rotates together afterward.
(a) Derive an expression, in terms of the given quantities, for the moment of inertia of the initial cross (without the disk) about the axle.
(b) Calculate the numerical value of the initial angular momentum of the cross about the axle.
(c) Determine the total moment of inertia of the cross–disk system after the disk has adhered.
(d) Using conservation principles, compute the angular speed of the cross–disk system immediately after the disk sticks.
(e) Find the ratio of the system’s final angular momentum to its initial angular momentum and explain physically why this ratio has that value.