Question

4) Figures 4A (side view) and 4B (overhead view) illustrates a uniform solid cylinder having mass M and radius R. The cylinder is positioned on a horizontal floor having sufficient friction to ensure that the cylinder can roll without slipping. The cylinder includes a mass-less yoke that is fixed to the symmetric axis of the cylinder and acts as a rolling friction-less pivot for the cylinder. An ideal spring having spring constant K is attached to the yoke at one end, and a fixed wall at the other end such that the spring is aligned with the rolling axis of the cylinder. An external agent rolls the cylinder away from the wall so as to stretch the spring. The cylinder is released from rest and the force from the spring acting on the yoke causes the cylinder to roll without slipping along the floor. a) Show that the center of mass of the cylinder undergoes simple harmonic motion; b) Determine an expression (K, M) for the period of oscillation. Hint: Construct an expression for the mechanical energy of the system and utilize the work-energy theorem to setup a familiar SHM equation of motion, keeping in mind that the derivative of a constant equals zero. Figure 4B Figure 4A Solid cylinder Yoke id cylinder Spring Yoke2 Spring Side view Top view

Answer 1

-a KT 2 K 1