Problem # p044

Rigid Body Motion, 2-D
The homogeneous semi-circular disk of mass m and radius R is released from rest in the position shown. The center of mass of the semi-circular disk is at cm and the mass moment of inertia with respect to its own center of mass, , is given by the equation below the figure.
Due to friction at point A, where the disk and the supporting floor touch, no slipping is observed as the disk begins to roll. The ultimate goal of this problem is to determine the necessary coefficient of friction and the initial acceleration.
Please, answer the questions below, they and the included hints will guide you through the problem.

  1. Sketch the Free-Body-Diagram of the disk.

  2. Find the equation for the moment of inertia of the disk with respect to point A.

  3. Determine the angular acceleration of the disk and from that the acceleration of the center of mass. HINT: At the instant shown the disk can be considered to rotate around the fixed point A.

  4. Determine the friction force F and normal force N necessary to achieve the motion. Hint : Look at the acceleration of the center of mass.

  5. What is the lowest value for the coefficient of static friction at which rolling (no-slipping) is observed ?