EXAMPLE PROBLEM ON TORQUE AND ANGULAR ACCELERATION: The Unwinding Cable

Question
A cable is wrapped around a uniform, solid cylinder of radius 'R' and mass 'M'. The cylinder rotates about its axis, and the cable unwinds without stretching or pulling. If the cable is pulled with a force of 'F' Newtons, what is its acceleration?

Hints

1. What is the moment of inertia for a uniform, solid cylinder, with the axis through its center?
2. What is the torque exerted?
3. What is the relationship between acceleration of the cable, a, and the angular acceleration, ?

For a uniform, solid cylinder of radius R and mass M, the moment of inertia is:     The torque exerted by a force F is found to be:     = R F since the force is perpendicular to the moment arm. (That is, =90, so sin(90)=1.) We also learned in this section that = I  Solving for , we get:

Figure 1 Diagram of the cable unwinding from a cylinder.