Rotational Motion: Angular Acceleration

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, ?
Solution
Drawing a diagram will aid us in solving this problem; refer to Figure 1.
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.


Explanation on Torque and Angular Acceleration
Self Test
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