Rolling Cylinder

A cylinder of mass M and radius R has a string wrapped around it, with the string coming off the top of cylinder. The string is pulled to the right with a force F. What is the acceleration if the cylinder rolls without slipping?

  1. a = F/M
  2. a < F/M
  3. a > F/M

To solve this problem, apply DID TASC

Step 1 -- Draw diagram and define coordinates:
Take +x to the right and +z into the screen (corresponding to clockwise rotation).

Step 5 -- Apply Newton's second law and constraints:
Forces | Torques
ΣFx = Ma | Στz = I α
F - fs = Ma | RF + Rfs = ½ MR2α

For rolling without slipping, &alpha = a/R

The torque equation becomes:   RF + Rfs = MR2a/2R  →  F + fs = ½ Ma

The force equation is:      F - fs = Ma

Subtracting the two equations gives:   2fs = - Ma/2,   or   fs = - Ma/4.

Surprise! The minus sign means the friction force points to the right!

Solving the force and torque equations gives:

a = 4F/3M    fs = - F/3