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?
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:
|ΣFx = Ma|||||Στz = I α|
|F - fs = Ma|||||RF + Rfs = ½ MR2α|
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