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:
Forces | | | Torques | ||
---|---|---|---|---|
ΣF_{x} = Ma | | | Στ_{z} = I α | ||
F - f_{s} = Ma | | | RF + Rf_{s} = ½ MR^{2}α |
The torque equation becomes: RF + Rf_{s} = MR^{2}a/2R → F + f_{s} = ½ Ma
The force equation is: F - f_{s} = Ma
Subtracting the two equations gives: 2f_{s} = - Ma/2, or f_{s} = - Ma/4.
Surprise! The minus sign means the friction force points to the right!
Solving the force and torque equations gives:
a = 4F/3M f_{s} = -
F/3