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The horizontal force P acts on the rim of the homogeneous cylinder of radius R and weight W. Determine the smallest coefficient of static friction that enables the cylinder to start rolling up the 30◦ incline.


Which FBD would be correct? should it be a vertical normal force with friction force going on left, or should it be the normal force facing perpendicularly the 30 degree include while the friction is along the 30 degree line also. which is the proper fbd?

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I've reopened your question with some cosmetic edits. For future reference, if you want to speed the process along, you can flag your question for moderator attention. –  David Z Feb 28 '12 at 2:07
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1 Answer

It would be with N perpendicular to the incline. The body is just leaving the (horizontal) ground, condition for leaving is $N_{ground}=0$. So the only N force possible would be $\perp$ incline.

Note that if the ball was at rest, there would be two normal forces, and thus two frictions.

The FBD would then be:

  1. $mg=W$ downwards
  2. $N \perp$ incline
  3. $P \parallel$ horizontal
  4. $f=\mu_{s,max}N \parallel$ incline. It's better to just keep it as $f$ till you find out its direction. Over here the direction seems to be downwards $\parallel$ incline, but it's not necessarily so. So keep $f$ as a variable, find it's value, and then equate it to $\mu_{s,max}N$. Whenever dealing with rolling friction, this is the best practice, as the direction of rolling friction need not be in the one that looks intuitively correct.
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