- #1
Hernaner28
- 263
- 0
Homework Statement
It's a cilinder of mass M and radius R rolling without slipping and I'm asked to find the maximum value of the static friction coefficient for the cilinder to roll without slipping.
Homework Equations
Non-slip conditions:
[tex] \displaystyle v=\omega R[/tex]
[tex] \displaystyle a=\alpha R[/tex]
Torque:
[tex] \displaystyle \tau =I\alpha [/tex]
The Attempt at a Solution
Net force in axis X (my axis is along with the inclined plane):
[tex] \displaystyle -mg\sin \theta -{{\mu }_{s}}N=ma[/tex]
Equation in axis y will give me the normal force:
[tex] \displaystyle N=mg\cos \theta [/tex]
So:
[tex] \displaystyle -mg\sin \theta -{{\mu }_{s}}mg\cos \theta =ma[/tex]
Solving for u:
[tex] \displaystyle {{\mu }_{s}}=\frac{-g\sin \theta -a}{g\cos \theta }=\frac{-g\sin \theta -\alpha R}{g\cos \theta }[/tex]
So I need to find out the angular aceleration:
[tex] \displaystyle \tau =I\alpha [/tex]
[tex] \displaystyle -{{\mu }_{s}}mg\cos \theta \cdot R=\frac{1}{2}m{{R}^{2}}\cdot \alpha [/tex]
[tex] \displaystyle \alpha =\frac{-2g{{\mu }_{s}}\cos \theta }{R}[/tex]
Replacing alpha in the equation I get that:
[tex] \displaystyle {{\mu }_{s}}\le \tan \theta [/tex]
But the option marked as correct is:
[tex] \displaystyle {{\mu }_{s}}\ge \frac{\tan \theta }{3}[/tex]
any help?
Thanks!