A hollow cylinder (radius $R$) is rolling against the wall at angular speed $\omega$. The coefficient of friction between the cylinder and the wall(ground) is $\mu$. After how many rotations the cylinder will stop rotating?

So I figured I need to find the time taken for cylinder to stop moving, and that would be $$ \beta=-\omega/t => t = -\omega/\beta $$ Where $\beta$ is angular acceleration, which is known from torques: $$ 2*F_f*R = I*\beta $$ That's where I got stuck... How do I know the friction? I'm familiar with such equation: $$ F_f = \mu*F_n $$ How do I find the normal force? Does it have anything to do with centripetal force?

  • $\begingroup$ The first thing to point is that the friction force at the wall and at the ground are generally not the same. $\endgroup$ – Ali Jul 24 '13 at 10:56

Hint: Look at the following diagram, and then solve the equations:

enter image description here

Or just notice that $F_w$ and $F_f$ do not depend on $\omega$, then use the Work-Energy principle.

Solution: $$\theta = \frac{R \omega^2 \left( 1+ \mu^2\right)}{2 g \mu (1+\mu)}$$

step by step solution:

Horizontal forces:$$F_f-N_w=0$$ Vertical forces: $$F_w+N_f-mg=0$$ We also know: $$F_f=\mu N_f$$ and $$F_w=\mu N_w \\$$ Solving them we find: $$ \\ F_f=\frac{mg\mu}{1+\mu^2} \\ F_w=\frac{mg\mu^2}{1+\mu^2} \\ \Rightarrow \left(F_f+F_w \right)R \theta = \frac{1}{2}I \omega^2 \\ \Rightarrow \theta = \frac{R \omega^2 \left( 1+ \mu^2\right)}{2 g \mu (1+\mu)} $$

  • $\begingroup$ So as I understand F_f will be equal to µmg (right?), but how do you find F_w? Also, using the work-energy principle, do we find that (F_w+F_f)*α = (I*w^2)/2 ? $\endgroup$ – eksponente Jul 24 '13 at 12:04
  • $\begingroup$ No(did you see the $F_w$ pointing up?), and no(there is an $R$ missing in your argument)! $\endgroup$ – Ali Jul 24 '13 at 12:08
  • $\begingroup$ We have some unknowns, and some relations between them. We have to solve the equations to find the unknowns. If it is still ambiguous, I may include the whole step by step solution in my answer; but that's not the point of "homework" questions. $\endgroup$ – Ali Jul 24 '13 at 12:11
  • $\begingroup$ I do, I do want to understand it!:) I got the part with the missing $R$, I just can't seem to find the relations with frictions... $\endgroup$ – eksponente Jul 24 '13 at 12:40
  • $\begingroup$ A big big thanks! Turns out I forgot the basics... $\endgroup$ – eksponente Jul 24 '13 at 13:10

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