Question

A uniform cylinder of radius r is rotating about its axis at an angular velocity ${\omega }_0$ . It is now placed into a corner as shown in the figure. The coefficient of friction between the wall and the cylinder as well as the ground and the cylinder is $\mu$ . If the number of turns n, the cylinder completes before it stops is given by
$n=\frac{{\omega }_0^{2}r(1+{\mu }^2)}{x\pi \mu g(1+\mu )}$. Find x. ___

Answer 1

Since the cylinder has no motion along x-axis as well as long y-axis
$E{F}_x=0$
$\Rightarrow N_2-\mu N_1=0$
$N_2=\mu N_1\left(1\right)$
$Also\int F_y=0$
$\mu N_2+N_1-mg=0\Rightarrow ({\mu }^2+1)N_1=mg$
$\left[from\right(1)putting{N}_2=\mu N_1]$
$\therefore N_1=\left[\frac{mg}{{\mu }^2+1}\right]and{N}_2=\frac{\mu mg}{{\mu }^2+1}\left(2\right)$
Now, angular retardation produced
$\propto =\frac{\mu N_{2}r+\mu N_{1}r}{l_0}=\frac{\mu r(N_2+N_2)}{l_0}=\frac{2\mu }{mr}(\mu +1)\frac{mg}{{\mu }^2+1}\left(3\right)$
$\therefore$ no. of turns completed by cylinder before it comes to rest,
$n=\frac{\theta }{2}\pi =\frac{{\omega }_0^{2}r({\mu }^2+1)}{8\pi \mu g(\mu +1)}$