Question

A solid, uniform cylinder with mass M and radius R is spinning with an angular velocity W on a thin, frictionless axle that passes along the cylinder axis. You design a simple friction brake to stop the cylinder by pressing the brake against the outer rim with a normal force. The coefficient of kinetic friction between the brake and rim is µ. Find the applied normal force to bring the cylinder to rest after it has turned through N revolution?

Answer 1

$$\theta \tau=-\mu nR\\0.5mR^2\theta \alpha=-\mu nR\\2\theta \alpha=-\omega^2\\\\0.25mR\omega^2=\mu n\\n=\frac{mR\omega^2}{4\mu}$$