Answer to Question #192286 in Mechanics | Relativity for norah

Question

A heavy disk of radius R is placed on an axle through its axis of symmetry perpendicular

to the disk. The disk is given angular velocity! about the axis and then is released. Because of

friction in the axle, the disk slows down and stops rotating after time T. How long will it take for

the same disk but of radius 2R to stop if it is given the same initial angular velocity? Assume the

disks are of uniform density, same thickness, and that the friction in the axle is the same in both

cases.

Answer 1

$I=\frac{mR^2}{2},$

$E=\frac{I\omega ^2}{2}=\frac{mR^2\omega^2}{4},$

$\omega=\varepsilon T,$

$E=\frac{mR^2\varepsilon^2 T^2}{4}\implies$ $T^2 \sim{~}\frac{1}{R^2}\implies T\sim \frac 1R,$

$T_1=\frac{RT}{2R}=0.5T.$