Question

A circular disc of mass m and radius R is rotating on a rough surface having a coefficient of friction $\mu$ with an initial angular velocity $\omega$. Assuming a uniform normal reaction on the entire contact surface, the time after which the disc comes to rest is :

• A. Disc will never stop rolling
• B. $\frac{3\omega R}{4\mu g}$
• C. $\frac{1\omega R}{2\mu g}$
• D. $\frac{\sqrt{3}\omega R}{2\mu g}$

Answer 1

The correct option is A Disc will never stop rolling

3 cases of rolling are shown in the attached figure:

Case (b):

$v=R\omega$

Point of contact is instantaneously at rest with respect to the surface and hence, no friction will act. Disc continues to roll forever.

Case (a):

$v>R\omega$

Friction acts opposite to the velocity as shown but aids angular velocity i.e. $v$ decreases and $\omega$ increases until $v=R\omega$ and again disc continues to roll forever.

Case (c):

$v<R\omega$

Friction acts in the direction of velocity as shown but opposes angular velocity i.e. $v$ increases and $\omega$ decreases until $v=R\omega$ and again disc continues to roll forever.