Question

A disc is given velocity V and angular velocity $\omega$ [where V = 2R$\omega$] on a surface with friction as shown in the figure. After what time will the friction on the disc become zero?

• A. $\frac{R\omega }{2g}$
• B. $\frac{R\omega }{3\mu g}$
• C. $\frac{2R\omega }{\mu g}$
• D. $\frac{R\omega }{\mu g}$

Answer 1

The correct option is D $\frac{R\omega }{\mu g}$


Friction will become zero when the disc starts pure rolling. Let the angular velocity of pure rolling be ${\omega }_0$

Using conservation of angular momentum about point of contact axis,
$mVR-\frac{m{R}^2\omega }{2}=\frac{3}{2}m{R}^2{\omega }_{\circ }Initialvelocity,V=2R\omega \Rightarrow \omega ={\omega }_{\circ }Finalvelocity,V_{\circ }=R{\omega }_{\circ }{V}_{\circ }=V-atR\omega =2R\omega -\mu gt(\because F=ma=\mu N=\mu mg\Rightarrow a=\mu g)\Rightarrow t=\frac{R\omega }{\mu g}$