Question

A cylinder of diameter $2R$ and mass $M$ rests on two rough pegs (coefficient of static friction $\mu$), the distance $2kR$ apart. A gradually increasing torque $\tau$ is applied as shown. The cylinder will topple before it slips if

• A. $\mu <\frac{k}{\sqrt{1-k^2}}$
• B. $\mu >\frac{k}{\sqrt{1-k^2}}$
• C. $\mu >\sqrt{1-k^2}$
• D. $\mu <\sqrt{1-k^2}$

Answer 1

The correct option is B $\mu >\frac{k}{\sqrt{1-k^2}}$

If cylinder doesn't slip then,

$\tau /R=f$ (frictional force)

If cylindr topple without slipping;

the torque about point P is;

$\tau >mg\times kR$

$\tau >mgR\sin \theta$

$\frac{\tau }{R}>mg\sin \theta$

Also, $\frac{\tau }{R}=f$

$f>mg\sin \theta$

$\mu mg\cos \theta >mg\sin \theta$

$\tan \theta \frac{kR}{\sqrt{\left(R^2\right)-(kR{)}^2}}$

$\mu >$$\frac{k}{\sqrt{1-\left(k^2\right)}}$

B is correct