Question

The body lies on an inclined plane with a title angle $\alpha =6^{\circ }$ . The plane rotates uniformaly around the vertical axis . The distance from the body to the axis of rotational is r = 10 cm . the smallest coefficient of friaction at which the body is just The angular velocity of rotational plane is $\mu =1\pi$ The angular velocity of rotational of the incline in nearly

• A. 4.5 rad/s
• B. `1 rad/s
• C. 1.5 rad/s
• D. 2 rad/s

Answer 1

The correct option is C 1.5 rad/s

Answer
$Mr{w}^2\cos \alpha +mg\sin \alpha =\mu N$
$N=ng-Mr{w}^2\sin \alpha$
$Mr{w}^2\cos \alpha +mg\sin \alpha =\mu (mg-Mr{w}^2\sin \alpha )$
$Mr{w}^2+Mg\tan \alpha =\mu (Mg-Mr{w}^2\tan \alpha )$
$r{w}^2+\mu r^2\tan \alpha =\mu g-g\tan \alpha$
$w=\sqrt{\frac{g(\mu -\tan \alpha )}{r\left(\right(1+r\tan \alpha \left)\right)}}\Rightarrow \sqrt{\frac{s\mu }{r}}$
$\Rightarrow \sqrt{\frac{9}{10}\times \frac{1}{\pi }\times 10}$
$\Rightarrow 1.5rad/s$