Question

A hollow sphere is released from the top of an inclined plane of inclination $\theta$. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding ? (b) Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part (a).

Answer 1

A hollow sphere is released from a top of an inclined plane a of inclination $\theta$. (a) To prevent sliding, the body will make only perfect rolling. In this condition, mgl sin $\theta$ - f = ma ...(1)

$\left(b\right)\left(\frac{1}{5}\right)tan\theta \left(mgcos\theta \right)R=\frac{2}{3}m{R}^2\alpha$

$\Rightarrow \alpha =\frac{3}{10}\left(\frac{gsin\theta }{R}\right)$

$ac=gsin\theta -\left(\frac{g}{5}\right)sin\theta$

$=\left(\frac{4}{5}\right)gsin\theta$

$\Rightarrow t^2=\frac{2s}{ac}$

$=2l\left(4g\frac{sin\theta }{5}\right)\left(\frac{5}{2gsin\theta }\right)$

$\therefore \text{Again,}\omega =\alpha t$

$\text{K.E.}=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

$=\frac{1}{2}m\left(2as\right)+\frac{1}{2}l\left({\alpha }^2{t}^2\right)$

$=\frac{1}{2}m\left(4g\frac{sin\theta }{5}\right)\times 2\times l+\frac{1}{2}\times \frac{2}{3}m{R}^2\times \frac{9}{100}$

$=\left(\frac{si{n}^2\theta }{R}\right)\times \left(\frac{5L}{2gsin\theta }\right)$

$=4mgl\frac{sin\theta }{5}+3mgl\frac{sin\theta }{40}$

$=\frac{7}{8}mglsin\theta$