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Solve the following :

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 a length I on the incline if the friction coefficient is half the value calculated in part (a).

Solution:

(a) $m g \sin \theta-f f=m a$

$f f=\mu N=\mu m g \cos \theta$

$\tau=I \alpha$ -(ii)

$f f=\left(\frac{-}{5} m R^{2}\right)\left(\frac{a}{R}\right)$

Solving (ii),(ii) and (iii)

$\mu=\frac{2}{5} \tan \theta$

(b) $\mu=\frac{\tan \theta}{5}$

$f f^{\prime}=\mu N=\frac{\tan \theta}{5}(m g \sin \theta)$

$f f^{\prime}=\frac{m g \sin \theta}{5}$

Translatory Motion Equation $m g \sin \theta-f f^{\prime}=m a$

$m g \sin \theta-\frac{m g \sin \theta}{5}=m a$

$m g \sin \theta-f f^{\prime}=m a$

$m g \sin \theta-\frac{m g \sin \theta}{5}=m a$

$a=\frac{4 g \sin \theta}{5}$

$u=0, s=l$

$v^{2}=u^{2}+2 a s$

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

Time to travel,$\quad s=u t+\frac{1}{2} a t^{2}$

$t^{2}=\frac{2 l}{a}$

Rotational Motion Equation $\tau=I \alpha$

ff $R=\left(\frac{2}{3} m R^{2}\right) \alpha$

$\frac{m g \sin \theta}{5}=\frac{2}{3} m R^{2} \alpha$ $\alpha=\frac{3 g \sin \theta}{10 R}$

$\omega=\alpha t$

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

$=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right)(\alpha t)^{2}+\frac{1}{2} m\left(\frac{8 g \sin \theta l}{5}\right)$ K.E $=\frac{7}{8} m g l \sin \theta$

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