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A uniform sphere of mass m and radius R is placed on a rough horizontal surface. The sphere is struck horizontally at a height h from the floor.

Match the following

a) $h=\frac{R}{2}$
i) sphere rolls without slipping with a constant velocity and no loss of energy

b) $h=R$ ii) sphere spins clockwise, loses energy by friction

c) $h=\frac{3R}{2}$ iii) sphere spins anti-clockwise, loses energy by friction

d) $h=\frac{7R}{5}$ iv) sphere has only a translational motion, loses energy by friction

Answers (1)

(a)-(iii), (c)-(ii), (d) -(i), (b)-(iv)

The sphere rolls without slipping when $\omega =\frac{v}{r}$

Let the velocity of the sphere after applying F be v

Then by the law of conservation of angular momentum

$mv(h-R)=I\; \omega$

$mv(h-R)=\frac{2}{5}mR^{2}\frac{v}{R}$

$h-R=\frac{2}{5}R$

$h=\frac{2}{5}R+R=\frac{7}{5}R$

Therefore, the sphere rolls without slipping with a constant velocity and no loss of energy. Thus (d) -(i)

Torque due to force $F=\tau =(h-R)\times F$

If $\tau =0$ , $h-R=0$ and thus $h=R$

In this case, the sphere will only have a translation motion and slip against the force of friction. Thus (b)-(iv)

For clockwise rotation of sphere $\tau >0$

$(h-R)\times\; F>0$

Or $h>R$ , Thus (c) - (ii)

For anti-clockwise rotation $\tau <0$

$(h-R)\times\; F<0$

$h<R,$ Thus (a) - (iii)

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