question_answer

A billiard ball is stuck with a cue stick horizontally as shown below.                   '

Now, choose the incorrect option.

A) When \[h=\frac{R}{2}\], sphere spins anti-clockwise and looses energy by friction

B)  When \[h=R,\]sphere translates without rotation and looses energy by friction

C) When \[h=\frac{3R}{2}\], sphere rolls without and slip and no loss of energy

D) When \[h=\frac{7R}{2}\], sphere rolls perfectly over the surface

Correct Answer: C

Solution :

Conservation of angular momentum gives, \[m\upsilon \left( h-R \right)=I\omega =\frac{2}{5}m{{R}^{2}}.\frac{\upsilon }{R}\]

\[\Rightarrow m\upsilon \left( h-R \right)=\frac{2}{5}m\upsilon R\] \[\Rightarrow \,\,\,\,\,\,\,\,\,h-R=\frac{2}{5}R\Rightarrow h=\frac{7}{5}R\]

So, option [d] is correct.

Torque due to applied force about centre of mass \[=\tau =F\left( h-R \right)\]

For \[\tau =0,h=R\] and no rotation exists. So, option [b] is correct.

For \[h>R\] sphere spins clockwise and looses energy. So, option [c] is incorrect.

For \[h<R,\tau <0\] and sphere spin anti-clockwise. So, option [a] is correct.