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. |
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