A string of length $3l$ is connected to a fixed cylinder whose top view is shown in figure. The string is initially slack. The other end of the string (connected to a marble) is moving at a constant velocity of $10m/s$ as shown. The string will get stretched at some instant and impulsive tension occurs in the string. If hinge is exerting a force of $40000N$ for $0.25ms$ on the cylinder to bear up the impact of impulsive tension, then mark the correct statements. (Take string to be light, breaking tension of the string is $2\times {10}^{5}N)$
 

 

The impulse exerted by the string on the cylinder is equal and opposite to the impulse exerted by  the cylinder on the string.

$\left|\overrightarrow{J}\right|=4000\times 0.25\times {10}^{-3}=10Ns$ along the string.

For marble, $J=0-(-2\times 10\cos \alpha )$

As final velocity of marble along the string is zero

Putting $J=10Ns,$ we get $\alpha ={60}^0.$

The string wraps up around the cylinder and marble moves in a circular path of varying radius  with speed $10\sin \alpha =5\sqrt{3}m/s.$ The speed remains constant as only the tension force is present  which is perpendicular to the velocity of the marble.For string to break, the impulsive tension gas to be $\ge 2\times {10}^{5}N$

$\therefore 40000\times 0.25\times {10}^{-3}\ge 2\times {10}^5\times \Delta t\Rightarrow t\le 0.05ms$