A string of length `3l` is connected to a fixed cylinder whose top view is shown in Fig. The string is initially slack. The other end of the string (connected to a marble) is moving at a constant velocity of `10 m//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 `40000 N` for `0.25 ms` 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 `2x10^5N)`

Options

A. The angle made by the velocity of marble with the length of string when it is just stretched is `60^(@)`
B. The marble will move in a circular path of varying radius with constant speed of `5 sqrt(3) m//s`., after the string is taut.
C. To answer above two options, the volume of `theta` must be given.
D. The string will break if impulse duration is less than `0.05 min`.

Answer

A, B, D

Explanation

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

`|vecJ|=40000xx0.25x10^(-3)=10N` s along the string. For marble,
`J=0-(2xx10cosalpha)`
As final velocity of marble along the string is zero
Putting `J=10Ns`, we get `alpha=60^(@)`
The string wraps up around the cylinder and marble moves in a circular path of varying radius with speed `10 sin alpha =5sqrt3 m//s`. The speed remain costasnt as only the tension force is present which is perpedicular to the velocity of the marble.
For string to break, the impulsive tension has to be
`ge2xx10^5N`
`40000x0.25xx10^(-3)ge2xx10^(5)xx/_\t`
`/_\tle0.05ms`