1.

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)

Answer»

The angle made by the VELOCITY of marble with the length of string when it is just stretched is`60^(@)`
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.
To answer above two options, the volume of `theta` must be given.
The string will break if impulse duration is less than `0.05 min`.

Solution :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`


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