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A smooth rope of mas m and length L lies...

A smooth rope of mas `m` and length `L` lies in a heap on a smooth horizontal floor, with one end attached to a block of mass `M`. The block is given a sudden kick and instantaneously acquires a horizontal velocity of magnitude `V_(0)` as shown in figure 1. As the block moves to right pulling the rope from heap, the rope being smooth, the heap remains at rest. At the instant block is at a distance `x` from point `P` as shown in figure `-2( P` is a point on the rope which has just started to move at the given instant `)`, choose correct options for next three question.

The magnitude of acceleration of block of mass `M` is

A

`(m^(3))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`

B

`(mM^(2))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`

C

`(m^(4))/(ML)(V_(0)^(2))/((M+(m)/(L)x)^(3))`

D

`(M^(2))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`

Text Solution

Verified by Experts

The correct Answer is:
B
The mass of moving material is `M+(m)/(L)x`
From conservation of momentum `MV_(0)=(M+(m)/(L)x)V`
`:. `Velocity of moving block and moving rope is
`V=(MV_(0))/(M+(m)/(L)x)`
`(B)` The acceleration of moving block is
`a=-v(dv)/(dx)=-(MV_(0))/((M+(m)/(L)x)^(2))xx(m)/(L)(dx)/(dt)`
`=-(m)/(L)(M^(2)V_(0)^(2))/((M+(m)/(L)x)^(3))`
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