In the diagram shown, the block A and B are of the same mass M and the mass of the block C is `(M_(1))`. Friction is present only under the block A. the whole system is suddenly released from the state of rest. The minimum coefficient of friction on keep the block A in the state of rest is equal to .

Options

A. `( M _(1))/( M )`
B. `( 2 M _(1))/( M )`
C. `( M _(1))/( 2 M )`
D. None of these

Answer

B

Explanation

When extension in spring is maximum, then maximum spring force will act on block A. In this situation required (calculated) friction to keep block A at rest will be minimum
Let x is maximum extension and at maximum extension kinetic energy of whole system is zero.
By COME
Gain in E.P.E. of spring = Loss in G.P. E. of block C
So, `(1)/(2) kx ^(2) = M _(1) gx`
` x = ( 2 m _(1) g )/( k) ...................(i)`

To keep block A at rest
`kx le mu N `
`ku le mu M g`
By putting value of x from equation (i)
`2 M _(1) g le mu M g `
` mu ge (2 M _(1))/( M )`
So` mu _(min) = (2 M _(1))/( M )`