A block of mass `m` attached with a massless spring of force constant k The block is placed over a rought inclined surface for which the coefficient of friction is `0.5M` is released from rest when the spring was unstretched The minimum to move the value of `M` required to move the block `m` up the plane is (neglect mass of spring and pulley and friction in pulley)

Options

A. `m//2`
B. `m//3`
C. `m//4`
D. None of these

Answer

a

Explanation

Let `m` start moving down and extension produced in spring be x at any time Value of x required to move the block `m`is

`kx = mu mg cos theta + mg sin theta`
`rArr kx = 0.5 mg (4)/(5)+ mg (3)/(5) = mg`
For minimum `M` is will stop after producing extension in the spring x
`Mgx = (1)/(2) kx^(2) rArr Mg = (1)/(2)kx`
`rArr Mg = (1)/(2) mg rArr M = (m)/(2)`