A block of mass m is attached with a massless spring of force constant k. The block is placed over a rough inclined surface for which the coefficient of friction is `mu=(3)/(4)`. The minimum value of M required to move the block up the plane is (Neglect mass of string, mass of pulley and friction in pulley)

Options

A. (a) `3/5m`
B. (b) `4/5m`
C. (c) `6/5m`
D. (d) `3/2m`

Answer

C

Explanation

As long as the block of mass m remains staionary, the block of mass M released from rest let it comes down by distance x, applying conservation of mechanical energy
`DeltaK+DeltaU=0implies0+(DeltaU_(gravity)+DeltaU_(spri ng))=0`
`(-Mgx+1/2kx^2)=0impliesx=(2Mg)/(k)`
Thus at the maximum extension in spring is
`kx=2Mg` ...(1)
For block of mass m to just move up the incline
`kx=mg sin theta+mu mg cos theta` ...(2)
`2Mg=mgxx3/5+3/4mgxx4/5`
or `M=3/5m`