In the adjacent figure, x-axis has been taken down the inclined plane. The coefficient of friction varies with `x` as `mu=kx`, where `k= tan theta`. A block is released at `O`.

The maximum velocity of block will be:

Options

A. `sqrt(g)`
B. `sqrt(g sin theta)`
C. `sqrt(g cos theta)`
D. `sqrt(g tan theta)`

Answer

B

Explanation

At any position `= g(sin theta-k x cos theta)`
`a = g sin theta (1-x) (v dv)/(d x) = g sin theta(1-x)`
`rArr overset(v)underset(0)(int)vdv = overset(x)underset(0)(int)g sin(1-x)dx (v^(2))/(2) = g sin theta (x -(x^(2))/(2))`
`v` will max when `a = 0 rArr x = 1 v_(max) = sqrt(gsintheta)`