A block B starts sliding down from the point A on a rough wedge. The angle that the incline plane makes with the horizontal is variable having value `(pi)/(3)` at A and 0 at B. Also the magnitude of rate of change of `theta` with x is 2 where x is the length measured along the incline plane starting from A. Assume total length of incline plane to be `(pi)/(6)m`. Based upon the above situation, answer the following question. `(x ge (pi)/(24)m)`

Find velocity of the block as it reaches down at the point B.

Options

A. `5 m//s`
B. `(5 + 5 sqrt3) m//s`
C. `sqrt(5 - 5 sqrt3) m//s`
D. Block will not reach at point B

Answer

D

Explanation

`F_("net")` along the incline plane `= mg sin theta - mu mg cos theta = ma`
`g sin theta - mu g cos theta - v (dv)/(dx)`
`g [ sin ((pi)/(3) - 2x) - 1 {cos ((pi)/(3) -2x)}] = (vdv)/(dx)`
`g [ intsin ((pi)/(3) - 2x) dx - int cos ((pi)/(3) - 2x) dx ] = underset(0)overset(v)int vdv`
`g [((- cos ((pi)/(3) - 2x))/(-2))_(0)^(l) - (sin. (((pi)/(3) - 2x))/(-2))_(0)^(l)] = (v^(2))/(2)`
`10 [(cos. ((pi)/(3) - 2l) - (1)/(2)) + (sin. ((pi)/(3) - 2l) - (sqrt3)/(2))] = v^(2)`
`10[(1 - (1)/(2)) + (0 - (sqrt3)/(2))] = v^(2)`
`sqrt(5 (1 - sqrt3)) = v`