In the arrangement shown in figure the ends of an in extensible string move downwards with uniform speed u. Pulleys A and B are fixed. Find the speed with which the mass Mmoves upwards.

Explanation

Suppose the distance of point O from the ceiling is y and the distance of point O from each pulley is x and the distance between the two pulleys is l:
`x^(2) =y^(2) + l^(2)/4`

Differentiating the above equation wr.t. to "t"
`2x(dx)/(dt) =2y(dy)/(dt) + 1/4 2l(dl)/(dt)`
But, `u=(-dx)/(dt), v=(dy)/(dt)` and `(dl)/(dt)`
`therefore -xu =yv` and `v=-u(x/y) = u sec theta`
`therefore` Velocity of mass =`v=u sec theta` (upwards)