In the arrangement shown in the Fig, the ends P and Q of an unstretchable string move downwards with uniform speed U. Pulleys A and B are fixed.
Mass M moves upwards with a speed

Options

A. `2Ucos theta`
B. `U//cos theta`
C. `2U//cos theta`
D. `U cos theta`

Answer

B

Explanation

This is a problem based on constraint motion. The motion of mass M is constraint with the motion of P and Q. Let `AN=x`, `NO=z`. Then velocity of mass is `(dz)/(dt)`. Also, let `OA=l`. then `(dl)/(dt)=U`
From `DeltaANO`, using pythagorus theorem
`:.` `x^2+z^2=l^2`
Here x is a constant.
Differentiating the above equation w.r.t to t
`0+2z(dz)/(dt)=2l(dl)/(dt)implieszv_M=lU`
`implies v_M=(l)/(z)U=(U)/(z//l)=(U)/(cos theta)` `( :' cos theta=z/l)`