question_answer

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

A) \[2U\cos \theta \]

B) \[U\cos \theta \]

C) \[2U/\cos \theta \]

D) \[U/\cos \theta \]

Correct Answer: D

Solution :

[d]

As P and Q move down, the length \[\ell \] decreases at the rate of U m/s.

From figure, \[{{\ell }^{2}}={{b}^{2}}+{{y}^{2}}\]

Differentiating with respect to time?

\[2\ell \frac{d\ell }{dt}=2y\frac{dy}{dt}\] (\[\therefore \] b is constant)

\[\therefore \]\[\frac{dy}{dt}=\frac{\ell }{y}\cdot \frac{d\ell }{dt}=\frac{1}{\cos \theta }\cdot \frac{d\ell }{dt}=\frac{U}{\cos \theta }\]