question_answer

In the arrangement shown in figure 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                              [IIT 1982]                      

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

B)             \[U\cos \theta \]            

C)             \[\frac{2U}{\cos \theta }\]                  

D)             \[\frac{U}{\cos \theta }\]            

Correct Answer: D

Solution :

                As P and Q fall down, the length l decreases at the rate of U m/s. From the figure, \[{{l}^{2}}={{b}^{2}}+{{y}^{2}}\]             Differentiating with respect to time             \[2l\times \frac{dl}{dt}=2b\times \frac{db}{dt}+2y\times \frac{dy}{dt}\]\[\left( \text{As }\frac{db}{dt}=0,\frac{dl}{dt}=U \right)\]             Þ\[\frac{dy}{dt}=\left( \frac{l}{y} \right)\times \frac{dl}{dt}\]\[\Rightarrow \frac{dy}{dt}=\left( \frac{1}{\cos \theta } \right)\times U=\frac{U}{\cos \theta }\]