Question

In the arrangement shown in the figure, the ends P and Q of an un-stretchable string move downwards with uniform speed. Pulleys A and B are fixed. The mass M moves upwards with a speed:

A. \[2U\cos \theta \]
B. \[\dfrac{U}{{\cos \theta }}\]
C. \[\dfrac{{2U}}{{\cos \theta }}\]
D. \[U\cos \theta \]

Answer 1

Hint: The above problem is resolved by taking the equilibrium condition at which the mass M is balanced and taking the velocity along the horizontal direction or the cozy component of velocity acting and the component of velocity acting on the thread's point. Moreover, the velocity component is identified by taking the necessary condition to cause the pulley's smooth motion.

Complete step by step answer:
Let the speed of the block of mass M is V.
The point on the thread just connected to mass M will have the same velocity.
The component of velocity V is applied along the thread, Such that the magnitude of the component is given as,
\[{V_1} = V\cos \theta \].
As, the point of the thread is moving with the speed of \[V\cos \theta \] , then as per the given condition the thread is moving with the velocity of U m/s.
Hence, the magnitude of velocity \[V\cos \theta \] will become equal to U at equilibrium condition.
 The mathematical form is,
\[\begin{array}{l}
U = V\cos \theta \\
V = \dfrac{U}{{\cos \theta }}
\end{array}\]
Therefore, the mass M moves upwards with a speed of \[\dfrac{U}{{\cos \theta }}\]

So, the correct answer is “Option B”.

Note:
To resolve the given problem, one must understand that there is some condition where the pulley system is working smoothly. That condition is taking place for the equilibrium, where all the necessary forces are balanced in some sense. Moreover, each component's weight must be appropriate to cause the motionlessness of the number of components attached to the pulley system. Furthermore, the pulley systems have a variety of applications in both the machines and industries.