First slide

Application of Newtons laws

Question

In figure, lower pulley is free to move in vertical direction only. Block A is given a uniform velocity u as shown, what is velocity of block B as a function of angle $θ$ .

Difficult

A

$u \cos θ$
B

$\frac{u}{\cos θ}$
C

$\frac{u [1 + \sin θ]}{\cos θ}$
D

$\frac{u [1 + \cos θ]}{\sin θ}$

Solution

Let in a small time dt, displacement of A is dx and displacement of B is dy. Increase in length l₁= dx.
Increase in length $l_{2} = d x \sin θ - d y \cos θ$
But net increase in length should be zero, so
$d x + d x \sin θ - d y \cos θ = 0$
$\begin{array}{l} \Rightarrow d y = \frac{(1 + \sin θ) d x}{\cos θ} \\ \Rightarrow \frac{d y}{d t} = (\frac{1 + \sin θ}{\cos θ}) \frac{d x}{d t} \Rightarrow v = (\frac{1 + \sin θ}{\cos θ}) u \end{array}$

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