First slide

Rectilinear Motion

Question

A particle slides from rest from the topmost point of a vertical circle of radius r along a smooth chord making an angle $θ$ with the vertical. The time of descent is

Difficult

A

$Least for θ = 0$
B

$Maximum for θ = 0$
C

$Least for θ = 45^{\circ}$
D

$Independent of θ$

Solution

$\begin{array}{l} a = g \sin a = g \sin (90^{\circ} - θ) (see figure) \\ = g \cos θ \\ l = 2 R \cos θ \end{array}$
$Now using s = u t + \frac{1}{2} a t^{2}, we get$
$\begin{array}{l} l = 0 t + \frac{1}{2} g \cos θ t^{2} \\ \Rightarrow 2 R \cos θ = \frac{1}{2} g \cos θ t^{2} \Rightarrow t = 2 \sqrt{\frac{R}{g}} \end{array}$
This is independent of $θ$ .

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