Question

A simple pendulum is released from rest with the string in horizontal position. The vertical component of the velocity of the bob becomes maximum, when the string makes an angle $\theta$ with the vertical. The angle $\theta$ is equal to

• A. $\frac{\pi }{4}$
• B. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• C. ${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

At angle theta , l = length of rod v = pendulum velocity u = vertical velocity

By energy conservation,

$mgl\cos \theta =\frac{m{v}^2}{2}=\frac{m{u}^2}{2{\sin }^2\theta }$

Differentiating wrt $\theta$ , and putting $\frac{du}{d\theta }=0$;

we get, $\theta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$