question_answer

A pendulum bob is released from angular position \[\theta ,\]such that the magnitude of its initial acceleration and acceleration at lowest position are equal.

The angular position \[\theta \] is

A) \[\theta ={{\cos }^{-1}}\left( \frac{3}{5} \right)\]

B) \[\theta =si{{n}^{-1}}\left( \frac{3}{5} \right)\]

C) \[\theta ={{\tan }^{-1}}\left( \frac{3}{5} \right)\]

D) \[\theta ={{\cos }^{-1}}\left( \frac{2}{5} \right)\]

Correct Answer: A

Solution :

Let \[{{a}_{1}}\]and \[{{a}_{2}}\]are the accelerations, at given positions. Then,

\[{{a}_{2}}=\frac{{{v}^{2}}}{l}=\frac{2gl\,(1-\cos \theta )}{l}\]

Also, \[{{a}_{1}}=g\,\sin \theta \]

As, \[{{a}_{1}}={{a}_{2}}\]

\[\Rightarrow \]\[g\sin \theta =2g\,(1-\cos \theta )\]

\[\Rightarrow \]\[\sin \theta =2\,(1-\cos \theta )\]

\[\Rightarrow \]\[5{{\cos }^{2}}\theta -8\cos \theta +3=0\]

\[\Rightarrow \]\[\cos \theta =\frac{3}{5}\]or \[\theta ={{\cos }^{-1}}\left( \frac{3}{5} \right)\]