question_answer

A particle is moving uniformly in a circular  path of radius r. When it moves through an angular displacement \[\theta ,\] then the magnitude of the corresponding linear displacement will be

A)  \[2r\,d\cos \left( \frac{\theta }{2} \right)\]

B)  \[2r\,\cot \left( \frac{\theta }{2} \right)\]

C)  \[2r\,\tan \left( \frac{\theta }{2} \right)\]

D)  \[2r\,\sin \left( \frac{\theta }{2} \right)\]

Correct Answer: D

Solution :

  In \[\Delta AOB\,\]\[\sin \frac{\theta }{2}=\frac{AB}{AO}\] \[(\because \,AO=r)\] \[AB=AO\,\,\sin \frac{\theta }{2}\Rightarrow AB=r\sin \frac{\theta }{2}\] \[AC=AB+BC\] \[(\because \,AB=BC)\] \[=r\sin \frac{\theta }{2}+r\sin \frac{\,\theta }{2}\] \[AC=2r\sin \frac{\theta }{2}\] So, the magnitude of the corresponding linear displacement will be \[\sin \frac{\theta }{2}.\]