Question

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. $ 2\,r\,\sin(\theta /2) $
• B. $ 2\,r\,\cos\,(\theta /2) $
• C. $ 2\,r\,{\tan}(\theta /2) $
• D. $ 2\,r\,{ \cot}(\theta /2) $

Answer 1

Answer: (A)

In $\Delta A O B \sin \frac{\theta}{2}=\frac{A B}{A O} \,\,\,(\because A O=r)$
$ A B =A O \sin \frac{\theta}{2} $
$\Rightarrow A B=r \sin \frac{\theta}{2} $
$ A C =A B+B C (\because A B=B C) $
$=r \sin \frac{\theta}{2}+r \sin \frac{\theta}{2}$
$ A C =2 r \sin \frac{\theta}{2} $
So, the magnitude of the corresponding linear displacement will be $2 r \sin \frac{\theta}{2}$.