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

Answer 1

Answer: (A)

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 $2r\sin \frac{\theta }{2}$.