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. $2r\cos \left(\frac{\theta }{2}\right)$
• B. $2r\cot \left(\frac{\theta }{2}\right)$
• C. $2r\tan \left(\frac{\theta }{2}\right)$
• D. $2r\sin \left(\frac{\theta }{2}\right)$

Answer 1

The correct option is D $2r\sin \left(\frac{\theta }{2}\right)$

The particle starts from point P and reaches at R after moving an angular displacement of $\theta$.

$\therefore$ The linear displacement of the particle $s=PR$

From the geometry, $PQ=r$ $sin(\frac{\theta }{2})$

$\therefore$ $s=2PQ=2r$ $sin(\frac{\theta }{2})$ $(\because PQ=\frac{1}{2}PR)$