Question

A particle move along a circular arc of radius $R$ making an angle of $\theta$ at center. The magnitude of displacement is:

• A. $2R\sin \theta /2$
• B. $2R\sin \theta$
• C. $R\sin \theta /2$
• D. $R\sin \theta$

Answer 1

The correct option is A $2R\sin \theta /2$

Let the particle move from $P$ to $Q$

The triangle in the figure is isosceles triangle.

$OP=OQ=R$

Let $\angle POQ=x$

Using sine law:

$\frac{OP}{\sin \left(\frac{180°-x}{2}\right)}=\frac{PQ}{\sin x}\sin 2A=2\sin A\cos A\therefore PQ=2R\sin \left(\frac{x}{2}\right)=2R\sin \left(\frac{\theta }{2}\right)$