Question

A particle travels from $A$ to $B$ along the circular path of radius $R$ as shown in figure. Distance and displacement of particle are

• A. $R\theta ,2R\sin \theta$
• B. $R\theta ,2R\cos \frac{\theta }{2}$
• C. $R\theta ,2R\sin \frac{\theta }{2}$
• D. $R\theta ,2R\cos \theta$

Answer 1

The correct option is C $R\theta ,2R\sin \frac{\theta }{2}$

$I$ Method:

Distance $=$ length of arc $AB=R\theta$

Displacement, $AB=AN+NB=2AN$

$(\therefore AN=NB)$

In $△ONA,$

$\frac{AN}{R}=\sin \frac{\theta }{2}\Rightarrow AN=R\sin \frac{\theta }{2}$

Displacement, $AB=2AN=2R\sin \frac{\theta }{2}$

Alternate method:

Distance $=$ length of arc $AB=R\theta$

Using cosine law for triangle $AOB,$

$AB=\sqrt{O{A}^2+O{B}^2-2OA\times OB\cos \theta }$

$AB=\sqrt{R^2+R^2-2R^2\cos \theta }$

$=\sqrt{2R^2-2R^2\cos \theta }$

$=\sqrt{2R^2(1-\cos \theta })$

$=\sqrt{2R^2.2{\sin }^2\frac{\theta }{2}}=2R\sin \frac{\theta }{2}$

Final answer: (d)