Question

A particle moves through angular displacement $\theta$ on a circular path of radius r. The linear displacement will be

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

Answer 1

The correct option is A 2 r sin $\left(\theta /2\right)$

The linear displacement, $\overrightarrow{△r}=\overrightarrow{r_2}-\overrightarrow{r_1};$ where $r_2=r_1=r$

Hence, $|△r|=\sqrt{r_2^2+r_1^2-2r_2{r}_{1}cos\theta }$

$⟹|\Delta r|=\sqrt{2r^2-2r^{2}cos\theta }$

$⟹|\Delta r|=\sqrt{2}r\sqrt{1-cos\theta }$

$⟹|\Delta r|=2rsin\left(\frac{\theta }{2}\right)$