Question

A particle moves with constant speed $v$ along a circular path of radius $r$ and completes the circle in time $T$. The acceleration of the particle is

• A. $2\pi v/T$
• B. $2\pi r/T$
• C. $2\pi r^2/T$
• D. $2\pi v^2/T$

Answer 1

Answer: (A)

$2\pi v/T$

We know that

$\overrightarrow{\Delta s}$$=\overrightarrow{\Delta Q}\times$$\overrightarrow{\Delta r}$

$\frac{\overrightarrow{\Delta s}}{\Delta t}$$=\frac{\overrightarrow{\Delta Q}}{\Delta t}\times$$\overrightarrow{\Delta r}$

$\overrightarrow{\Delta v}$$=\overrightarrow{\Delta w}\times$$\overrightarrow{\Delta r}\Rightarrow |v|=wr(\because \theta ={90}^0)$

$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}(\overrightarrow{w}\times \frac{d\overrightarrow{r}}{dt})+(\frac{d\overrightarrow{w}}{dt}\times \overrightarrow{r})$

$w=$ then $\frac{dw}{dt}=0$

$\overrightarrow{a}=\overrightarrow{w}\times \overrightarrow{v}$ or $a=wv$

$a=\frac{v}{r},v=\frac{v^2}{r},a=r{w}^2$

$a=r{w}^2$

$=r{\left[\frac{2\pi }{T}\right]}^2$

$=r.\frac{4{\pi }^2}{T^2}$

or

$a=vw$

$v.\frac{2\pi }{T}$

Hence, option $A$ is correct