Question

Figure shows the total acceleration and velocity of a particle moving clockwise in a circle of radius $2.5m$ at a given instant of time. At this instant, find:
(a) the radial acceleration,
(b) the speed of the particle and
(c) its tangential acceleration.

Answer 1

$a=25$ $m/s^2$

General form of velocity:

$\overrightarrow{V}=\dot{r}\hat{r}+r\dot{\theta }\hat{\theta }$

where, a dot above denotes time derivative

General form of acceleration:

$\overrightarrow{a}=(\ddot{r}-r{\dot{\theta }}^2)\hat{r}+(r\ddot{\theta }+2\dot{r}\dot{\theta })\hat{\theta }$

Since, the particle is moving in a circle,

$\dot{r}=\ddot{r}=0$

Now, $\left(a\right)$ radial acceleration:

$a_r=a\cos \theta$

$⟹a_r=25\times \cos {30}^o$

$⟹a_r=25\times \frac{\sqrt{3}}{2}$

$⟹a_r=21.650$

Also, $a_r=r{\dot{\theta }}^2=21.650$

Using magnitude:

${\dot{\theta }}^2=\frac{21.650}{r}$

$⟹{\dot{\theta }}^2=\frac{21.650}{2.5}$

$⟹{\dot{\theta }}^2=8.66$

$⟹\dot{\theta }=2.94$ $rad/s$

$\left(b\right)$ Speed of the particle:

$V=r\dot{\theta }$

$⟹V=2.5\times 2.94$

$⟹V=7.35$ $m/s$

$\left(c\right)$ Tangential acceleration:

$a_{\theta }=a\sin {30}^o$

$⟹a_{\theta }=25\times \frac{1}{2}$

$⟹a_{\theta }=12.5$ $m/s^2$