Question

In circular motion, assuming $\overline{v}=\overline{w}\times \overline{r}$, obtain an expression for the resultant acceleration of a particle in terms of tangential and radial component.

Answer 1

Let the particle be moving around a circular path of constant radius $r$. If it speeds up or slows down, its motion is non-uniform and both the angular speed $\left(\omega \right)$ and the linear speed $\left(v\right)$ change with time. Thus, at any instant, $\omega$ , $v$ and $r$ are related as :
$v=\omega r$
Hence, the angular acceleration of the particle is given by,
$\alpha =\frac{d\omega }{dt}$
and the tangential acceleration $a_t$ is given by,
$a_t=\frac{dv}{dt}=\frac{d\left(\omega r\right)}{dt}=r\frac{d\omega }{dt}$ [because $r$ is a constant]
Thus, $a_t=\alpha r$.