## Motion in a Circle With Non Constant Acceleration

Suppose a particle is moving on a circle of radius
$r$
with at any time
$t$
the ant clockwise angle of the particle from the positive
$x$
axis being given by
$\theta = \frac{t}{t+1}$
/>br /> the angular velocity is
$\omega = \frac{d \theta }{dt} =\frac{(t+1) \frac{d(t)}{dt}- t \frac{d(t+1)}{dt}}{(t+1)^2}= \frac{1}{(t+1)^2}$

using the quotient rule.
Similarly the angular acceleration is
$\alpha = - \frac{2}{(t+1)^2}$
.
The distance moved is a time
$t$
is
$s= r \theta = =\frac{rt}{t+1}$
, the speed is
$v= r \omega = \frac{r}{(t+1)^2}$
and the acceleration is
$a= r \alpha = - \frac{2r}{(r+1)^3}$
.
As
$t \rightarrow \infty, \; \theta \rightarrow 1$
and
$\theta , \alpha \rightarrow 0$
.