## Motion in a Circle With Non Constant Acceleration

\[r\]

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 \[\theta , \alpha \rightarrow 0\]

.