Formula: Non-Uniform (Accelerated) Circular Motion Angle Initial angular velocity Angular acceleration Time
$$\varphi ~=~ \omega_0 \, t ~+~ \frac{1}{2} \, \class{red}{\alpha} \, t^2$$
$$\varphi ~=~ \omega_0 \, t ~+~ \frac{1}{2} \, \class{red}{\alpha} \, t^2$$
$$\omega_0 ~=~ \left( \varphi - \frac{\class{red}{\alpha} \, t^2}{2} \right) \, \frac{1}{t}$$
$$\class{red}{\alpha} ~=~ \frac{2}{t^2} \, \left( \varphi - \omega_0 \, t \right)$$
$$t ~=~ \frac{\omega_0}{\class{red}{\alpha}} \pm \sqrt{ \frac{\omega_0^2}{\class{red}{\alpha}^2} + \frac{2\varphi}{\class{red}{\alpha}} }$$
Angle
$$ \varphi $$ Unit $$ \mathrm{rad} $$
Angle traversed within the time \(t\) during an accelerated circular motion. By accelerated is meant that the body turns faster and faster on the circular path.
Initial angular velocity
$$ \omega_0 $$ Unit $$ \frac{\mathrm{rad}}{\mathrm s} $$
Angular velocity at the beginning of acceleration, that is at time \( t = 0\). Angular velocity is analogous to the velocity \(v\) in a linear motion.
Angular acceleration
$$ \class{red}{\alpha} $$ Unit $$ \frac{\mathrm{rad}}{\mathrm{s}^2} $$
A constant angular acceleration that indicates the change (decrease / increase) in angular velocity per second. It is analogous to the acceleration \(a\) in a linear motion.
Time
$$ t $$ Unit $$ \mathrm{s} $$
Time at which the angle \(\varphi(t)\) was traversed during a circular motion.