Key Equations

Key Equations

Angular position	$\theta =\frac{s}{r}$
Angular velocity	$\omega =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{Δ}\theta }{\text{Δ}t}=\frac{d\theta }{dt}$
Tangential speed	$v_{\text{t}}=r\omega$
Angular acceleration	$\alpha =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{Δ}\omega }{\text{Δ}t}=\frac{d\omega }{dt}=\frac{d^2\theta }{d{t}^2}$
Tangential acceleration	$a_{\text{t}}=r\alpha$
Average angular velocity	$\bar{\omega }=\frac{{\omega }_0+{\omega }_{\text{f}}}{2}$
Angular displacement	${\theta }_{\text{f}}={\theta }_0+\bar{\omega }t$
Angular velocity from constant angular acceleration	${\omega }_{\text{f}}={\omega }_0+\alpha t$
Angular velocity from displacement and constant angular acceleration	${\theta }_{\text{f}}={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$
Change in angular velocity	${\omega }_{\text{f}}^2={\omega }_0^2+2\alpha (\text{Δ}\theta )$
Total acceleration	$\overrightarrow{a}={\overrightarrow{a}}_{\text{c}}+{\overrightarrow{a}}_{\text{t}}$
Rotational kinetic energy	$K=\frac{1}{2}\left({\displaystyle \sum _j{m}_j{r}_j^2}\right){\omega }^2$
Moment of inertia	$I={\displaystyle \sum _j{m}_j{r}_j^2}$
Rotational kinetic energy in terms of the moment of inertia of a rigid body	$K=\frac{1}{2}I{\omega }^2$
Moment of inertia of a continuous object	$I={\displaystyle \int r^{2}dm}$
Parallel-axis theorem	$I_{\text{parallel-axis}}=I_{\text{center of mass}}+m{d}^2$
Moment of inertia of a compound object	$I_{\text{total}}={\displaystyle \sum _i{I}_i}$
Torque vector	$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$
Magnitude of torque	$\left|\overrightarrow{\tau }\right|=r_{\perp }F$
Total torque	${\overrightarrow{\tau }}_{\text{net}}={\displaystyle \sum _i\left|{\overrightarrow{\tau }}_i\right|}$
Newton’s second law for rotation	${\displaystyle \sum _i{\tau }_i}=I\alpha$
Incremental work done by a torque	$dW=\left({\displaystyle \sum _i{\tau }_i}\right)d\theta$
Work-energy theorem	$W_{AB}=K_B-K_A$
Rotational work done by net force	$W_{AB}={\displaystyle \underset{{\theta }_A}{\overset{{\theta }_B}{\int }}\left({\displaystyle \sum _i{\tau }_i}\right)d\theta }$
Rotational power	$P=\tau \omega$