Key Equations

Angular position	$θ = \frac{s}{r}$
Angular velocity	$ω = lim_{Δ t \to 0} \frac{Δ θ}{Δ t} = \frac{d θ}{d t}$
Tangential speed	$v_{t} = r ω$
Angular acceleration	$α = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d ω}{d t} = \frac{d^{2} θ}{d t^{2}}$
Tangential acceleration	$a_{t} = r α$
Average angular velocity	$\bar{ω} = \frac{ω_{0} + ω_{f}}{2}$
Angular displacement	$θ_{f} = θ_{0} + \bar{ω} t$
Angular velocity from constant angular acceleration	$ω_{f} = ω_{0} + α t$
Angular velocity from displacement and constant angular acceleration	$θ_{f} = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$
Change in angular velocity	$ω_{f}^{2} = ω_{0}^{2} + 2 α (Δ θ)$
Total acceleration	$\vec{a} = {\vec{a}}_{c} + {\vec{a}}_{t}$
Rotational kinetic energy	$K = \frac{1}{2} (\sum_{j} m_{j} r_{j}^{2}) ω^{2}$
Moment of inertia	$I = \sum_{j} m_{j} r_{j}^{2}$
Rotational kinetic energy in terms of the moment of inertia of a rigid body	$K = \frac{1}{2} I ω^{2}$
Moment of inertia of a continuous object	$I = \int r^{2} d m$
Parallel-axis theorem	$I_{parallel-axis} = I_{center of mass} + m d^{2}$
Moment of inertia of a compound object	$I_{total} = \sum_{i} I_{i}$
Torque vector	$\vec{τ} = \vec{r} \times \vec{F}$
Magnitude of torque	$\| \vec{τ} \| = r_{⊥} F$
Total torque	${\vec{τ}}_{net} = \sum_{i} \| {\vec{τ}}_{i} \|$
Newton’s second law for rotation	$\sum_{i} τ_{i} = I α$
Incremental work done by a torque	$d W = (\sum_{i} τ_{i}) d θ$
Work-energy theorem	$W_{A B} = K_{B} - K_{A}$
Rotational work done by net force	$W_{A B} = \int_{θ_{A}}^{θ_{B}} (\sum_{i} τ_{i}) d θ$
Rotational power	$P = τ ω$