Main variables:

Main Equations:

Average angular speed:

\begin{equation*}
\omega_{av} = \frac{\theta_f – \theta_i}{t_f – t_i} = \frac{\Delta \theta}{\Delta t}.
\end{equation*}

Instantaneous speed:

\begin{equation*}
\omega_{ins} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}.
\end{equation*}

Magnitude of average angular acceleration:

\begin{equation*}
\alpha_{av} = \frac{\omega_f – \omega_i}{t_f – t_i} = \frac{\Delta \omega}{\Delta t}.
\end{equation*}

Magnitude of instantaneous angular acceleration:

\begin{equation*}
\alpha_{ins} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt}.
\end{equation*}

Angle as a function of time with constant acceleration:

\begin{equation*}
\vec{\theta}_f = \vec{\theta}_i + \vec{\omega}_i t + \frac{1}{2} \vec{\alpha} t^2.
\end{equation*}

Angle as a function of time with constant velocity:

\begin{equation*}
\vec{\theta}_f = \vec{\theta}_i + \vec{\omega}_i t.
\end{equation*}

Angular speed as a function of time:

\begin{equation*}
\vec{\omega}_f = \vec{\omega}_i + \vec{\alpha} t.
\end{equation*}

Angular speed as a function of the angle and the angular acceleration:

\begin{equation*}
\vec{\omega}_f^2 = \vec{\omega}_i^2 + 2  \vec{\alpha} \cdot (\vec{\theta}_f – \vec{\theta}_i).
\end{equation*}

Relation between the magnitude of the angular speed and the magnitude of the tangential speed:

\begin{equation*}
v = R \omega.
\end{equation*}

Relation between the magnitude of the angular acceleration and the magnitude of the tangential acceleration:

\begin{equation*}
a_{tan} = R \alpha.
\end{equation*}

Relation between the magnitude of the angular acceleration and the magnitude of the radial acceleration:

\begin{equation*}
a_{rad} = R \omega^2 .
\end{equation*}

Rotational motion Dynamics

Moment of inertia:

\begin{equation*}
I = \sum m_i r_i^2.
\end{equation*}

Moment of inertia in integral form:

\begin{equation*}
I = \int r_i^2 dm.
\end{equation*}

Moment of inertia for a disk:

\begin{equation*}
I_{\text{disk}} = \frac{1}{2} mR^2.
\end{equation*}

Moment of inertia for a rod, through the center:

\begin{equation*}
I_{\text{rod}} = \frac{1}{12} mL^2.
\end{equation*}

Moment of inertia for a rod rotated about end:

\begin{equation*}
I_{\text{rod}} = \frac{1}{3} mL^2.
\end{equation*}

Rotational kinetic energy:

\begin{equation*}
K = \frac{1}{2} I \omega^2.
\end{equation*}

Parallel-axis theorem:

\begin{equation*}
I = I_{cm} + Md^2.
\end{equation*}

Torque in terms of the position and the force:

\begin{equation*}
\vec{\tau} = \vec{r} \times \vec{F}.
\end{equation*}

Torque as the rotational analog of Newton’s second law:

\begin{equation*}
\sum \vec{\tau} = I \vec{\alpha}.
\end{equation*}

Torque when the system it’s rotating with a constant angular speed:

\begin{equation*}
\sum \vec{\tau} = 0.
\end{equation*}

Work done by a constant torque:

\begin{equation*}
W = \tau \Delta \theta.
\end{equation*}

Work done by the torque:

\begin{equation*}
W = \int_{\theta_1}^{\theta_2} \tau d\theta.
\end{equation*}

Angular momentum in terms of the linear momentum:

\begin{equation*}
\vec{L} = \vec{r} \times \vec{p}.
\end{equation*}

Angular momentum as a function of the linear velocity:

\begin{equation*}
\vec{L} =  \vec{r} \times (m \vec{v}).
\end{equation*}

Angular momentum as a function of the angular velocity:

\begin{equation*}
\vec{L} = I \vec{\omega}.
\end{equation*}

Rotational dynamics and angular momentum:

\begin{equation*}
\sum \vec{\tau} = \frac{ d\vec{L} }{dt}.
\end{equation*}

Rotational dynamics and angular momentum as a differential form:

\begin{equation*}
\sum \vec{\tau} = \frac{d \vec{L}}{dt}.
\end{equation*}