Moment of inertia
\[ I = \int r^2 dm \] \[ I=I_{com} + Mh^2 \]
Common formulae
Rotational Inertia through central axis of hoop
\[ I = MR^2 \]
RI through central axis of an annular cylinder
\[ I = \frac{1}{2}M(R_1^2+R_2^2) \]
Solid cylinder about central axis
\[ I = \frac{1}{2}MR^2 \]
Solid cylinder about central diameter
\[ I = \frac{1}{2}MR^2 + \frac{1}{12}ML^2 \]
Thin rod about central diameter
\[ I = \frac{1}{12}ML^2 \]
Solid cylinder about diameter
\[ I = \frac{2}{5}MR^2 \]
Thin spherical shell about any diameter
\[ I = \frac{2}{3}MR^2 \]
Hoop about diameter
\[ I = \frac{1}{2}MR^2 \]
Slab about center
\[ I = \frac{1}{12}M(l^2 + w^2) \]
Newton’s second law for rotation
\[ \tau_{net} =I\alpha \]
Work and rotational kinetic energy
\[ W = \int_{\theta_i}^{\theta_f} \tau \dd{\theta} \]
if \(\tau\) is constant, its just
\[ W=\tau(\theta_f-\theta_i) \]
\[ P=\tau \omega \]