Rotational Motion

Moment of inertia

$I = \sum m r^{2} = \int r^{2} d m,$ $I = M K^{2} (K =$ radius of gyration $)$
Axis theorems
$I = I_{c m} + M d^{2}$ (parallel) $;$ $I_{z} = I_{x} + I_{y}$ (perpendicular, plane bodies)
Standard (about the central axis)
ring $M R^{2};$ disc $\frac{1}{2} M R^{2};$ solid sphere $\frac{2}{5} M R^{2};$ hollow sphere $\frac{2}{3} M R^{2}$
rod about centre $\frac{1}{12} M L^{2};$ about end $\frac{1}{3} M L^{2};$ ring about diameter $\frac{1}{2} M R^{2};$ disc about diameter $\frac{1}{4} M R^{2}$

Without slipping
$v_{c m} = ω R,$ $K = \frac{1}{2} M v^{2} (1 + \frac{K ^{2}}{R ^{2}})$
Down an incline
$a = \frac{g sin θ}{1 + K ^{2} / R ^{2}},$ $μ_{m i n} = \frac{tan θ}{1 + R ^{2} / K ^{2}}$