Translational motion versus rotational motion

It should be clear, by now, that there is a strong analogy between rotational motion and standard translational motion. Indeed, each physical concept used to analyze rotational motion has its translational concomitant. Likewise, every law of physics governing rotational motion has a translational equivalent. The analogies between rotational and translational motion are summarized in Table 3.

Table 3: The analogies between translational and rotational motion.

Translational motion		Rotational motion
Displacement	$d{\bf r}$	Angular displacement	$d\mbox{\boldmath$\phi$}$
Velocity	${\bf v} = d{\bf r}/dt$	Angular velocity	$\mbox{\boldmath$\omega$}= d\mbox{\boldmath$\phi$}/dt$
Acceleration	${\bf a} = d{\bf v}/dt$	Angular acceleration	$\mbox{\boldmath$\alpha$}= d\mbox{\boldmath$\omega$}/dt$
Mass	$M$	Moment of inertia	$I = {\scriptstyle\int \rho \vert\hat{\mbox{\boldmath$\omega$}}\times {\bf r}\vert^2 dV}$
Force	${\bf f} = M {\bf a}$	Torque	$\mbox{\boldmath$\tau$}\equiv{\bf r}\times {\bf f}= I \mbox{\boldmath$\alpha$}$
Work	$W = \int {\bf f}\!\cdot\!d{\bf r}$	Work	$W = \int \mbox{\boldmath$\tau$}\!\cdot\!d\mbox{\boldmath$\phi$}$
Power	$P = {\bf f}\!\cdot\!{\bf v}$	Power	$P = \mbox{\boldmath$\tau$}\!\cdot\!\mbox{\boldmath$\omega$}$
Kinetic energy	$K = M v^2/2$	Kinetic energy	$K = I \omega^2/2$