Rotational Kinetic Energy

Figure B.4: Point-mass $m$ rotating in a circle of radius $R$ with tangential speed $v=R\omega$ , where $\omega =\dot {\theta }$ denotes the angular velocity in rad/s.

The rotational kinetic energy of a rigid assembly of masses (or mass distribution) is the sum of the rotational kinetic energies of the component masses. Therefore, consider a point-mass $m$ rotating^B.13 in a circular orbit of radius $R$ and angular velocity $\omega$ (radians per second), as shown in Fig.B.4. To make it a closed system, we can imagine an effectively infinite mass at the origin $\underline {0}$ . Then the speed of the mass along the circle is $v=R\omega$ , and its kinetic energy is $(1/2)mv^2=(1/2)mR^2\omega^2$ . Since this is what we want for the rotational kinetic energy of the system, it is convenient to define it in terms of angular velocity $\omega$ in radians per second. Thus, we write

$$E_R \eqsp \frac{1}{2} I \omega^2, \protect$$ (B.7)

where

$$I \eqsp mR^2 \protect$$ (B.8)

is called the mass moment of inertia.