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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.
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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$ rotatingB.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

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

where

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

is called the mass moment of inertia.
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Mass Moment of Inertia
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Translational Kinetic Energy