Rotational Kinetic Energy Revisited

If a point-mass is located at
and is rotating about an
axis-of-rotation
with angular velocity
, then the
distance from the rotation axis to the mass is
,
or, in terms of the vector cross product,
. The tangential velocity of the mass is
then
, so that the kinetic energy can be expressed as
(*cf.* Eq.
(B.23))

where

In a collection of masses having velocities , we of course sum the individual kinetic energies to get the total kinetic energy.

Finally, we may also write the rotational kinetic energy as half the
*inner product* of the angular-velocity vector and the
angular-momentum vector:^{B.27}

where the second form (introduced above in Eq. (B.7)) derives from the vector-dot-product form by using Eq. (B.20) and Eq. (B.22) to establish that .

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