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
![$\displaystyle I \eqsp m\left\Vert\,\underline{\tilde{\omega}}\times\underline{x}\,\right\Vert^2.
$](http://www.dsprelated.com/josimages_new/pasp/img2933.png)
In a collection of masses
having velocities
, we of
course sum the individual kinetic energies to get the total kinetic
energy.
Finally, we may also write the rotational kinetic energy as half the inner product of the angular-velocity vector and the angular-momentum vector:B.27
![$\displaystyle E_R \eqsp \frac{1}{2}\, \underline{\omega}\cdot \underline{L}\eqsp \frac{1}{2} I \omega^2,
$](http://www.dsprelated.com/josimages_new/pasp/img2935.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ \underline{\omega}\cdot\underline{L}=\underline{\omega}\cdot \mathbf{I}\underl...
...^2 \underline{\tilde{\omega}}^T\mathbf{I}\underline{\tilde{\omega}}= I \omega^2$](http://www.dsprelated.com/josimages_new/pasp/img2936.png)
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Principal Axes of Rotation