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I know the KE equation for combined translational and rotational motion, which is $K=Mv^2/2+I\omega^2/2$, and here $v$ is the velocity of COM and $I$ is the moment of inertia about the COM. But what if the body rotates about a point $\mathcal{P}$ other than the COM? Can I use the the velocity of $\mathcal{P}$ and moment of inertia about $\mathcal{P}$ in the equation?

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  • $\begingroup$ Use the parallel axis theorem to determine the value for $I$ about $P$. $\endgroup$
    – joseph h
    Commented Mar 28 at 6:57
  • $\begingroup$ Yes but can I use the same equation with respect to P rather than using it wrt COM? Because my book has only mentioned the formula wrt com $\endgroup$
    – yyzr
    Commented Mar 28 at 7:01

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Translation

the position vector to point P is:

$$\mathbf R_P=\mathbf R_C+\mathbf S\,\mathbf u_{CP}$$

where $~\mathbf S~$ is the transformation matrix between body fixed coordinate system and inertial system

take the time derivative you obtain the velocity

$$\mathbf v_P=\mathbf v_C+ \dot{\mathbf{S}}\,\mathbf u$$

with $~\dot{\mathbf{S}}=\mathbf S\,\mathbf\omega^\times$

$$\mathbf v_P=\mathbf v_C+ \mathbf S\left(\omega\times\,\mathbf u\right)$$

the KE

$$ K_{ET}=\frac 12\,\mathbf v_P^T\,M\,\mathbf v_P$$

Rotation

the KE

$$ K_{ER}=\frac 12 \mathbf \omega^T\,\mathbf I_P\,\mathbf \omega$$

where $$ \mathbf I_P=\mathbf I_C-M\,\mathbf u^\times\,\mathbf u^\times\\\\ \mathbf u^\times= \left[ \begin {array}{ccc} 0&-u_{{z}}&u_{{y}}\\ u_{ {z}}&0&-u_{{x}}\\ -u_{{y}}&u_{{x}}&0\end {array} \right] $$

  • $~C~$ center of mass
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  • $\begingroup$ So what is the final expression for total KE? $\endgroup$
    – yyzr
    Commented Apr 3 at 3:08
  • $\begingroup$ The sum of translation and rotation $\endgroup$
    – Eli
    Commented Apr 3 at 6:04
  • $\begingroup$ Does the expression look like this? $KE= (M v²_{P})/2+ (I_{P} \omega²)/2$ $\endgroup$
    – yyzr
    Commented Apr 4 at 6:11
  • $\begingroup$ Yes but v and $~\omega ~$ are vectors not scalars $\endgroup$
    – Eli
    Commented Apr 4 at 6:17
  • $\begingroup$ Ok, but that means for calculating translational KE, I have to concentrate the total mass at 'P' but actually the total mass can only be concentrated at COM. Plz explain this $\endgroup$
    – yyzr
    Commented Apr 4 at 14:16