A body in free motion does not necessarily rotate about the center of mass.A body in free motion does not necessarily rotate about the center of mass. The center of mass might have straight linear motion in addition any rotation. The general motion is a screw motion with a rotation about some instantaneous axis and parallel translation at the same time.
Consider an arbitrary body rotating by $\vec{\omega}$ and at some instant the center of mass (point C) has linear velocity $\vec{v}_C$.
I can prove that there is always a point A where the linear velocity of the extended rigid body is only parallel to the rotation axis defined by $\vec{\omega}$. The combined motion is a rotation about A with a parallel translation of $\vec{v}_A = h \vec{\omega}$. The scalar $h$ is called pitch. If the body is purely rotating without translation then $h=0$ and if the body is purely translates then $h=\infty$ and $\|\vec{\omega}\|=0$.
The motion of an arbitrary rigid motion is decomposed as such:
- Speed of rotation $$\omega = \| \vec{\omega} \|$$
- Direction of rotation $$\hat{e} = \frac{\vec{\omega}}{\omega}$$
- Location of rotation axis $$\vec{r}_A = \vec{r}_C + \frac{\vec{\omega}\times \vec{v}_C}{\omega^2}$$
- Screw pitch $$h = \frac{\vec{\omega}\cdot\vec{v}_C}{\omega^2}$$
NOTES: $\cdot$ is the vector inner product, and $\times$ is the vector cross product.
Proof
The linear velocity at A is found by the frame transformation law $$\vec{v}_A = \vec{v}_C + \vec{\omega} \times (\vec{r}_A-\vec{r}_C)$$ Using the location expression from above is
$$ \vec{v}_A = \vec{v}_C + \frac{\vec{\omega} \times(\vec{\omega}\times \vec{v}_C)}{\omega^2}$$
Using the Vector Triple Product
$$ \vec{v}_A = \vec{v}_C + \frac{\vec{\omega}(\vec{\omega}\cdot\vec{v}_C)-\vec{v}_C (\vec{\omega}\cdot\vec{\omega})}{\omega^2}$$
With the simplification that $(\vec{\omega}\cdot\vec{\omega}) = \omega^2$ and the definition for screw pitch $\vec{\omega}\cdot\vec{v}_C = h \omega^2$ the above is
$$ \vec{v}_A = \vec{v}_C + \frac{\vec{\omega}(h \omega^2)-\vec{v}_C (\omega^2)}{\omega^2} = h \vec{\omega}$$
So the velocity at A is parallel to the rotation $\vec{\omega}$
Reverse Proof
You can start from a general screw motion at a known point A, with direction $\hat{e}$, speed $\omega$ and pitch $h$ (two parameters for the point as the location along the line does not count, two parameters for the direction as the magnitude does not mater, one for the speed and one for the pitch equals six independent parameters that describe the motion. These will be transformed to the more familiar six motion parameters $\vec{\omega}$ and $\vec{v}_C$ below:
- Rotational Vector $$\vec{\omega} = \omega \hat{e}$$
- Linear Vector $$\begin{align} \vec{v}_C &= \vec{v}_A - \vec{\omega} \times (\vec{r}_A-\vec{r}_C) \\ & = h \vec{\omega} + (\vec{r}_A-\vec{r}_C) \times \vec{\omega} \end{align}$$
All six motion parameters are defined now at the center of mass. Sometimes the above is combined into one expression
$$ \begin{bmatrix} \vec{v}_C \\ \vec{\omega} \end{bmatrix} = \omega \begin{bmatrix} h \hat{e} + \vec{r}\times \hat{e} \\ \hat{e} \end{bmatrix} $$ which clearly decomposes the motion into the magnitude (speed) in the first part, and the screw axis (geometry) in the second part. The two vectors defining the screw axis are called plücker line coordinates.
Example
A cylindrical body is rotating about its axis, and translating perpendicular to the axis (red vectors). This motion is described by a pure rotation about the screw axis (purple vectors).
