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Physics
Classical Physics
Mechanics
Why do objects always rotate about their centre of mass?
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[QUOTE="etotheipi, post: 6358020"] Many others have said similar things, but I thought that I'll add one bit. If a rigid body has angular velocity ##\vec{\omega}## (w.r.t. some coordinate system) then [I]any two points[/I] on the rigid body rotate about each other at angular velocity ##\vec{\omega}##. Let the position of a particle on the rigid body in some reference configuration (e.g. at ##t=0##) be ##\vec{\xi}##, such that there exists some mapping to the position vector of that particle at any given time ##\vec{r}(t) = \chi(\vec{\xi}, t)## as a function of this fixed label. The vector between any two points on the rigid body is of fixed magnitude and can be expressed as a rotation of this such vector at ##t=0##, i.e. ##\vec{r}_{AB}(t) = \hat{R}(t)(\vec{\xi}_A - \vec{\xi}_B)##. If you differentiate and use that ##\dot{\hat{R}}(t) \vec{\alpha} dt \approx \vec{\omega} \times \hat{R}(t) \vec{\alpha} dt## for some vector ##\vec{\alpha}##, you finally end up with $$\vec{v}_{AB}(t) = \vec{\omega} \times \hat{R}(t)(\vec{\xi}_A - \vec{\xi}_B) = \vec{\omega} \times \vec{r}_{AB}(t)$$It is often convenient to take "point B" to be the centre-of-mass, because you can also use things like König's theorem for KE and angular momentum that make life much simpler. But there is nothing stopping you conceptually from transforming into another non-inertial, body-fixed frame and considering the rotation of the rigid body to be about that point :smile:. [/QUOTE]
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Mechanics
Why do objects always rotate about their centre of mass?
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