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Physics
Classical Physics
Mechanics
Why do objects always rotate about their centre of mass?
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[QUOTE="vanhees71, post: 6358022, member: 260864"] Take the rigid body. The center-mass-motion theorem (Noether's theorem for Galilei boosts) then tells you that $$M \ddot{\vec{R}}=\vec{F}_{\text{tot},\text{ext}}.$$ Here $$\vec{R}=\frac{1}{M} \int_V \mathrm{d}^3 x \rho_m(\vec{x}) \vec{x}).$$ In the case of homogeneous gravity (as in the usual approximation for motion close to Earth) you get the potential of the external force as $$V=-\int_{V} \mathrm{d}^3 x \rho_m(\vec{x}) \vec{g} \cdot \vec{x}=-m \vec{g} \cdot \vec{R},$$ and indeed the center of mass moves as a single particle in free fall, $$M \ddot{\vec{R}}=M \vec{g}.$$ For the case of a homomgeneous electric field you rather get $$V=-\int_{V} \mathrm{d}^3 x \rho_q(\vec{x}) \vec{E} \cdot \vec{x},$$ here you don't get a simple linear function in ##\vec{R}##, if the charge distribution in the (assumed to be charged) body is not proportional to the mass distribution. [/QUOTE]
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Mechanics
Why do objects always rotate about their centre of mass?
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