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the total mass, and v the velocity of the centre of mass. But take the case of a gyroscope, where the force acting on the centre of mass is gravity, assuming it is uniform over the body. The gyroscope doesn't topple over, but precesses, which imples that the centre of mass theorem doesn't apply to rotational bodies. Why not? The Theorem seems to be correct for any system of particles, whether it is rotaing or not, if one looks at the derivation of the Centre of Mass Theorem:-

For each particle in a body, f = md^2r/dt^2.

If I sum all the forces, including the external forces applied to the body,

and make use of Newton's third law, then the internal forces all cancel, so I'm left with just the external force F.

F = sum(k=1, k=n)[mkd^2rk/dt^2] where mk and rk is the mass and radius

vector f the kth particle.

So F = d^2/dt^2[sum(k=1, k=n)[mk x rk]] . If the centre of mass is defined as R = (sum(k=1, k=)[mk x rk] )/M

Then external F = Md^2R/dt^2.

Which of the above lines is false for a rotating body?

Thanks