# Centre of Mass Theorem false for a rotating body?

1. Feb 16, 2006

### jmc8197

For a distributed mass, F = M dv/dt where F is the total external force, M
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

2. Feb 16, 2006

### phucnv87

For rotating body, you have to use the moment of momentum

3. Feb 17, 2006

### Galileo

Why not? If the gyroscope doesn't topple then it just means the net external force on the system is not given by the gravitational force alone.

4. Feb 17, 2006

### rachmaninoff

jmc8197-

Your theorem certainly applies to gyroscopes, tops, any rotating solids. Why do you think it doesn't? The center of mass of a precessing top doesn't move at all (or moves in a uniform straight line) - everything else moves around it. Be precise.

Centre of Mass theorems still hold.

The only external forces in this particular system are gravity, and the constraint force of the surface on which the gyroscope is sitting.

5. Feb 17, 2006

### Staff: Mentor

You seem to assume that the only force on a precessing gyroscope is gravity, which, of course, acts through the center of mass. But this ignores the force that supports the gyroscope, which does not act through the center of mass and thus exerts a torque. (As others in this thread have already pointed out.)

6. Feb 17, 2006

### jmc8197

But surely the centre of mass traces out a circle as it precesses, the com being in the centre of the fly wheel?

7. Feb 17, 2006

### jmc8197

Let's consider things simply. The forces acting on the centre of mass are gravity, and the reaction of the table to the support. So another torque must come from the fact that it is precessing, or maybe from spinning? or a combination of the two? I never knew that a gyroscope could generate an external force for itself...

8. Feb 17, 2006

### Staff: Mentor

I think your question is: The center of mass of a precessing gyroscope moves in a circle, so what provides the centripetal force?

The answer: The force of the support must provide a centripetal component in order for the gyroscope to precess about its support point.

Good question!