Rotation of a Point Particle

Question

I wonder if there is a meaning of rotation for a point particle. Does a point particle have angular momentum and does he reply to torque?

Answer 1

The one-word answer is "no".

Consider a cube with side $a$ made of material with density $\rho$. The mass of the cube is $\rho a^3$. The moment of inertia about the center is $\rho a^5/6$.

As $a \to 0$, the moment of inertia gets smaller much faster than the mass, because of the $a^5$ factor compared with $a^3$.

Another way to see this is to think about the rotational inertia of a uniform rod of length $l$. You can find the moment of inertia about the center by the "standard" formula, or you can cut the rod into $n$ equal pieces and find the moment of inertia using the parallel axis theorem. If you do that, you find that as $n$ increases, the contribution from the moment of inertia of each piece about its own center is negligible compared with the "$\text{mass}\times\text{length}^2$" contributions from the parallel axis theorem. So as $n \to \infty$ the moment of inertia of each piece (i.e. each "point particle") about its own center is $0$.