# Difference between rotational and translational KE for a point particle on disk spinning around a stationary axis

My textbook states that a particle spinning around a stationary axis has rotational kinetic energy only. For a rolling object, KE consists of both translational and rotational kinetic energy, which means that they describe different quantities of motion. For a single particle in a disk spinning around a stationary axis, why doesn't the particle have an expression for translational kinetic energy?

I think you (or the textbook author) is mixing up the ideas of a point particle and a rigid body.

A point particle only has translational KE. Since its "radius of a point particle" is zero by definition, there is no way to tell (in Newtonian mechanics) if it is rotating or not.

For a rigid body which is not a point, the total KE is the sum of the KEs of every point particle in the body. However, you can split the KE into two parts: (1) the KE of a point particle with the total mass of the body, at the center of mass of the body and (2) the translational KE of the particles moving relative to the center of mass.

So the rotational KE of a rigid body is really part of the translational KE of all the point particles in the body.

If the center of mass of the rigid body is not moving, the rotational KE of the body is the total translational KE of all the point particles, since part (1) defined above is zero.

For a rigid body rotating about a fixed axis, the rotational energy is the sum of all of the tangential kinetic energies of all of the particles in the body. For your point particle, you can talk about the energy associated with its rotation about the axis, or its translational kinetic energy. They are both the same energy.