3.18 Rotational Kinetic Energy

The simplest model of physical matter is the BODY which is an approximation of the selected matter as being localized at a point, that is, to have mass but no extent. For mass without extent, rotation makes no sense. Otherwise stated, a BODY does not rotate.

Our next model, a step more complex than the BODY, will consist of two identical masses. We assume the masses are locked side-by-side, and rigid at a constant distance from each other. Each mass is a "BODY" but they are usually called PARTICLES and their configuration is assumed rigid.

Basic Thermodynamics ~ J. Pohl ©

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4.18 Rotational Kinetic Energy

The simplest model of physical matter is the BODY which is an approximation of the selected matter as being localized at a point, that is, to have mass but no extent. For mass without extent, rotation makes no sense. Otherwise stated, a BODY does not rotate.

Our next model, a step more complex than the BODY, will consist of two identical masses. We assume the masses are locked side-by-side, and rigid at a constant distance from each other. Each mass is a "BODY" but they are usually called PARTICLES and their configuration is assumed rigid.

Two-Particle Rigid Body  This is the simplest model (approximation of physical matter) for which rotation is meaningful.

The simplest model of "RIGID BODY" is the idea of matter (a system) composed of two BODIES (or particles) of equal masses located close together in a rigid arrangement. Some call these "dumbbell molecules" since they resemble weight lifter equipment. The BODIES (particle pair) have a center of mass but each mass is frozen at a fixed distance from the other.

While neither BODY moves toward or away from the other, the BODIES move. We take the center of mass as a reference. Rigid too-particle BODIES exhibit rotational behavior and have rotational energy. Let us forgo how force is applied to a rigid body pair. The BODIES are named "A" and "B" and have (at the instant shown) the vector positions and velocities.

In these expressions, e_AB(t) is a time dependent unit vector. It suits our purposes to restrict this discussion to an "in a plane" rotation which will show the results more readily.

For the two BODIES the kinetic energy is written as:

(1) 1

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(2) 2

The velocity of BODY A equals the velocity of the center of mass plus the velocity of itself relative to the center of mass.

(3) 20

The length of the vector from the center of mass of the pair to either of them is l. That length times the rotation rate of each BODY with respect to the c.m. equals its velocity relative to the c.m. Next write these velocities into the kinetic energy expression.

(3) 3

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(4) 4

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(5) 5

Of course, the final conclusion is that rigid bodies have both Translational Kinetic Energy and Rotational Kinetic Energy.

RIGID BODY: It is important to realize the results above could be extended to three rigid particles, four and so on. Admittedly the vector algebra would become tedious. So long as the idea RIGID suits the physical event modeled, the mass and moment of inertia are sufficient properties for the translational and rotational kinetic energies to be specified. More will be stated in this regard later.

Real masses are distributed, of course. Here we consider a two-particle rigid body as a more detailed (and consequently more complicated) model of physical matter; just one step more complicated that a particle.

In having mass at a point. Such systems can possess translation kinetic energy. But since they have no extent, but having no "extent," rotation they since the particle