Kinetic Energy of a Mechanical System

The kinetic energy of a mechanical system is the sum of the kinetic energies of all its particles. It is a scalar quantity, so the energies of individual parts are added algebraically.

General definition

For a system of $n$ particles:

$$T=\sum_{i=1}^{n}\frac{m_iv_i^2}{2}.$$

System kinetic energy is nonnegative and is measured in joules.

König's theorem

The kinetic energy of a system can be decomposed into the kinetic energy of translational motion of its center of mass and the kinetic energy of motion relative to the center of mass:

$$T=\frac{Mv_C^2}{2}+T_C.$$

This decomposition is especially useful for rigid bodies.

Rigid-body translation

In pure translation, all points have the same velocity, so:

$$T=\frac{Mv_C^2}{2}.$$

Rotation about a fixed axis

If a rigid body rotates with angular velocity $\omega$ about a fixed $z$ axis:

$$T=\frac{I_z\omega^2}{2},$$

where $I_z$ is the mass moment of inertia about the rotation axis.

Plane motion

For plane motion of a rigid body:

$$T=\frac{Mv_C^2}{2}+\frac{I_C\omega^2}{2},$$

where $I_C$ is the mass moment of inertia about the axis through the center of mass perpendicular to the plane of motion.

Example

A 4 kg disk has center-of-mass speed 3 m/s, central mass moment of inertia 0.5 kg·m², and angular speed 4 rad/s. Then $T=4\cdot3^2/2+0.5\cdot4^2/2=18+4=22$ J.

Common mistakes

• using only translational energy for a body that is also rotating;
• using a mass moment of inertia about the wrong axis;
• adding velocities instead of kinetic energies;
• forgetting that relative kinetic energy is zero in pure translation.