Work-Energy Theorem for a Mechanical System

The work-energy theorem for a mechanical system relates the change in total kinetic energy to the work of forces acting on the particles of the system.

Differential form

For each particle, the elementary change in kinetic energy equals the elementary work of the applied forces. Summing over the system gives:

$$dT=\sum dA^{e}+\sum dA^{i},$$

where superscripts $e$ and $i$ denote external and internal forces.

Integral form

Between configurations 1 and 2:

$$T_2-T_1=\sum A_{1\to2}^{e}+\sum A_{1\to2}^{i}.$$

Unlike the momentum and angular-momentum theorems, the work of internal forces is not zero in a general mechanical system.

Internal forces in a rigid body

For an ideal rigid body, internal interaction forces do no net work because distances between body particles remain unchanged. In a deformable system, internal forces may perform work associated with changes in internal or potential energy.

Ideal constraints

Reactions of ideal fixed constraints often do no work when the point of application has no displacement in the reaction direction. This can eliminate unknown reactions from the energy equation.

Application to rigid bodies

First evaluate $T_1$ and $T_2$ using the expression appropriate to translation, rotation, or plane motion, then equate their difference to the total work of forces and couples.

Work of a moment in rotation

For a moment $M_z$ acting during rotation about a fixed axis:

$$A=\int_{\varphi_1}^{\varphi_2}M_z\,d\varphi.$$

For a constant moment, $A=M_z(\varphi_2-\varphi_1)$.

Example

A system starts from rest and the total work of its external and internal forces is 50 J. Its final kinetic energy is therefore 50 J. Determining individual velocities then depends on the type of motion and mass distribution.

Common mistakes

• automatically neglecting work of all internal forces for every mechanical system;
• using the wrong kinetic-energy expression for the type of motion;
• including reactions that do no work or omitting reactions of moving constraints that do work;
• confusing the work of a moment with the moment itself.