Work & Energy of Rigid Bodies

The Work-Energy Principle extends to rigid bodies by accounting for rotational kinetic energy and the work done by moments/couples.
$T_1+\sum U_{1\to 2}=T_2$

Using work-energy equations allows us to ignore vector force/acceleration components and allows fast calculation of speed at a given position.

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Kinetic Energy of a Rigid Body

For general plane motion, kinetic energy has both a translational and rotational component:
$T=\frac{1}{2}m{\mathbf{v}}_G^2+\frac{1}{2}{I}_G{\omega }^2$

Special Cases

Translation only ($\omega =0$): $T=\frac{1}{2}m{\mathbf{v}}_G^2$

Rotation about a fixed axis $(O)$: $T=\frac{1}{2}{I}_O{\omega }^2$ (equivalent to combined form via parallel axis theorem).

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Work of a Force

A force only does work if its point of application moves in the direction of the force. For a rigid body, a force’s point of application may not be $G$, so the displacement must be measured wherever the force acts.

Concept bq

A force does zero work if its point of application does not move, or moves perpendicular to the force.

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Work of a Couple Moment

A couple moment $M$ does work as the body rotates through an angle:
$U_M=\int Md\theta$
For a constant moment:
$U_M=M\Delta \theta$

Comment bq

$U_M$ is $+\text{ve}$ when $M$ and $\Delta \theta$ act in the same rotational direction.

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