Kinetics of Rigid Bodies: Work and Energy

The principle of work and energy for rigid bodies includes rotational kinetic energy and the work done by couples.

Kinetic Energy

The total kinetic energy of a rigid body in general plane motion is the sum of translational and rotational kinetic energy:

$T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$

Translation Only: $\omega = 0 \Rightarrow T = \frac{1}{2} m v^2$
Rotation about Fixed Axis $O$ : $v_G = r_G \omega$ $T = \frac{1}{2} I_O \omega^2$ (Note: Using $I_O$ simplifies the expression).

Work of a Force: Same as particles, $U = \int \mathbf{F} \cdot d\mathbf{r}$ .
Work of a Couple Moment ( $M$ ): $U_M = \int_{\theta_1}^{\theta_2} M \, d\theta$ If $M$ is constant: $U_M = M(\theta_2 - \theta_1)$ .

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