Kinetic energy and work of rigid body

Kinetic energy of a rigid body in plane motion

Let the body have mass $m$, angular velocity $\omega$, and let the mass centre G have velocity $\mathbf{v}_G$. A particle of mass $dm$ at position $\boldsymbol\rho$ relative to G has velocity $\mathbf{v}_G + \boldsymbol\omega\times\boldsymbol\rho$ (velocity of G plus velocity relative to G, magnitude $\omega\rho$).

$$T = \int \tfrac12 |\mathbf{v}_G + \boldsymbol\omega\times\boldsymbol\rho|^2\,dm = \tfrac12 v_G^2\int dm + \mathbf{v}_G\cdot\left(\boldsymbol\omega\times\int\boldsymbol\rho\,dm\right) + \tfrac12\omega^2\int\rho^2\,dm$$

Since G is the mass centre, $\int\boldsymbol\rho\,dm = 0$, so the middle term vanishes, and $\int\rho^2\,dm = I_G$:

$$T = \tfrac12 m v_G^2 + \tfrac12 I_G\,\omega^2$$

Special cases: pure translation $T = \tfrac12mv^2$; rotation about a fixed point O, $T = \tfrac12 I_O\omega^2$; rolling without slipping, $T = \tfrac12 I_C\omega^2$ about the contact point.

Principle of work and energy for a rigid body

$$T_1 + U_{1\to2} = T_2$$

where $U_{1\to2}$ is the work of all external forces and couples (and internal ones that do work, such as springs or friction between connected parts) between positions 1 and 2. For a system of connected bodies, add the kinetic energies of all bodies.

Work of forces and couples

• Force $\mathbf{F}$: $U = \int \mathbf{F}\cdot d\mathbf{r}$ at its point of application; weight: $U = -W\Delta h$; spring: $U = -\tfrac12 k(x_2^2 - x_1^2)$.
• Couple $M$: $U = \int M\,d\theta$, which is $M\Delta\theta$ for constant $M$.

Forces that do no work

• Reactions at smooth fixed pins and smooth surfaces (perpendicular to motion).
• The friction force on a body rolling without slipping, because its point of application (the contact point) is instantaneously at rest, so no energy is dissipated. The normal reaction on a rolling body also does no work.
• Internal forces between particles of a rigid body (equal, opposite, no relative displacement).

If the body slips, the friction force does work $-\mu N\,s$ (energy lost as heat).