11.7: Summary
In this chapter we have extended the energy methods from Ch. 7 to rigid bodies. It was derived how moments contribute to the work on the rigid body and how the angular velocity of rigid bodies increases their kinetic energy. By including these effects, the principle of work and energy and the energy conservation concepts can also be applied to systems that include rigid bodies.
- Work of an external force on a rigid body
\[W_{B, 1 \rightarrow 2}=\int_{\overrightarrow{\boldsymbol{r}}_{1}}^{\overrightarrow{\boldsymbol{r}}_{2}} \overrightarrow{\boldsymbol{F}}_{B, \mathrm{ext}} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}} \tag{11.42} \label{11.42}\]
- Internal forces in a rigid body do not perform work.
- Work of a couple on a rigid body
\[W_{\text {couple }, 1 \rightarrow 2}=\int_{\phi_{1}}^{\phi_{2}} \overrightarrow{\boldsymbol{M}}_{\text {couple }} \cdot \mathrm{d} \overrightarrow{\boldsymbol{\phi}} \tag{11.43} \label{11.43}\]
- Potential energy of a couple
\[V_{\text {couple }}=\frac{1}{2} \kappa \phi^{2} \tag{11.44} \label{11.44}\]
- Kinetic energy of a rigid body
\[\begin{align} T_{C, 2 D} & =T_{C, \text { trans }}+T_{C, \text { rot }, 2 D} \tag{11.45} \label{11.45}\\[4pt] T_{C, \text { trans }} & =\frac{1}{2} m_{\mathrm{tot}}\left|\overrightarrow{\boldsymbol{v}}_{G}\right|^{2} \tag{11.46} \label{11.46}\\[4pt] T_{C, \text { rot }, 2 D} & =\frac{1}{2} I_{G, z z} \omega_{C}^{2} \tag{11.47} \label{11.47}\end{align}\]
- Principle of work and energy
\[W_{\mathrm{tot}}=\Delta T_{\mathrm{tot}} \tag{11.48} \label{11.48}\]
- Conservation of energy when only conservative forces act
\[\Delta V_{\mathrm{tot}}+\Delta T_{\mathrm{tot}}=0 \tag{11.49} \label{11.49}\]