7.5: Summary
In this chapter we have introduced the concepts of work, kinetic energy and potential energy and shown how they can provide a simpler route for solving certain problems in dynamics. The law of conservation of energy provides insight in the source of forces, and shows that if work is done, this drains other energy resources. We summarise the most important concepts and equations from this chapter:
- Work and energy
- Work: \(W_{12}=\int_{s_{1}}^{s_{2}} F_{t} \mathrm{~d} s=\int_{\overrightarrow{\boldsymbol{r}}_{1}}^{\overrightarrow{\boldsymbol{r}}_{2}} \overrightarrow{\boldsymbol{F}} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}}\)
- Kinetic energy: \(T=\frac{1}{2} m v^{2}\)
- Principle of work and energy: \(W_{12}=\Delta T\)
- Conservation of energy
- A force \(\overrightarrow{\boldsymbol{F}}_{c}(\overrightarrow{\boldsymbol{r}})\) is conservative if it does not perform work along a closed path.
- For a system on which only conservative forces act, the internal energy is conserved: \(U=T+V=\) constant.
- The change in potential energy is the negative of the work done by a conservative force: \(\Delta V=-W_{12}\).
\(-\overrightarrow{\boldsymbol{F}}_{c}=-\overrightarrow{\boldsymbol{\nabla}} V\)
\(-\Delta T+\Delta V+\Delta Q+\Delta W_{\text {ext }}=0\)
- Power: \(P=\frac{\mathrm{d} W}{\mathrm{~d} t}=\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{v}}\)
- Efficiency \(\eta=\frac{E_{\text {in }}}{E_{\text {out }}} \times 100 \%=\frac{P_{\text {in }}}{P_{\text {out }}} \times 100 \%\)
- Potential energy expressions
\(-V_{g}=m g y+C\)
\(-V_{k}=\frac{1}{2} k x^{2}\)
- Normal contact forces are conservative.
- Tangential (friction) contact forces are non-conservative.
- The centre of mass and gravity: \(\overrightarrow{\boldsymbol{r}}_{G}=\frac{1}{m} \sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i}=\int_{V} \overrightarrow{\boldsymbol{r}} \rho \mathrm{d} V\)