8 Potential Energy and Conservation of Energy
8 Chapter Review
Key Terms
- conservative force
- force that does work independent of path
- conserved quantity
- one that cannot be created or destroyed, but may be transformed between different forms of itself
- energy conservation
- total energy of an isolated system is constant
- equilibrium point
- position where the assumed conservative, net force on a particle, given by the slope of its potential energy curve, is zero
- exact differential
- is the total differential of a function and requires the use of partial derivatives if the function involves more than one dimension
- mechanical energy
- sum of the kinetic and potential energies
- non-conservative force
- force that does work that depends on path
- non-renewable
- energy source that is not renewable, but is depleted by human consumption
- potential energy
- function of position, energy possessed by an object relative to the system considered
- potential energy diagram
- graph of a particle’s potential energy as a function of position
- potential energy difference
- negative of the work done acting between two points in space
- renewable
- energy source that is replenished by natural processes, over human time scales
- turning point
- position where the velocity of a particle, in one-dimensional motion, changes sign
Key Equations
| Difference of potential energy | [latex]\Delta {U}_{AB}={U}_{B}-{U}_{A}=\text{−}{W}_{AB}[/latex] |
| Potential energy with respect to zero of
potential energy at |
[latex]{\mathbf{\overset{\to }{r}}}_{0}\Delta U=U(\mathbf{\overset{\to }{r}})-U({\mathbf{\overset{\to }{r}}}_{0})[/latex] |
| Gravitational potential energy near Earth’s surface | [latex]U(y)=mgy+\text{const}.[/latex] |
| Potential energy for an ideal spring | [latex]U(x)=\frac{1}{2}k{x}^{2}+\text{const}.[/latex] |
| Work done by conservative force over a closed path | [latex]{W}_{\text{closed path}}=\oint {\mathbf{\overset{\to }{E}}}_{\text{cons}}\cdot d\mathbf{\overset{\to }{r}}=0[/latex] |
| Condition for conservative force in two dimensions | [latex](\frac{d{F}_{x}}{dy})=(\frac{d{F}_{y}}{dx})[/latex] |
| Conservative force is the negative derivative of potential energy | [latex]{F}_{l}=-\frac{dU}{dl}[/latex] |
| Conservation of energy with no non-conservative forces |
[latex]0={W}_{nc,AB}=\Delta {(K+U)}_{AB}=\Delta {E}_{AB}.[/latex] |
Summary
8.1 Potential Energy of a System
- For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.
- Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.
- The potential energies for Earth’s constant gravity, near its surface, and for a Hooke’s law force are linear and quadratic functions of position, respectively.
8.2 Conservative and Non-Conservative Forces
- A conservative force is one for which the