Work and energy for one-dimensional systems For one-dimensional motion work is path independent. 7. One-dimensional systems This follows from the fact that It is possible to introduce potential energy: 1 Equilibrium: Stable equilibrium: Unstable equilibrium:

2 Example (Stability of a cube balanced on a cylinder): If r>b then equilibrium is stable If r<b then equilibrium is unstable h 2b r

3 Integration of equation of motion in one-dimension Two arbitrary constants in this solutions are energy E and initial time t 0 Limits of motion (turning points): A finite motion in 1-D is oscillatory – motion between two turning points. Period:

4 Example (Period of oscillation of a simple pendulum): l Complete elliptic integral of the first kind

5 Spherical coordinates

6 Central forces Definition: Example: A central force that is conservative is spherically symmetric A central force that is spherically symmetric is conservative