3. Conservative Forces For a force to be conservative, the work it does must be independent of the path taken between the endpoints. The total amount of work done depends only on the endpoints. Example 1: Any force that acts in 1-D and depends only on the position x. Denote the force as f(x). Clearly this depends only on the endpoints. Example 2: A mass changes position near the surface of the earth.

4. Depends only on endpoinjts

5. Potential Energy Formulae Let’s stick to 1-D Only differences in energy can be measured, so we can pick the value of U(a) to cancel Then we have: Essentially an indefinite integral

6. For the spring we will always use x = 0 as the reference point, although in principle we could use other points.. For gravity we can choose the y = 0 point wherever convenient.

7. Force from Potential Energy We can find the various components of the force by taking the appropriate derivative.

8. Exercise: A 3-D potential energy is given by: Find F x at the point (1, 2, 4).

9. Graphs of Potential Energy Given the total energy and a graph of U vs x, one can qualitatively describe the types of motions available to the system E Turning points U K Negative slope of curve is force

10. Exercise: Describe possible motions for different total energies. Find force at x = 1 m