1. Short Version :7. Conservation of Energy

2. 7.1. Conservative & Non-conservative Forces F is conservative if for every closed path C. is path-independent i.e., is path-dependent  W AB W BA + W AB = 0  W AB =  W BA = W AB F is non-conservative if there is a closed path C such that W BA W AB Mathematica

3. Example: Work done on climber by gravity Going up: W 1 = (  m g ) h =  m g h Going down: W 2 = (  m g ) (  h) = m g h Round trip: W = W 1 + W 2 = 0 Horizontal displacement requires no work. Gravity is conservative.

4. Example: Work done on trunk by friction Going right: W 1 = (   m g ) L =   m g L Going left: W 2 = (  m g ) (  L) =   m g L Round trip: W = W 1 + W 2 =  2  m g L  0 Friction is non-conservative.

5. GOT IT? 7.1. If it takes the same amount of work to push a trunk across a rough floor as it does to lift a weight to the same distance straight upward. How do the amounts of work compare if the trunk & weight are moved along curved paths between the same starting & end points? Ans. Work is greater for the trunk.

6. 7.2. Potential Energy Conservative force: Potential energy = stored work =  ( work done by force ) Note: only difference of potential energy matters. 1-D case: Constant F:

7. Gravitational Potential Energy Horizontal component of path does not contribute. Vertical lift:  m g

8. Elastic Potential Energy Ideal spring :  parabolic x  x 0 x = x 0 x  x 0 U is always positive x 0 = equilibrium position Let Setting x 0 = 0 :

9. 7.3. Conservation of Mechanical Energy Mechanical energy: Law of Conservation of Mechanical Energy : if ( no non-conservative forces )

10. Example 7.5. Spring & Gravity A 50-g block is placed against a spring at the bottom of a frictionless slope. The spring has k = 140 N/m and is compressed 11 cm. When the block is released, how high up the slope does it rise? Initial state: Final state: 

11. Example 7.6. Sliding Block A block of mass m is launched from a spring of constant k that is compressed a distance x 0. The block then slides on a horizontal surface of frictional coefficient . How far does the block slide before coming to rest? Initial state: Work done against friction:  Final state: Launch: Conservation of energy :

12. 7.4. Potential Energy Curves Frictionless roller-coaster track How fast must a car be coasting at point A if it’s to reach point D? Criterion: turning points potential barrier potential well

13. Example 7.7. H 2 Near the bottom of the potential well of H 2, U = U 0 + a ( x  x 0 ) 2, where U 0 =  0.760 aJ, a = 286 aJ / nm 2, x 0 = 0.0741 nm. ( 1 aJ = 10  18 J ) What range of atomic separation is allowed if the total energy is  0.717 aJ? Turning points: 

14. Force & Potential Energy Force ~ slope of potential curve ( x along direction of F )

15. Gaussian Gun 1 2 1 1 2 Assume fields of the induced dipoles negligible compared to that of the magnet. Video