7. Conservation of Energy

1. 7. Conservation of Energy Conservative & Non-conservative Forces Potential Energy Conservation of Mechanical Energy Potential Energy Curves

2. How many different energy conversions take place as the Yellowstone River plunges over Yellowstone Falls? P.E.  K.E., sound, & heat

3. Work against conservative force (path independent), e.g., gravity: Work can be stored  potential energy. Work against non-conservative force (path dependent), e.g., friction: Work is dissipated  heat& entropy.

4. 7.1. Conservative & Non-conservative Forces F is conservative if for every closed path C. WBA+WAB = 0  WAB =  WBA = WAB WAB i.e., is path-independent WBA WAB F is non-conservative if there is a closed path C such that  is path-dependent

5. Example: Work done on climber by gravity Going up: W1 = ( m g ) h =  m g h Going down: W2 = ( m g ) ( h) = m g h Round trip: W = W1 + W2 = 0 Gravity is conservative.

6. Example: Work done on trunk by friction Going right: W1 = ( m g ) L =  m g L Going left: W2 = (  m g ) ( L) =   m g L Round trip: W = W1 + W2 =  2  m g L  0 Friction is non-conservative.

7. 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.

8. 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:

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

10. Example 7.1. Riding the Elevator • A 55 kg engineer takes an elevator from her office on the 33rd floor to the 59th floor. • Later, she descends to street level. • If she takes her office as the zero of potential energy, • and if the distance between floors is 3.5 m, • what’s her potential energy • in her office. • on the 59th floor. • at street level ? 59 33 1 (a) (b) (c)

11. Application: Pumped Storage Excess electric energy stored by pumping water to higher ground. (see Prob. 29) Northfield Mountain Pumped Storage Project, MA, USA

12. Elastic Potential Energy x0 = eqilibrium position Ideal spring: Let  parabolic U is always positive Setting x0 = 0 : x x0 x = x0 x x0

13. Example 7.2. Springs vs Gasoline A car’s suspension consists of springs with overall effective k = 120 kN/m. How much these springs need be compressed to store the same amount of energy as in 1 gram of gasoline? From Appendix C: Energy contents of gasoline = 44 MJ / kg. Springs can’t compete with gasoline as energy source.

14. Example 7.3. Climbing Rope Climbing ropes are springy to cushion falls. Consider rope with F =  k x + b x2, where k = 223 N/m, b = 4.10 N/m2. Find the potential energy when it’s stretched 2.62 m, taking U = 0 at x = 0. which is  3% less than U = ½ k x2 = 765 J.

15. 7.3. Conservation of Mechanical Energy Mechanical energy: Law of Conservation of Mechanical Energy: ( no non-conservative forces ) if

16. Example 7.4. Tranquilizing an Elephant A biologist uses a spring-loaded gun to shoot tranquilizer darts into an elephant. The gun’s spring has k = 940 N/m, & is compressed x = 25 cm before firing a 38-g dart. Assuming the gun points horizontally, at what speed does the dart leave the gun? Initial state: Final state:  Problem is harder to solve by 2nd law.

17. 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: 

18. GOT IT? 7.3. Bowling Ball A bowling ball is tied to the end of a long rope and suspended from the ceiling. A student holds the ball to her nose, then releases it from rest. Should she duck as it swings back?

19. Example 7.6. Sliding Block A block of mass m is launched from a spring of constant k that is compressed a distance x0. The block then slides on a horizontal surface of frictional coefficient . How far does the block slide before coming to rest? Initial state: Launch: Work done against friction: Final state:  Conservation of energy :

20. 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? turning points Criterion: potential barrier potential well

21. Example 7.7. H2 Near the bottom of the potential well of H2, U = U0 + a ( x x0 )2 , where U0= 0.760 aJ, a = 286 aJ / nm2 , x0 = 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: 

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

23. GOT IT?. 7.4. • Below is the potential energy curve for an electron in a microelectronic device. • Find the point where the force on the electron is greatest. • Find the rightmost position possible if the electron has total energy E1. • Find the leftmost position possible if the electron has total energy E2& starts out to the right of D. • Find a point where the force on the electron is zero. • Find a point where the force on the electron points to the left. • In some cases, there may be multiple answers. (B) (E) (C) (A,D) (B,E)