This Week

1. This Week • 7/12 Lecture – Chapter 8 • 7/13 Recitation – Bungee • Problems: 8.4, 8.23, 8.44 • 7/14 Lab – Kinematics in 1-D • Homework #4 Due @ 5pm • 7/15 Lecture – Chapter 9 • 7/16 Recitation – ??? • Problems: ???

2. Chapter 8 Potential Energy and Conservation of Energy

3. Kinetic Energy: Work (const force): Work (variable force): Work-KE theorem Review:

4. Potential Energy • What happens to the energy I use when I do work on an object? • Already know about kinetic energy… • In certain circumstances, it is stored as potentialenergy • In the case of non-conservative forces, the energy can be lost (usually as heat)

5. Non-Conservative Forces • Non-conservative if the force does not reverse the energy transfer when the path is reversed • Path dependent! • Examples: • Friction • Air resistance • John Kerry ????

6. Conservative Forces • Conservative force does reverse the energy transfer when the path is reversed • Path independent, work depends only upon position • Examples: • Gravity • Springs

7. Gravitational Potential Energy Recall the work done lifting an object against gravity: Lifting: Dropping: DUgrav = -W = +mgh (if lifting) DUgrav = -W = -mgh (if dropping) d h Note: We get the same energy out as we put in (conservative)

8. Gravitational Potential Energy An easier way to get the sign right… So the gravitational potential energy can be written as: But isn’t the choice of y = 0 arbitrary?

9. Potential Energy from Spring Energy associated with distortion of spring from equilibrium length F x=0 x=xf

10. Important Points About Potential Energy 1. U depends on the position/arrangement of objects DU between two arrangements does not depend on path 2. Only DU has physical meaning -- the numerical value of U itself is arbitrary This means you decide for yourself where U = 0 (like you decide where x = 0)

11. Important Points about Potential Energy 3. U is only defined for "conservative" forces, which do not dissipate energy No U for frictional forces 4. Can rewrite conservation of energy: Old: Kinitial + Wtot = Kfinal New: Kfinal – Kinitial = Wtot = Wc + Wnc But: Wc = -ΔU = Uinitial – Ufinal  Kinitial + Uinitial + Wnc = Kfinal + Ufinal  Einitial + Wnc = Efinal Define E = K + U

12. Conservation of Mechanical Energy If there are no non-conservative forces: Kinitial + Uinitial + Wnc = Kfinal + Ufinal This gives us conservation of mechanical energy: Kinitial + Uinitial = Kfinal + Ufinal Einitial = Efinal

13. vmax ymax E U Energy K vmax = 2gymax y Example: Ball Thrown Upwards (neglect air resistance) E = K + U = ½mv2 + mgy = constant Set U = 0 at floor y = ymax E = U = mgymax y = 0  E = K = ½mvmax2

14. vf = 2gh Example: Pig on a Curved Track No friction Starts from rest How fast is it going at the bottom? h Kinitial + Uinitial = Kfinal + Ufinal Only endpoints matter, curve of track doesn't! Kfinal = Uinitial - Ufinal = - DU Kfinal = mgh ½mvf2 = mgh

15. E U Pendulum: Qualitative View Turning Points q energy K q K can never be negative Motion is bounded

16. L q Pendulum Problem If I release the pendulum at rest from an angle q, how fast is it going at the bottom? Lcosθ L-Lcosθ At bottom: y = 0 (set U = 0 there) At angle q: y = L – Lcos q Kinitial + Uinitial = Kfinal + Ufinal v = 2gL(1 – cos q) mg(L – Lcos q) = ½mv2

17. Traditional political custom: Prague Defenestration n. [Lat., de-,”out of”; fenstra, “window”.]An act of throwing something or someone out of a window. A popular uprising led by the priest Jan Zelivsky included the throwing of the city councilors from the windows of the New Town Hall. 1419 The governors of Bohemia attempted to crush Czech Protestantism. They were thrown from the windows of the council room in Hradcany. This event helped precipitate the Thirty Years War. 1618

18. Potential Energy Example If the royal councilors were given the heave-ho at 5 m/s who is going fastest when they hit the ground? a b c Ei=Ef Ei = Ki+Ui = ½mvi2 + mgh ½mvi2 + mgh = ½mvf2 Ef = Kf + Uf = ½mvf2 vf2 = vi2 + 2gh They all hit the ground with the same velocity!

19. U bounds E1 trapped in well E2 x Graphical Representation of Energy For a closed system: E = K + U = constant Can plot U(x) to see how system evolves Since K cannot be negative, motion is bounded by E

20. Force From Potential Energy -DU  Work done by force In the limit of small Dx, we get: Why the minus sign?

21. U F F F = 0 x dU/dx > 0 F < 0 dU/dx < 0 F > 0 Force From Potential Energy Force pushes in direction to decrease potential energy, i.e., "downhill"

22. R How High to “Loop the Loop”? Ei = mgh = Ef = mg(2R) + ½mv2 Centripetal: v2/R = g (minimum v, barely loops) v mgh = 2mgR + ½mRg h h= 2R + 0.5R = 2.5R (minimum)

23. mk = 0.3 v0 = 4 m/s q = 30 Pig Sliding Up a Frictional Plane How high does he get? Set U = 0 at bottom Kinitial + Uinitial + Wfriction = Kfinal + Ufinal

24. d Example (continued) h Kinitial + Wfriction = Ufinal q= 30 Kinitial = ½mv02 Ufinal = mgh = mgdsinq Wfriction = -|Ff|d = -mkNd = -mk (mgcosq) d ½mv02– mk mgd cosq = mgd sinq

25. d h q= 30 Example (continued)

26. If F•dx is always zero, no work can be done! Constraint Forces The normal force and tension, in many situations, exert forces which are perpendicular to the motion Motion on a fixed surface (e.g. track) Tension from a rope with one end fixed (e.g. pendulum)

27. Constraint Forces No motion in direction of constraint No work No potential energy Example - pole vaulting Originally invented by Dutch farmers

28. Energy in the Pole Vault More detail: pole acts as a spring Pole mgh Kinetic Energy Time

29. Estimating the Record Fast sprinter travels ~10 m/s Vaulter running: Erun=½mv2 At top of motion: Etop=mgh ½mv2=mgh h=v2/2g About 5 m. Record: 6.14 m Sergey Bubka, 1994. Remember-center of mass is about 1m up at start.

30. Another Cultural Moment…

31. Cow-a-pult 400 kg Say that a spring of constant k = 104 N/m is stretched 2 m to launch the cow. What is the max range of the cow if it is released 5m above the ground? Uspring,i+Ugrav,i+Ki = Uspring,f+Ugrav,f+Kf

32. Cow-a-pult (continued) Maximum range when q = 45:

33. d = 6 m Example: q= 30 A 2 kg piglet on rough plane is compressing a spring by x = 0.1 meters and is released from rest on this plane (mk = 0.5). If v = 4 m/s after traveling 6m, what is k of the spring?

34. d = 6 m q= 30 Example (continued) Kinitial + Uinitial + Wfriction = Kfinal + Ufinal Ugravity = 0 at start Uspring = ½kx2 at start Uspring = 0 at end Ugravity = mgd sinq at end Wfriction = -|Ff|d = -mkNd = -mkmgd cosq Kinitial = 0 Kfinal = ½mv2

35. d = 6 m q= 30 Example (continued) Kinitial + Uinitial + Wfriction = Kfinal + Ufinal

36. F x A More Complicated Potential The bow provides a non-constant force to the arrow What is the kinetic energy of the arrow? What is the final speed of the arrow?

37. Arrows of Outrageous Fortune… Area = Uinitial = 50 J Uinitial=Kfinal= ½mv2 An arrow has a mass ~ 0.03 kg…

38. Work Due to Gravity Near the Earth Away from the Earth

39. The End of the World as We Know It?

40. Extinction! 70 Million years ago Dinosaurs ruled the Earth They disappeared at the boundary between the Cretaceous and Tertiary periods (K-T boundary) Luis Alvarez (a Nobel Prize winner in Physics) suggested an asteroid impact might be responsible

41. The Impact Site Alvarez calculated the asteroid would need to be 10 km across and would leave a crater 150 km in diameter A huge crater off the Yucatan peninsula of Mexico has been identified as a possible impact site. Research on this crater has shown it is the result of an extra-terrestrial impact.

42. Assume an asteroid started at rest in the middle of the inner Oort cloud (~5000 RE-S) Assume it is acted on primarily by the Sun Assume mass ~1016 kg (10 km rock) 1 Ton TNT=4109J Asteroid Impact=2109 MT TNT Over 80,000 MPH!

43. 2 1 d = h/sinθ h θ The Ball Race Ball 1 falls straight down, ball 2 rolls down a plane Which reaches the bottom first? Which is traveling fastest at the bottom? Ball 1: h=½gt2 t1= 2h/g Ball 2: a||=g sinθ  t2= 2d/(gsinθ) = 2h/(gsin2θ) = t1/sinθ > t1 Balls 1,2: same Ki=0, Ui=mgh, Uf=0  same Kf  same vf