1. Physics 218, Lecture XIII1 Physics 218 Lecture 13 Dr. David Toback

2. Physics 218, Lecture XIII2 Checklist for Today Things due for Last Thursday: –Read Chapters 7, 8 & 9 Things that were due Last Monday: –Chap 5&6 turned in on WebCT Things that were due for Wednesday’s Recitation: –Problems from Chap 7 Things due for this coming Monday: –Problems from Chap 7 on WebCT –Chaps 5&6 if you haven’t done them already

3. Physics 218, Lecture XIII3 The Schedule This week: (2/25) HW on Chaps 5&6 on WebCT 3 rd and 4 th lectures (of six) on Chapters 7, 8 & 9 Chapter 7 in recitation Next week: (3/3) Chapter 7 due in WebCT 5 th and 6 th lectures (of six) on Chapters 7, 8 & 9 Chapter 8 in recitation Following week: (3/10) Spring Break!!! Following Week: (3/17) Chapter 8 due in WebCT Reading for Chapters 10 & 11 Lecture on Chapters 10 & 11 Chapter 9 and Exam 2 Review in recitation Following Week: (3/24) Chapter 9 due in WebCT Exam 2 on Tuesday Recitation on Chapters 10 & 11 Reading for Chapters 12 & 13 for Thursday Lecture 12 & 13 on Thursday

4. Physics 218, Lecture XIII4 Last time: –Work and Energy –The Work-Energy relationship This time and next time: –Potential Energy –Conservation of Mechanical Energy –Conservation of Energy –Lots of problems Chapters 7, 8 & 9 Cont

5. Physics 218, Lecture XIII5

6. 6 Different Style Than the Textbook I like teaching this material using a different style than the textbook 1.Teach you the concepts 2.Give you the important equations 3.Then we’ll do lots of problems

7. Physics 218, Lecture XIII7 Potential Energy Things with potential: COULD do work –“This woman has great potential as an engineer!” Here we kinda mean the same thing E.g. Gravitation potential energy: –If you lift up a brick it has the potential to do damage

8. Physics 218, Lecture XIII8 Example: Gravity & Potential Energy You lift up a brick (at rest) from the ground and then hold it at a height Z How much work has been done on the brick? How much work did you do? If you let it go, how much work will be done by gravity by the time it hits the ground? We say it has potential energy: U=mgZ –Gravitational potential energy

9. Physics 218, Lecture XIII9 Other Potential Energies: Springs Last week we calculated that it took ½kx 2 of work to compress a spring by a distance x How much potential energy does it now how have? U(x) = ½kx 2

10. Physics 218, Lecture XIII10 Force and Potential Energy If we know the potential energy, U, we can find the force This makes sense… For example, the force of gravity points down, but the potential increases as you go up

11. Physics 218, Lecture XIII11 Force and Potential Energy Draw some examples… –Gravity –Spring

12. Physics 218, Lecture XIII12 Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E ≡ K+U

13. Physics 218, Lecture XIII13 Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved (more on this next week) E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick K 2 +U 2 = K 1 +U 1 Conservation of Mechanical Energy E 2 =E 1

14. Physics 218, Lecture XIII14 Problem Solving What are the types of examples we’ll encounter? –Gravity –Things falling –Springs Converting their potential energy into kinetic energy and back again E = K + U = ½mv 2 + mgy

15. Physics 218, Lecture XIII15 Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams

16. Physics 218, Lecture XIII16 Conservation of Energy Problems Before…

17. Physics 218, Lecture XIII17 After

18. Physics 218, Lecture XIII18 Quick Problem We drop a ball from a height D above the ground Using Conservation of Energy, what is the speed just before it hits the ground?

19. Physics 218, Lecture XIII19 Potential Energy A brick held 6 feet in the air has potential energy Subtlety: Gravitational potential energy is relative to somewhere! Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the floor?  U = U 2 -U 1 = W ext = mg (h 2 -h 1 ) Write U = mgh U=mgh + Const Only change in potential energy is really meaningful

20. Physics 218, Lecture XIII20 Z Z Before After C Falling onto a Spring We want to measure the spring constant of a certain spring. We drop a ball of known mass m from a known height Z above the uncompressed spring. Observe it compresses a distance C. What is the spring constant?

21. Physics 218, Lecture XIII21 Quick Problem A refrigerator with mass M and speed V 0 is sliding on a dirty floor with coefficient of friction . Is mechanical energy conserved?

22. Physics 218, Lecture XIII22 Non-Conservative Forces We’ve talked about three different types of forces: 1.Gravity: Conserves mechanical energy 2.Normal Force: Conserves mechanical energy (doesn’t do work) 3.Friction: Doesn’t conserve mechanical energy Since Friction causes us to lose mechanical energy (doesn’t conserve mechanical energy) it is a Non-Conservative force!

23. Physics 218, Lecture XIII23 Law of Conservation of Energy Mechanical Energy NOT always conserved If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc. Energy = Kinetic Energy + Potential Energy + Heat + Others… –Total Energy is what is conserved!

24. Physics 218, Lecture XIII24 Conservative Forces If there are only conservative forces in the problem, then there is conservation of mechanical energy Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another –Good examples: Gravity and Springs Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost. –Good example: Friction (like on Roller Coasters)

25. Physics 218, Lecture XIII25 Law of Conservation of Energy Even if there is friction, Energy is conserved Friction does work –Can turn the energy into heat –Changes the kinetic energy Total Energy = Kinetic Energy + Potential Energy + Heat + Others… –This is what is conserved Can use “lost” mechanical energy to estimate things about friction

26. Physics 218, Lecture XIII26 Roller Coaster with Friction A roller coaster of mass m starts at rest at height y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

27. Physics 218, Lecture XIII27 Energy Summary If there is net work on an object, it changes the kinetic energy of the object (Gravity forces a ball falling from height h to speed up  Work done.) W net =  K If there is a change in the potential energy, some one had to do some work: (Ball falling from height h speeds up → work done → loss of potential energy. I raise a ball up, I do work which turns into potential energy for the ball)  U Total = W Person =-W Gravity

28. Physics 218, Lecture XIII28 Energy Summary If work is done by a non-conservative force it does negative work (slows something down), and we get heat, light, sound etc. E Heat+Light+Sound.. = -W NC If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K 1 +U 1 = K 2 +U 2 +E Heat… K 1 +U 1 = K 2 +U 2 -W NC

29. Physics 218, Lecture XIII29 Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed V o and compresses it a total distance D. Determine 

30. Physics 218, Lecture XIII30 l l Bungee Jump You are standing on a platform high in the air with a bungee cord (spring constant k) strapped to your leg. You have mass m and jump off the platform. 1.How far does the cord stretch, l in the picture? 2.What is the equilibrium point around which you will bounce?

31. Physics 218, Lecture XIII31 Coming up… Lectures: –Last lectures on Chaps 7, 8 and 9 HW due in WebCT on Monday –Chapter 7 Reading for Lecture next week –Chaps 10 & 11: Momentum Recitation next week –Chapter 8

32. Physics 218, Lecture XIII32

33. Physics 218, Lecture XIII33 Roller Coaster You are in a roller coaster car of mass M that starts at the top, height Z, with an initial speed V 0 =0. Assume no friction. a)What is the speed at the bottom? b)How high will it go again? c)Would it go as high if there were friction? Z

34. Physics 218, Lecture XIII34 Energy Potential Energy & Conservation of Energy problems The relationship between potential energy and Force Energy diagrams and Equilibrium

35. Physics 218, Lecture XIII35 Energy Review If there is net work on an object, it changes the kinetic energy of the object (Gravity forces a ball falling from height h to speed up  Work done.) W net =  K If there is a change in the potential energy, some one had to do some work: (Ball falling from height h speeds up → work done → loss of potential energy. I raise a ball up, I do work which turns into potential energy for the ball)  U Total = W Person =-W Gravity

36. Physics 218, Lecture XIII36 Energy Review If work is done by a non-conservative force it is negative work (slows something down), and we get heat, light, sound etc. E Heat+Light+Sound.. = -W NC If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K 1 +U 1 = K 2 +U 2 +E Heat… K 1 +U 1 = K 2 +U 2 -W NC

37. Physics 218, Lecture XIII37 Potential Energy Diagrams For Conservative forces can draw energy diagrams Equilibrium points –Motion will move “around” the equilibrium –If placed there with no energy, will just stay (no force)

38. Physics 218, Lecture XIII38 Stable vs. Unstable Equilibrium Points The force is zero at both maxima and minima but… –If I put a ball with no velocity there would it stay? –What if it had a little bit of velocity?

39. Physics 218, Lecture XIII39 Roller Coaster with Friction A roller coaster car of mass m starts at rest at height y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

40. Physics 218, Lecture XIII40 Roller Coaster with Friction A roller coaster car of mass m starts at rest at height y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). An odometer tells us that the total scalar distance traveled is d. Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path? Assuming that the magnitude and angle of the force of friction, F, between the car and the track is constant, find |F|.

41. Physics 218, Lecture XIII41 Bungee Jump A jumper of mass m sits on a platform attached to a bungee cord with spring constant k. The cord has length l (it doesn’t stretch until it has reached this length). How far does the cord stretch  y? l

42. Physics 218, Lecture XIII42 A football is thrown A 145g football starts at rest and is thrown with a speed of 25m/s. 1.What is the final kinetic energy? 2.How much work was done to reach this velocity? We don’t know the forces exerted by the arm as a function of time, but this allows us to sum them all up to calculate the work

43. Physics 218, Lecture XIII43 Robot Arm A robot arm has a funny Force equation in 1- dimension where F 0 and X 0 are constants. What is the work done to move a block from position X 1 to position X 2 ?