1. Physics 218, Lecture XII1 Physics 218 Lecture 12 Dr. David Toback

2. Physics 218, Lecture XII2 This week… This week we will finish up Chapters 6 & 7 –Last set of topics for Exam 2 Exam 2: Thurs, October 26 th Covers chapters 1-7

3. Physics 218, Lecture XII3 The Schedule This week: (10/9) Finish up Chapters 6&7 in lecture Chapter 6 in recitation Next week: (10/16) Chapter 6 HW due Chapter 8 in lecture (reading questions due) Chapter 7 in recitation Following week: (10/23) HW 7 due Chapter 9 in lecture on Tuesday (reading questions due) Chapter 8 in Recitation Exam 2 on Thursday October 26 th

4. Physics 218, Lecture XII4 Energy Conservation of Mechanical Energy problems Conservative Forces Conservation of Energy

5. Physics 218, Lecture XII5

6. 6 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

7. Physics 218, Lecture XII7 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

8. Physics 218, Lecture XII8 Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams

9. Physics 218, Lecture XII9 Conservation of Energy Problems Before…

10. Physics 218, Lecture XII10 After

11. Physics 218, Lecture XII11 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?

12. Physics 218, Lecture XII12 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

13. Physics 218, Lecture XII13 Non-Conservative Forces In this problem there are three different types of forces acting: 1.Gravity: Conserves mechanical energy 2.Normal Force: Conserves mechanical energy 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!

14. Physics 218, Lecture XII14 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!

15. Physics 218, Lecture XII15 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)

16. Physics 218, Lecture XII16 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

17. Physics 218, Lecture XII17 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?

18. Physics 218, Lecture XII18 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

19. Physics 218, Lecture XII19 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

20. Physics 218, Lecture XII20 Next time… More problems on Chapters 6 & 7 Recitation on Chapter 6 problems

21. Physics 218, Lecture XII21

22. Physics 218, Lecture XII22 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 ). 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?

23. Physics 218, Lecture XII23 What if the Roller Coaster had Friction? If there were no friction, the roller coaster would go back up to height Z and come to a stop (then come back down again)

24. Physics 218, Lecture XII24 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 energy at the top? b)What is the speed at the bottom? c)How much work is done by gravity in going from the top to the bottom? Z

25. Physics 218, Lecture XII25 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 by an amount D. Determine 

26. Physics 218, Lecture XII26 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