1. Physics 218, Lecture XIV1 Physics 218 Lecture 14 Dr. David Toback

2. Physics 218, Lecture XIV2 Notes Exam Next Tuesday: –Here. Usual class time: Covering: Exam 1 material Chapter 3(9) Chapter 4(1-8) Chapter 5(1-3) Chapter 6(1-8) Chapter 7(1-4) Calculus 2 Today: Chapter 8 NOT ON EXAM

3. Physics 218, Lecture XIV3

4. 4 Conservation of Energy Potential Energy Mechanical Energy Conservation of Mechanical Energy Conservation of Energy

5. Physics 218, Lecture XIV5 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

6. Physics 218, Lecture XIV6 Example: Gravity & Potential Energy If we lift up a brick to a height h, how much work is done on the brick? If we lift up a brick to a height h, how much work are we doing? If we let it go, how much work will be done by gravity by the time it hits the ground? We say it has a potential energy: U=mgh –It has gravitational potential energy

7. Physics 218, Lecture XIV7 Potential Energy Subtlety: 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

8. Physics 218, Lecture XIV8 Roller Coaster A Roller Coaster of mass M=1000kg starts at point A. We set Y(A)=0. What is the potential energy at height A, U(A)? What about at B and C? What is the change in potential energy as we go from B to C? If we set Y(C)=0, then what is the potential energy at A, B and C? Change from B to C

9. Physics 218, Lecture XIV9 Other Potential Energies: Springs How much work do we exert 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 XIV10 Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E≡K+U

11. Physics 218, Lecture XIV11 Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved In other words, at time t 1 the mechanical energy is the same as at some later time t 2 –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

12. Physics 218, Lecture XIV12 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

13. Physics 218, Lecture XIV13 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 h above the uncompressed spring and observe that it compresses a distance Y. What is k?

14. Physics 218, Lecture XIV14 Roller Coaster II A Roller Coaster of mass M starts at the top, height h, with a speed of V 0 =0. 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? d)At what height is it at half the max speed? h

15. Physics 218, Lecture XIV15 What if the Roller Coaster had Friction? If there were no friction, the roller coaster would go back up to height h and come to a stop (then come back down again). Would it go as high, if there were friction? Three different types of forces acting: –Gravity: Conserves mechanical energy –Normal Force: Conserves mechanical energy –Friction: Doesn’t conserve mechanical energy Friction is a Non-Conservative force!

16. Physics 218, Lecture XIV16 Law of Conservation of Energy What about non-conservative forces? Is Energy still conserved? Friction turns the energy into heat E = Kinetic Energy + Potential Energy + Heat + Others… –This is what is really conserved!

17. Physics 218, Lecture XIV17 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

18. Physics 218, Lecture XIV18 Thursday No exam! Exam is next Tuesday! Next time: More on Energy. –Don’t forget we’ve switched the schedules so you need to do the reading that was originally scheduled for next week by Thursday

19. Physics 218, Lecture XIV19 Example: Gravity Work by Gravity

20. Physics 218, Lecture XIV20 Potential Energy in General Is the potential energy always equal to the work done on the object? –No, non-conservative forces –Other cases? What about for conservative forces?

21. Physics 218, Lecture XIV21 Force and Potential Energy Derivation Example

22. Physics 218, Lecture XIV22 Water Slide Who hits the bottom with a faster speed?

23. Physics 218, Lecture XIV23 Mechanical Energy Consider a Conservative System W net =  K (work done ON an object)  U Total = -W net Combine  K = W net = -  U Total =>  K +  U = 0 Conservation of Energy

24. Physics 218, Lecture XIV24 Conservation of Energy Define E=K+U  K +  U = 0 => (K 2 -K 1 ) +(U 2 -U 1 )=0 K 2 +U 2 = K 1 +U 1 Conservation of Mechanical Energy E 2 =E 1

25. Physics 218, Lecture XIV25 Conservative vs. Non-Conservative Forces Nature likes to “conserve” certain types of things Keep them the same Kinda like conservative politicians Conservationists

26. Physics 218, Lecture XIV26 Conservative Forces Physics has the same meaning. Except nature ENFORCES the conservation. It’s not optional, or to be fought for. “A force is conservative if the work done by a force on an object moving from one point to another point depends only on the initial and final positions and is independent of the particular path taken” (We’ll see why we use this definition later)

27. Physics 218, Lecture XIV27 Closed Loops Another definition: A force is conservative if the net work done by the force on an object moving around any closed path is zero This definition and the previous one give the same answer. Why?

28. Physics 218, Lecture XIV28 Is Friction a Conservative Force?