Physics 114A - Mechanics Lecture 19 (Walker: Ch. 8.3-5) Energy Conservation I February 11, 2014

1. Physics 114A - MechanicsLecture 19 (Walker: Ch. 8.3-5)Energy Conservation IFebruary 11, 2014 John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu

2. Announcements • HW#5 is due at 11:59 PM on Thursday, February 13.HW#6 is due at 11:59 PM on Thursday, February 20. • Clicker scores as of Friday (02/07/2014) have been posted on Catalyst. • We will have Exam 2 on Friday, February 14. (Happy Valentine’s day!) On Thursday, February 13 we will have a pre-exam review. Bring questions.Exam 2 will cover Chapters 5-8 and will be similar to Exam 1 in its structure. Bring a Scantron sheet and a calculator with good batteries. • There will again be assigned seating. If you have not already done so and would like to request a left-handed seat, right-handed aisle seat, or front row seat, E-mail your request to me ASAP. Physics 114A - Lecture 19

3. Lecture Schedule (Part 2) We are here. Physics 114A - Lecture 19

4. Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently: Physics 114A - Lecture 19

5. Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve: Physics 114A - Lecture 19

6. Conservation ofMechanical Energy Physics 114A - Lecture 19

7. Strategy: Conservationof Mechanical Energy • MODEL: Choose a system without friction or other losses of mechanical energy. • VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you’re trying to find. • SOLVE: The mathematical representation is based on the law of conservation of mechanical energy. • ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Physics 114A - Lecture 19

8. Example: Graduation Fling At the end of a graduation ceremony, the graduates fling their caps into the air. Suppose a 0.120 kg cap is thrown straight upward with a speed of 7.85 m/s and that frictional forces can be ignored. (a) Use kinematics to find the speed of the cap when it has risen 1.18 m above the fling point. (b) Show that the total mechanical energy of the cap is unchanged. Physics 114A - Lecture 19

9. x2 Energy and Kinematics Conservation of Energy: Physics 114A - Lecture 19

10. Example: Catching a Home Run At the bottom of the 9th inning,a player hits a 0.15 kg baseball overthe outfield fence. The ball leaves the bat with a speed of 36.0 m/sand a fan in the bleachers catchesit 7.2 m above the point where itwas hit. Neglect air resistance. (a) What is the kinetic energy Kf of the ball when caught? (b) What is the speed vf of the ball when caught. Physics 114A - Lecture 19

11. Speed and Path Energy is a scalar. The speed of the cap is vi at height yi and its speed is vf at height yf, independent of the path between the two heights. Thus the angle at which the cap is launched does not change this result, as long a vi is large enough to carry the cap to height yf. Physics 114A - Lecture 19

12. m 2m h Clicker Question 1 When a ball of mass m is dropped from height h, its kinetic energy just before landing is K. If a 2nd ball of mass 2m is dropped from height h/2, what is its kinetic energy just before landing? (a) K/4 (b) K/2 (c) K (d) 2K (e) 4K Physics 114A - Lecture 19

13. Divers and Energy Conservation Question: Which diver hits the water with the greatest speed? Same U and K Same U and K Physics 114A - Lecture 19

14. Basic Energy Model • There are (at least) two kinds of energy, the kinetic energy K associated with motion of a particle and the potential energy U associated with its position . • Kinetic energy can be transformed into potential energy, and potential energy can be transformed into kinetic energy. • Under some circumstances, the mechanical energy Emech = K + U is a conserved quantity. Its value at the end of a process is the same as at the beginning. (Energy loss»0) Q1: Under what circumstances is Emech conserved? Q2: What happens to the energy when Emech is not conserved? Q3: How do you calculate U for forces other than gravity? Physics 114A - Lecture 19

15. Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9) Note: The signs here are tricky. Study p. 235 until you understand them. Physics 114A - Lecture 19

16. Example: Find the Diver’s Depth A 95.0 kg diver steps off a diving board and drops into the water, 3.00 m below. At some depth d below the water’s surface, the diver comes to rest. If the nonconservative work done on the diver is Wnc =-5,120 J, what is the depth d? Physics 114A - Lecture 19

17. 1 Example: Judging a Putt A golfer badly misjudges a putt, giving the ball an initial speed v1, which sends the ball a distance d that is only one quarter of the distance to the hole. If the nonconservative force F due to the resistance of the grass is constant, what initial speed v2 would have been needed to putt the ball from its initial position to the hole? Physics 114A - Lecture 19

18. Example: Landing with a Thud A block of mass m1 = 2.40 kg is on a horizontal table with a coefficient of friction mk = 0.450 between them and is connected to a hanging block of mass m2= 1.80 kg as shown. When the blocks are released, they move a distance d = 0.50 m, and then m2 hits the floor. Find the speed of the blocks just before m2 hits. Physics 114A - Lecture 19

19. Example: Marathon Man An 80.0 kg jogger starts from rest and runs uphill into a stiff breeze. At the top of the hill the jogger has done work Wnc1 = +18,000 J, air resistance has done workWnc2 = -4420 J, and thejogger’s speed is 3.50 m/s. Find the height of the hill. Physics 114A - Lecture 19

20. Potential Energy Curvesand Equipotentials The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy: Physics 114A - Lecture 19

21. Potential Energy Curve The potential energy U and kinetic energy K add to the total energy E0 (dashed line) at all x values. K vanishes at A and B, which are the turning points of the motion. Physics 114A - Lecture 19

22. Total mechanical energy Potential Energy Curvesand Equipotentials The potential energy curve for a mass and spring: Physics 114A - Lecture 19

23. Example: A Potential Problem A 1.60 kg object in a conservative system moves along the x axis, where the potential energy is as shown. A physical example would be a bead sliding along a wire shaped like the red curve. If the object’s speed at x = 0 is 2.30 m/s, what is its speed at x = 2.00 m? Physics 114A - Lecture 19

24. Highestpotentialenergy Lowestpotentialenergy Potential Energy Curvesand Equipotentials Contour maps are also a form of potential energy curve: Physics 114A - Lecture 19

25. End of Lecture 19 • Before next Tuesday, read Walker Chapter 9.1-3. • HW#5 is due at 11:59 PM on Thursday, February 13. HW#6 is due at 11:59 PM on Thursday, February 20. • We will have Exam 2 on Friday, February 14. It will cover Chapters 5-8 and will be similar to Exam 1 in its structure. Bring a Scantron sheet, a straight-edge, and an calculator with good batteries. • There will again be assigned seating. If you have not already done so and would like to request a left-handed seat, right-handed aisle seat, or front row seat, E-mail your request to me ASAP. Physics 114A - Lecture 19