Physics Mechanics. Lecture 18 Energy Conservation I

Transcription

1 Physics Mechanics Lecture 18 Energy Conservation I 1

2 Conservation of Mechanical Energy Definition of mechanical energy: Using this definition and considering only conservative forces, we find: Or equivalently: 2

3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve: 3

4 Conservation of Mechanical Energy 4

5 Energy and Kinematics Conservation of Energy: x2 5

6 Speed and Path Energy is a scalar. The speed of the cap is v i at height y i and its speed is v f at height y f, 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 v i is large enough to carry the cap to height y f. 6

7 Example: Graduation Fling At the end of a graduation ceremony, the graduates fling their caps into the air. Suppose a 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. 7

8 Example: Catching a Home Run At the bottom of the 9 th inning, a player hits a 0.15 kg baseball over the outfield fence. The ball leaves the bat with a speed of 36.0 m/s and a fan in the bleachers catches it 7.2 m above the point where it was hit. Neglect air resistance. (a) What is the kinetic energy K f of the ball when caught? (b) What is the speed v f of the ball when caught. 8

9 Basic Energy Model 1. 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. 2. Kinetic energy can be transformed into potential energy, and potential energy can be transformed into kinetic energy. 3. Under some circumstances, the mechanical energy E mech = 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 E mech conserved? Q2: What happens to the energy when E mech is not conserved? 9

10 Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, 10

11 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 W nc = 5,120 J, what is the depth d? 11

12 Example: Judging a Putt A golfer badly misjudges a putt, giving the ball an initial speed v 1, which sends the ball a distance d that is only one quarter of the distance to the hole. 1 If the nonconservative force F due to the resistance of the grass is constant, what initial speed v 2 would have been needed to putt the ball from its initial position to the hole? 12

13 Example: Landing with a Thud A block of mass m 1 = 2.40 kg is on a horizontal table with a coeff. of friction µ k = 0.450, and is connected to a hanging block of mass m 2 = 1.80 kg. When the blocks are released, they move a distance d = 0.50 m, and then m 2 hits the floor. Find the speed of the blocks just before m 2 hits the floor. 13

14 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 W nc1 = +18,000 J, air resistance has done work W nc2 = 4420 J, and the jogger s speed is 3.50 m/s. Find the height of the hill. 14

15 Potential Energy Curves and Equipotentials The potential energy U and kinetic energy K add to the total energy E 0 (dashed line) at all x values. K vanishes at A and D, which are the turning points of the motion. 15

16 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? 16

17 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve: Highest potential energy Lowest potential energy 17

18 Strategy: Conservation of 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. 18