Chapter 8 Potential Energy and Conservation of Energy. Copyright Dr. Weining Man and Pearson

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1 Chapter 8 Potential Energy and Conservation of Energy

2 Units of Chapter 8 Conservative and Nonconservative Forces Potential Energy and the Work Done by Conservative Forces Conservation of Mechanical Energy Work Done by Nonconservative Forces Potential Energy Curves and Equipotentials

3 Kinetic Energy is like your checking account. W external, W push, pull, wind, man, wave, elevator, W mg Kinetic Energy K = ½mv 2 W spring W kinetic-friction = f k d

4 What s special about work done by gravity? From height h to ground, For the red ball, W mg = mgh For the green ball, W mg = mg d cosθ = mgh For the blue ball, after a complicated path, W mg is also = mgh Work done by gravity is only determined by the initial and final height.

5 When you lift a box and place it at height h, you did work mgh, where did the energy go? Later the box can drop and gain kinetic energy back. Copyright Dr. Weining Man

6 8-1 Conservative and Nonconservative Forces Conservative force: the work it does is stored in the form of energy that can be released at a later time Example of a conservative force: gravity Example of a nonconservative force: friction Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force

7 8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero: up

8 8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path:

9 8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero: After that a lot of negative work was doen.

10 8-2 The Work Done by Conservative Forces A ball with initial velocity v 0, raised h, Work done by gravity: W mg = mg * h*cos(180) = - mg h Gravity force did negative work. Change of kinetic energy: K = Kf Ki = 0 ½ m v 02 = W mg = - mg h Lost Kinetic Energy of amount ½ m v 02 = mg h Q: Where did those energy go? Copyright Dr. Weining Man

11 A: The energy is stored as potential energy. PE is like your saving account. Potential energy gain (mg h) during the rising part. We can get that energy back as kinetic E if the ball falls back off. During falling, Kinetic Energy will increase mg h. Potential energy will reduce mg h. Positive Work done by conservative force reduces Potential Energy and increase Kinetic Energy Negative Work done by conservative force reduces Kinetic Energy and increase Potential Energy

12 8-2 The Gravitational potential energy PE or U, is due to being under gravitational influence of another mass. (Earth). U=mgy OR PE =mgy y is the height above ground. You can define any y=0 ground. What matters is the Change in height. Falling from height y to ground, Gravitational potential energy reduced from mgyto 0.

13 8-2 Potential Energy stored in Spring This is the potential E stored in a deformed spring. k is spring constant. (not Kinetic Energy) x is the compressed or stretched length of the spring.

14 Kinetic Energy is like your checking account. W external, W push, pull, wind, man, wave, elevator, W mg Kinetic Energy K = ½mv 2 W spring W kinetic-friction = f k d

15 KE is like your checking account. PE is like your saving account. When you consider the total amount of K+U, no need to worry about transitions between K and U. NO NEED TO calculate W done by mg and spring anymore, When you consider K+U together. W external, W push, pull, wind, man, wave, elevator, Kinetic Energy K = ½mv 2 W mg U = mgy Potential Energy W spring U = ½k( x) 2 W kinetic-friction = f k d

16 When only mg or spring force (conservative forces) do work, (No friction, no resistance (in air, on ice, not water ), no external forces or energy source does work, (no human, animal, motor, lift, push ) K+U will stay unchanged. Kinetic Energy K = ½mv 2 U = mgy Potential Energy U = ½k( x) 2 Got Conservation of Mechanical Energy, when W nc =0

17 8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve: When there is no work done by non-conservative forces, Ki + Ui =Kf +Uf; (happily) You don t need to calculate work here, because work done by gravity and spring forces are included in the change of potential energy. K = - U= Work of gravity or spring

18 8-3 Conservation of Mechanical Energy You can solve v easily if h is known. Q1: If initial height was lower, the final speed to hit the ground would be more or less? Q2: Had the initial height was doubled, the final speed to hit the ground would become Half, Twice,, or 4 times? Similarly, if an object was thrown UPWARD with v 0, maximum height can be easily found.

19 Figure 8-9 When no work is done by non-conservative forces, Ki + Ui =Kf +Uf; When we look for change of KE or PE, only initial and final height matters. ½mv 02 +mgh 0 =½mv f2 +mgh f No matter which direction velocities are. If no friction, K + U does not change. It s true for all middle positions, not only the start and the end. You have ½ mv 02 +mgh 0 =½mv f2 +mgh f =½mv 12 +mgh 1 =½mv 22 +mgh 2, or any middle position at 1,2,3,. regardless velocity directions.

20 For A object launched at angle θ and V 0, through the entire motion, for all points in between. ½mv 02 +mgh 0 =½mv f2 +mgh f =½mv 12 +mgh 1 =½mv 22 +mgh 2 At maximum height, ½mv 02 +mgh 0 =½mv top2 +mgh max Is v top = 0? Sure, v top_y = 0, But v top = v top_x = v 0 cosθ; a x =0. But for a pendulum, or an object sliding up alone a incline, track etc, total v IS zero at maximum height. Copyright 2010 Pearson Education, Inc. Copyright Dr. Weining Man

21 Example, A truck slide down for distance L from rest, on an incline. (θ) If there is no friction, L After distance L what will be its final speed? θ h Ki + Ui =Kf +Uf; 0+ mgh 0 =½mv f2 + 0 h 0 =L sinθ mg L sinθ =½mv f 2 g L sinθ =½v f 2 v f 2 = 2g L sinθ Don t memorize conclusions like v 2 =2gh It only works without friction and when initial or final v=0 Learn the physics, UNDERSTAND it, UNDERSTAND when to use the sweet Copyright 2010 Pearson Education, Inc.

22 Total Energy conservation: Energy cannot be created or destroyed. It only change forms. When Energy seems to be created/ destroyed, it actually is being converted from/to other forms. When friction does work, it reduces kinetic energy and convert that into Heat. Copyright Dr. Weining Man

23 When nonconservative forces do work, the total mechanical energy is not conserved: we need to calculate work done by Non-Conservative forces. (Wnc= sum of all work except for mg and spring s work) W external, W push, pull, wind, man, wave, elevator, Kinetic Energy K = ½mv 2 U = mgy Potential Energy U = ½k( x) 2 W kinetic-friction = f k d

24 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance: By comparing Ef and Ei You can find Wnc, which is otherwise hard to calculate since water resistance may not be constant force.

m=1000kg, v 0 =0, µ k =0.2 φ=30 ο, d=0.5m,find v f N=mgcosφ f k =µ k N=µ k mgcosφ f k =0.2 9800 cos30=1697(ν) W NC = W fk = f k d = 1697*0.

25 m=1000kg, v 0 =0, µ k =0.2 φ=30 ο, d=0.5m,find v f N=mgcosφ f k =µ k N=µ k mgcosφ f k = cos30=1697(ν) W NC = W fk = f k d = 1697*0.5= 849 J ; K f + U f = K i + U i +W NC New Problem solving strategy : 1.Ananlyze all forces and decide whether Non- Conservative force does work or not. 2. Only compute total work down by NC forces 3. Set equation Yeah! No need to calculate ALL work any more, when I understand conservative and non-conservative forces. ½mv f2 + 0 = 0 + mgh f k d ; h=d sin φ ½mv f2 = mg d sin30 f k d =1601J v f =1.79m/s

26 Summary of Chapter 8 Conservative forces conserve mechanical energy. They do work to convert energy between Kinetic Energy and Potential Energy. Conservative force does zero work on any closed path Work done by a conservative force is independent of path Conservative forces: gravity, spring force. Nonconservative forces convert mechanical energy into other forms.

27 Summary of Chapter 8 Work done by nonconservative force on closed path is not zero, and depends on the path Nonconservative forces: friction, air resistance, pull, push, most other forces Energy in the form of potential energy can be converted to kinetic or other forms Work done by a conservative force is the negative of the change in the potential energy Gravity: U = mgy Spring: U = ½ k( x) 2

28 Summary of Chapter 8 Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces to do work. Nonconservative forces change a system s mechanical energy Work done by nonconservative forces equals change in a system s mechanical energy (K+U) Gravitational Potential energy at distance r from earth center: U= GM earth m/r

29 Copyright Dr. Weining Man

30 Copyright Dr. Weining Man

8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential

8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential energy, e.g. a ball in your hand has more potential energy