1. NAZARIN B. NORDIN nazarin@icam.edu.my

2. What you will learn: Define work, power and energy Potential energy Kinetic energy Work-energy principle Conservation of mechanical energy

3. The Work-Energy Principle The equation we derived is one form of the work- energy theorem (principle). It states that the work done by a net force on an object is equal to the change in the object’s kinetic energy. More generally, If the work is positive, the kinetic energy increases. Negative work decreases the kinetic energy.

4. The Work-Energy Principle A hand raises a book from height h 0 to height h f, at constant velocity. Work done by the hand force, F: Work done by the gravitational force:

5. Potential Energy An object can have potential energy by virtue of its surroundings. Familiar examples of potential energy: A wound-up spring A stretched elastic band An object at some height above the ground

6. Potential Energy In raising a mass m to a height h, the work done by the external force is We therefore define the gravitational potential energy: (6-5a) (6-6)

7. Potential Energy This potential energy can become kinetic energy if the object is dropped. Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces). If, where do we measure y from? It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.

8. Potential Energy Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

9. Potential Energy The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring. (6-8)

10. Work Done by a Varying Force For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up. As the pieces become very narrow, the work done is the area under the force vs. distance curve.

11. Potential Energy The force increases as the spring is stretched or compressed further. We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written: (6-9)

12. Kinetic Energy, and the Work-Energy Principle Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

13. Kinetic Energy, and the Work-Energy Principle If we write the acceleration in terms of the velocity and the distance, we find that the work done here is We define the kinetic energy: (6-2) (6-3)

14. Kinetic Energy, and the Work-Energy Principle This means that the work done is equal to the change in the kinetic energy: If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases. (6-4)

15. Kinetic Energy, and the Work-Energy Principle Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules.

16. The Work-Energy Principle Total (net) force exerted on the book: zero. Total (net) work done on the book: zero. Change in book’s kinetic energy: zero.

17. The Work-Energy Principle Now, we let the book fall freely from rest at height h f to height h 0. Net force on the book: mg. Work done by the gravitational force:

18. The Work-Energy Principle Calculate the book’s final kinetic energy kinematically: The book gained a kinetic energy equal to the work done by the gravitational force (per the work-energy theorem).

19. Conservative Forces The gravitational force is an example of a conservative force:  The work it does is path-independent.  A form of potential energy is associated with it (gravitational potential energy). Other examples of conservative forces:  The spring force  The electrical force

20. Nonconservative Forces The frictional force is an example of a nonconservative force:  The work it does is path-dependent.  No form of potential energy is associated with it. Other examples of nonconservative forces:  normal forces  tension forces  viscous forces

21. Total Mechanical Energy A man lifts weights upward at a constant velocity. He does positive work on the weights. The gravitational force does equal negative work. The net work done on the weights is zero. But …

22. Conservation of Mechanical Energy The gravitational potential energy of the weights increases: The work done by the nonconservative normal force of the man’s hands on the bar changed the total mechanical energy of the weights:

23. Conservation of Mechanical Energy Work done on an object by nonconservative forces changes its total mechanical energy. If no (net) work is done by nonconservative forces, the total mechanical energy remains constant (is conserved):

24. Conservation of Mechanical Energy This equation is another form of the work-energy theorem. Note that it does not require both kinetic and potential energy to remain constant – only their sum. Work done by a conservative force often increases one while decreasing the other. Example: a freely-falling object.

25. Conservation of Every Kind of Energy “Energy is neither created nor destroyed.” Work done by conservative forces conserves total mechanical energy. Energy may be interchanged between kinetic and potential forms. Work done by nonconservative forces still conserves total energy. It often converts mechanical energy into other forms – notably, heat, light, or noise.

26. Mechanical Energy and Its Conservation The principle of conservation of mechanical energy: If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved.

27. Problem Solving Using Conservation of Mechanical Energy In the image on the left, the total mechanical energy at any point is:

28. Problem Solving Using Conservation of Mechanical Energy Example 3: Falling rock. If the original height of the rock is y 1 = h = 3.0 m, calculate the rock’s speed when it has fallen to 1.0 m above the ground.

29. Problem Solving Using Conservation of Mechanical Energy Example 4: Roller-coaster car speed using energy conservation. Assuming the height of the hill is 40 m, and the roller-coaster car starts from rest at the top, calculate (a) the speed of the roller-coaster car at the bottom of the hill, and (b) at what height it will have half this speed. Take y = 0 at the bottom of the hill.

30. Problem Solving Using Conservation of Mechanical Energy Conceptual Example 5: Speeds on two water slides. Two water slides at a pool are shaped differently, but start at the same height h. Two riders, Paul and Joanna, start from rest at the same time on different slides. (a) Which rider, Paul or Kathleen, is traveling faster at the bottom? (b) Which rider makes it to the bottom first? Ignore friction and assume both slides have the same path length.

31. Problem Solving Using Conservation of Mechanical Energy Which to use for solving problems? Newton’s laws: best when forces are constant Work and energy: good when forces are constant; also may succeed when forces are not constant

32. Problem Solving Using Conservation of Mechanical Energy Example 6: Pole vault. Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter’s center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself.

33. Problem Solving Using Conservation of Mechanical Energy For an elastic force, conservation of energy tells us: Example 8-7: Toy dart gun. A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (with spring stiffness constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length ( x = 0), what speed does the dart acquire?

34. Problem Solving Using Conservation of Mechanical Energy Example 8-8: Two kinds of potential energy. A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring, which it compresses an amount Y = 15.0 cm. Determine the spring stiffness constant of the spring. Assume the spring has negligible mass, and ignore air resistance. Measure all distances from the point where the ball first touches the uncompressed spring ( y = 0 at this point).

35. Problem Solving Using Conservation of Mechanical Energy Example 9: A swinging pendulum. This simple pendulum consists of a small bob of mass m suspended by a massless cord of length l. The bob is released (without a push) at t = 0, where the cord makes an angle θ = θ 0 to the vertical. (a) Describe the motion of the bob in terms of kinetic energy and potential energy. Then determine the speed of the bob (b) as a function of position θ as it swings back and forth, and (c) at the lowest point of the swing. (d) Find the tension in the cord, T. Ignore friction and air resistance.