Work and Energy

1. Work and Energy Chapter 7

2. Conservation of Energy • Energy is a quantity that can be converted from one form to another but cannot be created or destroyed. • The total amount of energy present in any process remains the same. Physics chapter 7

3. Work • Has a very specific definition in physics. • Involves a force moving a body through a displacement. • If the force is in the same direction as the motion, Physics chapter 7

4. Vector equation for work • If the force is directed at an angle to the displacement, then only the component of the force that is parallel to the displacement does work. F Physics chapter 7

5. Vector equation for work • Important – work is a scalar quantity, not a vector, even though it is derived from two vectors. • Work does not have a direction. Physics chapter 7

6. Work • If q is between 0° and 90°, cos q is positive, so work done is positive. • If q is between 90° and 180°, cos q is negative, so work done is negative. • If q is 90°, cos q is zero, so work done is zero. Physics chapter 7

7. Example • Lifting, carrying, and lowering a book. Physics chapter 7

8. Units on work • The unit for work is a joule. Physics chapter 7

9. Example • During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled 5 m. • Find the amount of work you do Physics chapter 7

10. On your own • A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. • Find the work done by the crane on the truck Physics chapter 7

11. Work and energy with varying forces • The equations we’ve developed so far for work involve constant forces and motion along a straight line. • Now let’s look at varying forces. • We could also look at motion along curved lines, but we would need to use integrals, which you don’t know how to do yet. Physics chapter 7

12. The farther you stretch a spring, the harder it is to stretch. The force you must apply follows the following equation, known as Hooke’s Law Where k is called the force constant of the spring or the spring constant. k has units of N/m The spring force Physics chapter 7

13. Work done on a spring • Where X is the total elongation of the spring. • See page 151. Physics chapter 7

14. Work done on a spring • Where x1 is the initial position of the spring, x2 is the final position of the spring • x = 0 is the equilibrium position of the spring (when it is neither stretched nor compressed.) Physics chapter 7

15. Example • A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm. • Find the work done on the block as the block moves to its equilibrium position x2 = 0 Physics chapter 7

16. On your own • To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. How much work must be done to compress this spring 4.00 cm from its unstretched length? Physics chapter 7

17. Work and Kinetic Energy • The total work done on a body is related to changes in the speed of the body. Physics chapter 7

18. Work and Kinetic Energy Physics chapter 7

19. The quantity ½ mv2 is called the kinetic energy and represented by the letter K. It is a scalar quantity and can never be negative. It is zero when an object is not moving. Work-Kinetic Energy Theorem Physics chapter 7

20. Units on kinetic energy • Looking at the equation, we have Physics chapter 7

21. Work-kinetic energy theorem • The work-kinetic energy theorem still works for varying forces. • See the last paragraph on page 153 for a mathematical explanation of this. Physics chapter 7

22. Motion along a curved path • Where dl is a distance along the path • The work-kinetic energy theorem still holds true. Physics chapter 7

23. Example • During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled for 5 m. • Find • The work you do • The final speed of the sled Physics chapter 7

24. On your own • A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. • Find • The work done by the crane on the truck • The total work done on the truck • The upward speed of the truck after the 2 m if it started from rest Physics chapter 7

25. Example • A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm. • Find • The work done on the block as the block moves to its equilibrium position x2 = 0 • The speed of the block at x2=0 Physics chapter 7

26. Where does kinetic energy come from? • If an object falls off a cliff, gravity does work on it as it falls, and increases its kinetic energy. • But where does that energy come from? Physics chapter 7

27. Potential Energy • Energy seems to be stored in some form related to height. • This energy is related to the position of a body, not its motion. • It is called potential energy. – measures potential or possibility for work to be done. • (Some kinds of potential energy are related to things other than height.) Physics chapter 7

28. Work done by gravity • Work done by gravity as a body falls from height y1 to height y2. • Also works if object is rising – then y2 is greater than y1, so the work is negative, as it should be. Physics chapter 7

29. Gravitational Potential Energy • We define the gravitational potential energy as • So, the work done by gravity is Physics chapter 7

30. If only gravity does work Physics chapter 7

31. Conservation of Mechanical Energy • When only gravity does work, the total mechanical energy is constant. Physics chapter 7

32. Where do I measure y from? • Wherever you want. • It is only the change in potential energy that we are interested in, not the value of U at a particular point. • So choose your zero potential energy point wherever it is convenient. Physics chapter 7

33. Example • Standing near the edge of the roof of a 12-m high building, you kick a ball with an initial speed of vi = 16 m/s at an angle of 60° above the horizontal. Neglecting air resistance, use conservation of energy to find • How high above the height of the building the ball rises • Its speed just before it hits the ground • 9.79 m 22.2 m/s Physics chapter 7

34. You try • You throw a 0.200-kg ball straight up in the air, giving it an initial upward velocity of 20.0 m/s. Use conservation of energy to find how high it goes. • 20.4 m Physics chapter 7

35. Effect of other forces • If forces other than gravity are acting on a body Physics chapter 7

36. Example • A child of mass 40 kg goes down an 8.0 m long slide inclined at 30° above the horizontal. The coefficient of kinetic friction between the slide and the child is 0.35. If the child starts from rest at the top of the slide, how fast is she traveling when she reaches the bottom? • 5.60 m/s Physics chapter 7

37. Curved path • The shape of the path makes no difference for gravitational potential potential energy. • You can still use the same equations for conservation of energy. Physics chapter 7

38. q0 On your own • A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled aside so that the string make an angle q0 with the vertical, and is released from rest. Find an expression for its speed as it passes through the bottom of the arc. Physics chapter 7

39. Elastic potential energy • Think slingshot. • The farther you stretch it, the more kinetic energy the projectile can gain. Physics chapter 7

40. Springs • For springs, the elastic potential energy is given by • The work done by a spring is • The equations work whether the spring is stretched or compressed. Physics chapter 7

41. Where is x = 0? • We don’t get to choose this time • x = 0 is always at the equilibrium position of the spring – when it is neither stretched nor compressed. Physics chapter 7

42. Conservation of energy • The conservation of energy works the same for springs as it does for gravity. Physics chapter 7

43. Conservation of energy • If we have both gravitational and elastic potential energies, then Physics chapter 7

44. Example • A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N/m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from its initial position? (The spring and the cheese are not attached.) • 1.72 m Physics chapter 7

45. On your own • A 2000-kg elevator with broken cables is falling at 25 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 3.00 m as it does so. During the motion, a safety clamp applies a constant 17,000 N frictional force to the elevator. • What is the force constant of the spring? • 1.41 x 105 N/m Physics chapter 7

46. Conservative forces • The work done by conservative forces: • Can always be expressed as the difference between the initial and final values of a potential energy function. • Is reversible • Is path-independent (It only depends on the starting and ending points.) • Equals zero when the starting and ending points are the same. Physics chapter 7

47. Conservative forces • Gravitational (without air resistance) • Spring (without friction) Physics chapter 7

48. Nonconservative forces • Depend on path • Are not reversible • Cannot be expressed in terms of a potential energy function • Do work even when the starting and ending points are the same Physics chapter 7

49. Nonconservative forces • Friction • Chemical reaction forces Physics chapter 7

50. Nonconservative forces • Increase or decrease the internal energy of a system • Frequently, this happens through heat. Physics chapter 7