1. Work

2. Energy has the ability to do work; it can move matter. Work may be useful or destructive. Introduction

3. Work is defined as the product of the force component that is parallel to an object’s motion and the distance that the object is moved. Work

4. Mechanical work is done by a force on a system. W ≡ Fd cos θ Work is done by a force F through a displacement d. Work

5. W ≡ Fd cos θ θ is the smallest angle (≤180°) between the force and displacement vectors when they are placed tail-to- tail. Work

6. W ≡ Fd cos θ Work is a scalar. Work can be positive, negative, or zero, depending on the angle θ. Work

7. θ < 90°: Work is positive. 90° < θ < 180°: Work is negative. θ = 90°: Work is zero. Units: Joules (J) 1 J ≡ 1 N × 1 m Work

8. This is the unit used for both work and energy. It must not be confused with the N · m, used for torque; joules are never used for torque. Joule (J)

9. Any kind of force can do work. No work is done if no object moves (since d = 0). Example 9-1: Why is the angle 0°? Calculating Work

10. Force-distance graph The area “under the curve” of a force-distance graph approximates the work done on a system by the force. Determining Work Graphically

11. For a constant force, the “area” is rectangular and simple to calculate. Be sure to select the appropriate units for your result (typically N × m = J). Determining Work Graphically

12. An external force to stretch a spring is an example of a varying force. Determining Work Graphically

13. Equilibrium position: the normal or relaxed length of the spring F ex : an external force d = Δx = x 2 – x 1 x 1 is equilibrium position. Springs

14. F ex = k d k is a proportionality constant called the spring constant. Work done on a spring by an external force is positive. Hooke’s Law

15. no mass value of k is truly constant throughout its range of displacements exemplifies a Hooke’s Law force Ideal Springs

16. W ex = ½k(Δx)². This is consistent with its force-distance graph. Ideal Springs How much work is done to stretch a spring from its equilibrium position by Δx?

17. How much work is done by the spring? According to Newton’s 3 rd Law: Ideal Springs F s = - F ex F s = -k d

18. Work done by the spring is negative because the displacement is opposite the spring’s force. This is true whether the spring is stretched or compressed. Ideal Springs

19. The force-distance graph of the work done by the spring is below the x-axis. In Example 9-3, the two forces are opposites of each other. Ideal Springs

20. Defined: the time-rate of work done on a system Average power: the work accomplished during a time interval divided by the time interval Power

21. Average power: Power P = W ΔtΔt P = Fv cos θ Fd cos θ ΔtΔt = Power is a scalar quantity.

22. The unit of power is the Watt (W). 1 W = 1 J/s Power

23. Energy

24. Kinetic Energy mechanical energy associated with motion positive scalar quantity measured in joules

25. Work-Energy Theorem states that the total energy done on a system by all the external forces acting on it is equal to the change in the system’s kinetic energy W total = ΔK = K 2 – K 1

26. Kinetic Energy can be defined as: K = ½mv² Note that kinetic energy must mathematically be a positive quantity.

27. Potential Energy energy due to an object’s condition or position relative to some reference point assumed to have zero potential energy measured in joules

28. Potential Energy takes various forms: gravitational elastic electrical results from work done against a force

29. Conservative Forces One of the following things must be true: The net work done by the force on a system as it moves between any two points is independent of the path followed by the system.

30. Conservative Forces One of the following things must be true: The net work done by the force on a system that follows a closed path (begins and ends at the same point in space) is zero.

31. Conservative Forces Examples of conservative forces: gravitational force any central force any Hooke’s law force

32. Conservative Forces energy expended when doing work against them is stored as potential energy and can be regained as kinetic energy if not, it is called a nonconservative force

33. Conservative Forces Examples of nonconservative forces: kinetic frictional force internal resistance forces fluid drag

34. Conservative Forces When work is done against nonconservative forces, the energy is not stored as potential energy but is converted into other forms of mechanically unusuable energy.

35. work required to move masses apart against the force of gravity near earth’s surface, work done lifting against gravity: Gravitational Potential Energy W lift = |mg|Δh

36. Work must be done against a force in order to increase the potential energy of a system with respect to that force. Gravitational Potential Energy W g = -ΔU g

37. requires a well-defined reference point for height The U g = |mg|h formula is still in effect, where h is the distance the object can fall. Relative Potential Energy

38. defined as the potential energy per kilogram at a specified distance r from a zero reference distance near the earth’s surface: Gravitational Potential U g (r) = |g|h

39. for any object of mass m at any distance r from mass M: Gravitational Potential The units are J/kg U g (r) = -G M r

40. Gravitational potential will always be negative, but when the objects are moved farther apart, it is a positive change in potential energy. Gravity can do work! Gravitational Potential

41. Work must be done against a force in order to increase the potential energy of a system with respect to that force. Elastic Potential Energy

42. ΔU s = change in spring’s potential energy Elastic Potential Energy ΔU s = ½k(d 2x 2 – d 1x 2 )

43. Total Mechanical Energy

44. All mechanical work on a system can be subdivided into the work done by conservative forces (W cf ) and the work done by nonconservative forces (W ncf ). W total = W cf + W ncf = ΔK

45. The work done by nonconservative forces is equal to the change of the system’s total energy. Total mechanical energy is the sum of a system’s kinetic and potential energies. E ≡ K + U

46. We can also say that the work accomplished by all nonconservative forces on a system during a certain process is equal to the change of total mechanical energy of a system. W ncf = ΔE

47. If mechanical energy is conserved, we obtain: ΔK = -ΔU K 1 + U 1 = K 2 + U 2

48. If mechanical energy is not conserved, we obtain: K 1 + U 1 = K 2 + U 2 + W ncf