Work

1. Work

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

3. Work • 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.

4. Work • 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.

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

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

7. Work • 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

8. Joule (J) • 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.

9. Calculating Work • 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°?

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

11. Determining Work Graphically • 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).

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

13. Springs • Equilibrium position: the normal or relaxed length of the spring • Fex: an external force • d = Δx = x2 – x1 • x1 is equilibrium position.

14. Hooke’s Law • Fex = kd • k is a proportionality constant called the spring constant. • Work done on a spring by an external force is positive.

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

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

17. Ideal Springs • How much work is done by the spring? • According to Newton’s 3rd Law: Fs = -Fex Fs = -kd

18. Ideal Springs • 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.

19. Ideal Springs • 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.

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

21. W Fdcosθ P = = Δt Δt Power • Average power: P = Fvcosθ • Power is a scalar quantity.

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

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 Wtotal = ΔK = K2 – K1

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

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

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

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

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

40. Gravitational Potential • 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!

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

42. Elastic Potential Energy • ΔUs = change in spring’s potential energy ΔUs = ½k(d2x2 – d1x2)

43. Total Mechanical Energy

44. All mechanical work on a system can be subdivided into the work done by conservative forces (Wcf) and the work done by nonconservative forces (Wncf). Wtotal = Wcf + Wncf = 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. Wncf = ΔE

47. If mechanical energy is conserved, we obtain: ΔK = -ΔU K1 + U1 = K2 + U2

48. If mechanical energy is not conserved, we obtain: K1 + U1 = K2 + U2 + Wncf