2. 1 Chapter 6 Energy and Energy Transfer

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4. 3 Introduction to Energy The concept of energy is one of the most important topics in science Every physical process that occurs in the Universe involves energy and energy transfers or transformations Energy is not easily defined

5. 4 Energy Approach to Problems The energy approach to describing motion is particularly useful when the force is not constant A global approach to problems involving energy and energy transfers will be developed This could be extended to biological organisms, technological systems and engineering situations

6. 5 6.1 Systems A system is a small portion of the Universe We will ignore the details of the rest of the Universe This is a simplification model A critical skill is to identify the system

7. 6 Identifying Systems A system may be a single object or particle be a collection of objects or particles be a region of space vary in size and shape

8. 7 Environment There is a system boundary around the system The boundary is an imaginary surface It does not necessarily correspond to a physical boundary The boundary divides the system from the environment The environment is the rest of the Universe

9. 8 6.2 Work The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude, F, of the force, the magnitude  r of the displacement of the point of application of the force, and cos  where  is the angle between the force and the displacement vectors

10. 9 Work, cont. W = F  r cos  The displacement is that of the point of application of the force A force does no work on the object if the force does not move through a displacement The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application

11. 10 Fig 6.1

12. 11 Work Example The normal force, n, and the gravitational force, m g, do no work on the object cos  = cos 90° = 0 The force does do work on the object Fig 6.2

13. 12 Units of Work Work is a scalar quantity The unit of work is a joule (J) 1 joule = 1 newton. 1 meter J = N · m

14. 13 More About Work The system and the environment must be determined when dealing with work The environment does work on the system Work by the environment on the system The sign of the work depends on the direction of relative to Work is positive when projection of onto is in the same direction as the displacement Work is negative when the projection is in the opposite direction

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17. 16 Fig 6.5

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19. 18 6.3 Scalar Product of Two Vectors The scalar product of two vectors is written as It is also called the dot product  is the angle between A and B Fig 6.6

20. 19 Scalar Product, cont The scalar product is commutative The scalar product obeys the distributive law of multiplication

21. 20 Dot Products of Unit Vectors Using component form with and :

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25. 24 6.4 Work Done by a Varying Force Assume that during a very small displacement,  x, F is constant For that displacement, W 1  F  x For all of the intervals, Fig 6.7

26. 25 Work Done by a Varying Force, cont Therefore, The work done is equal to the area under the curve Fig 6.7

27. 26 Work Done By Multiple Forces If more than one force acts on a system and the system can be modeled as a particle, the total work done on the system is the work done by the net force

28. 27 Work Done by Multiple Forces, cont. If the system cannot be modeled as a particle, then the total work is equal to the algebraic sum of the work done by the individual forces

29. 28 Hooke’s Law The force exerted by the spring is F s = - kx x is the position of the block with respect to the equilibrium position (x = 0) k is called the spring constant or force constant and measures the stiffness of the spring This is called Hooke’s Law Fig 6.8

30. 29 Hooke’s Law, cont. When x is positive (spring is stretched), F is negative When x is 0 (at the equilibrium position), F is 0 When x is negative (spring is compressed), F is positive Fig 6.8

31. 30 Hooke’s Law, final The force exerted by the spring is always directed opposite to the displacement from equilibrium F is called the restoring force If the block is released it will oscillate back and forth between –x max and x max

32. 31 Work Done by a Spring Identify the block as the system Calculate the work as the block moves from x i = - x max to x f = 0 The total work done as the block moves from –x max to x max is zero Fig 6.8

33. 32 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 6.8

34. 33 Spring with an Applied Force Suppose an external agent, F app, stretches the spring The applied force is equal and opposite to the spring force F app = -F s = -(-kx) = kx Work done by F app is equal to -1/2 kx 2 max Fig 6.9

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36. 35 Fig 6.10

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39. 38 6.5 Kinetic Energy Kinetic Energy is the energy of a particle due to its motion K = 1/2 mv 2 K is the kinetic energy m is the mass of the particle v is the speed of the particle A change in kinetic energy is one possible result of doing work to transfer energy into a system

40. 39 Kinetic Energy, cont Calculating the work: Fig 6.11

41. 40 Work-Kinetic Energy Theorem The Work-Kinetic Energy Theorem states  W = K f – K i =  K In the case in which work is done on a system and the only change in the system is in its speed, the work done by the net force equals the change in kinetic energy of the system. We can also define the kinetic energy K = 1/2 mv 2

42. 41 Work-Kinetic Energy Theorem – Example The normal and gravitational forces do no work since they are perpendicular to the direction of the displacement W = F  x W =  K = 1/2 mv f 2 - 0 Fig 6.12

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47. 46 Fig 6.13

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53. 52 6.6 Nonisolated System A nonisolated system is one that interacts with or is influenced by its environment A new analysis model An isolated system would not interact with its environment The Work-Kinetic Energy Theorem can be applied to nonisolated systems

54. 53 Energy Transfer Work is a method of energy transfer Work has the effect of transferring energy between the system and the environment If positive work is done on the system, energy is transferred to the system Negative work indicates that energy is transferred from the system to the environment

55. 54 Internal Energy The energy associated with an object’s temperature is called its internal energy, E int The friction does work and increases the internal energy of the surface Fig 6.14

56. 55 Ways to Transfer Energy Into or Out of A System Work – transfers by applying a force and causing a displacement of the point of application of the force Mechanical Waves – allow a disturbance to propagate through a medium Heat – is driven by a temperature difference between two regions in space

57. 56 More Ways to Transfer Energy Into or Out of A System Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it Electrical Transmission – transfer is by electric current Electromagnetic Radiation – energy is transferred by electromagnetic waves

58. 57 Examples of Ways to Transfer Energy (Fig 6.15) a) Work b) Mechanical Waves c) Heat

59. 58 Examples of Ways to Transfer Energy, cont. (Fig 6.15) d) Matter transfer e) Electrical Transmission f) Electromagnetic radiation

60. 59 Conservation of Energy Energy is conserved This means that energy cannot be created or destroyed If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer

61. 60 Conservation of Energy, cont. Mathematically,  E system =  E system is the total energy of the system T is the energy transferred across the system boundary Established symbols: T work = W and T heat = Q Others do not have standard symbols The Work-Kinetic Energy theorem is a special case of Conservation of Energy

62. 61 Continuity Equation The conservation of energy equation is an example of an continuity equation Specifically, it is the continuity equation for energy A continuity equation arises in any situation in which the change in a quantity in a system occurs solely because of transfers across the boundary

63. 62 Conservation of Energy, Completed The primary mathematical representation of the energy analysis of a nonisolated system is  K +  E int = W + T MT + T ET + T ER If any of the terms on the right are zero, the system is an isolated system The Work-Kinetic Energy Theorem is a special case of the more general equation above

64. 63 Fig 6.16

65. 64 6.7 Problems Involving Kinetic Energy When kinetic friction is involved in a problem, you must use a modification of the work-kinetic energy theorem  W other forces – ƒ k d =  K The term ƒ k d is the work associated with the frictional force Also,  e int = ƒ k d when friction is the only force acting in the system

66. 65 Problems Involving Kinetic Energy, cont A friction force transforms the kinetic energy in a system to internal energy For a system in which the frictional force alone acts, the increase in the internal energy of the system is equal to its decrease in kinetic energy

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70. 69 Fig 6.17(a)

71. 70 Fig 6.17(b)

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81. 80 6.8 Power The time rate of energy transfer is called power The average power is given by when the method of energy transfer is work

82. 81 Instantaneous Power The instantaneous power is the limiting value of the average power as  t approaches zero This can also be written as

83. 82 Power Generalized Power can be related to any type of energy transfer In general, power can be expressed as dE/dt is the rate rate at which energy is crossing the boundary of the system for a given transfer mechanism

84. 83 Units of Power The SI unit of power is called the watt 1 watt = 1 joule / second = 1 kg. m 2 / s 2 A unit of power in the US Customary system is horsepower 1 hp =550 ft. lb/s = 746 W Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x10 6 J

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86. 85 Fig 6.18

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92. 91 6.9 Horsepower Ratings of Automobiles The strength of the frictional force exerted on a car by the roadway is related to the rate at which energy is transferred to the wheels to set them into rotation From Newton’s Second Law, the driving force is proportional to the acceleration Therefore, there is a close relationship between the power rating of a vehicle and its possible acceleration

93. 92 Horsepower and Acceleration

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