1. Chapter 5 Work and Energy

2. Chapter 5: Work and Energy
5.1 Work Done by a Constant Force 5.2 Work Done by a Variable Force 5.3 The Work-Energy Theorem: Kinetic Energy 5.4 Potential Energy 5.5 The Conservation of Energy 5.6 Power

3. 5.1 Work Done by a Constant Force

4. 5.1 Work Done by a Constant Force
Definition: Work done by a constant force is equal to the product of the magnitudes of the displacement and the component of the force parallel to the displacement.

5. 5.1 Work Done by a Constant Force

6. 5.1 Work Done by a Constant Force
W = (F cosӨ)d, where
F is the magnitude of the force vector
d is the magnitude of the displacement vector
Ө is the angle between the two vectors (Warning – this angle is not necessarily measured from the horizontal)
When the angle is zero, cos Ө = 1, and W = F·d
When the angle is 180°, cos Ө = -1 and W = - F·d (yes, negative work!)
example: the force of brakes to slow down a car.
Although force and displacement are vectors, work is a scalar quantity.
The SI unit of work is the N·m, which is called a joule (J).

7. 5.1 Work Done by a Constant Force
If there is no displacement, no work is done: W = 0.
For a constant force in the same direction as the displacement, W = Fd.
For a constant force at an angle to the displacement, W = (FcosӨ) d

8. 5.1 Work Done by a Constant Force: Check for Understanding

9. 5.1 Work Done by a Constant Force: Check for Understanding

10. 5.1 Work Done by a Constant Force: Check for Understanding

11. 5.1 Work Done by a Constant Force: Check for Understanding

12. Homework for Section 5.1
See Handout provide in class.

13. 5.6 Power

14. Warmup: Rock-It

16. 5.6 Work

17. 5.6 Power
A common British unit of power is the horsepower (hp) and 1 hp = 746 W.

18. 5.6 Power

19. 5.6 Power

20. 5.6 Power
Example 5.11: A 1500 kg car accelerates from 0 to 25 m/s in 7.0 s. What is the average power delivered to the car by the engine? Ignore all frictional and other losses.

21. 5.6 Power: Check for Understanding

22. 5.6 Power: Check for Understanding

23. 5.6 Power: Check for Understanding

24. 5.6 Power: Check for Understanding

25. Homework for Power
See handout

26. The Work Energy Theorem: The Basic Model

27. Energy Energy is the capacity of a physical system to perform work.
Energy exists in several forms such as heat, kinetic or mechanical energy, light, potential energy, electrical, or other forms.
The SI unit of energy is the joule (J) or newton-meter (N * m). The joule is also the SI unit of work.

28. Law of Conservation of Energy
According to the law of conservation of energy, the total energy of a system remains constant, though energy may transform into another form.
Two billiard balls colliding, for example, may come to rest, with the resulting energy becoming sound and perhaps a bit of heat at the point of collision.

29. Basic Energy Model
Within a system, energy can be transformed between various forms.
Example #1: In a pendulum or swing (with the absence of frictional forces)
Potential Energy  Kinetic Energy  Potential Energy
Example #2: Potential energy of oil or gas is changed into energy to heat a building.
In a closed system, the total energy in a system is constant and only transforms from on form of energy to another.

30. Basic Energy Model
In an open system (most systems are open), energy can be transferred into and out of a system in 2 ways:
Work: The transfer of energy by mechanical forces.
Example: a golfer uses a club and gets a stationary golf ball moving when he or she hits the ball. The club does work on the golf ball as it strikes the ball. Energy leaves the club and enters the ball. This is a transfer of energy.
Heat: The non-mechanical transfer of energy from a hotter object to a cooler object.
Example: Brake on a car creating heat while stopping and the area around the brakes getting hot.

31. The Work Energy Theorem: Types of Mechanical Energy

32. K = kinetic energy (Joules – J)
I. Kinetic Energy
Kinetic Energy – Energy of motion
KE = K = ½ mv2
K = kinetic energy (Joules – J)
m = mass (kg)
v = velocity (m/s)
Note: KE cannot be negative, because…
You cannot have a negative mass
Velocity is always squared

33. I. Kinetic Energy

34. I. Kinetic Energy

35. II. Elastic Potential Energy
Elastic potential energy is potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring.
It is equal to the work done to stretch the spring, which depends upon the spring constant k as well as the distance stretched.

36. II. Elastic Potential Energy
PEe = Us = ½ kx2
PEe & Us = Elastic potential energy = Joules (J)
k = spring constant (measures how stiff and strong the spring is) = N/m
x = displacement of the spring from the rest or equilibrium position. (m)
Note: PEe or Us cannot be negative because…
k cannot have a negative value
x is squared

37. II. Elastic Potential Energy

38. II. Elastic Potential Energy
A child pulls back horizontally on a rubber band that has an unstretched length of 0.1 m and it stretches to a length of 0.25 m. If the spring constant of the rubber band is 10N/m, how much energy is stored in the spring?

39. III. Gravitational Potential Energy
GPE is the energy associated with a gravitational field like the one we live in on Earth. AKA – Energy of position
PEg = U = mgh
PEg or U = gravitational potential energy – Joules (J)
m = mass (kg)
g = acceleration due to gravity (m/s2)
h = height above ground (m)
Note: GPE can be negative, but only if the object is below the horizontal line of the coordinate plane.

40. III. Gravitational Potential Energy

41. III. Gravitational Potential Energy
A 10.0 kg object is moved from the second floor of a house 3.00 m above the ground to the first floor 0.30 m above the ground. What is the change in gravitational potential energy?

42. 5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

43. 5.4 The Work-Energy Theorem: Potential Energy

44. 5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

45. 5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

46. 5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding
Or, use opp = hyp (sin 30) = 0.5 m

47. Homework for Section Kinetic & Potential Energy Problems

48. B. Work Done by a Variable Force

49. Variable Forces: Spring Forces & Energy
AP Physics I

50. I. Spring Forces
Back and forth motion that is caused by a force that is directly proportional to the displacement. The displacement centers around an equilibrium position.

51. Springs – Hooke’s Law
One of the simplest type of simple harmonic motion is called Hooke's Law. This is primarily in reference to SPRINGS.
The negative sign only tells us that “F” is what is called a RESTORING FORCE, in that it works in the OPPOSITE direction of the displacement.

52. Example
A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?
110 N
1000 N/m

53. Hooke’s Law from a Graphical Point of View
Suppose we had the following data:
x(m)
Force(N)
0.1 12
0.2 24
0.3 36
0.4 48
0.5 60
0.6 72
k =120 N/m

54. We have seen F vs. x Before!!!!
Work or ENERGY = FDx
Since WORK or ENERGY is the AREA, we must get some type of energy when we compress or elongate the spring. This energy is the AREA under the line!
Area = ELASTIC POTENTIAL ENERGY
Since we STORE energy when the spring is compressed and elongated it classifies itself as a “type” of POTENTIAL ENERGY, Us. In this case, it is called ELASTIC POTENTIAL ENERGY.

55. B. Work Done by a Variable Force
The work done by an external force in stretching or compressing a spring (overcoming the spring force) is
Work-Energy Thereom - WORK = D ENERGY (in a system)
so, W = Us, which makes W = ½ kx2
where x is the stretch or compression distance and k is the spring constant. F
F = kx
work = area under the curve
area of the triangle = ½ (base x height)
work = ½ (x)(kx) = ½ kx2
slope = k x

56. Elastic Potential Energy
The graph of F vs.x for a spring that is IDEAL in nature will always produce a line with a positive linear slope. Thus the area under the line will always be represented as a triangle.
NOTE: Keep in mind that this can be applied to WORK or can be conserved with any other type of energy.

57. Conservation of Energy in Springs

58. Example
A slingshot consists of a light leather cup, containing a stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot?
3000 N/m
60 J
49 m/s

59. B. Work Done by a Variable Force
The reference position xo is chosen for convenience.
xo may be chosen to be at the end of the spring in its unloaded position.
b) xo may be at the equilibrium position when a mass is suspended on a spring. This is convenient when the mass oscillates up and down on the spring.

60. B. Work Done by a Variable Force
Example:
A spring of spring constant 20 N/m is to be compressed by 0.10 m.
What is the maximum force required?
What is the work required?
Solution:
Given: k = 20 N/m x = m
Unknown: a) Fs(max) b) W
From Hooke’s law, the maximum force corresponds to the maximum compression.
Fs(max) = -kx = -(20 N/m)(-0.10 m) = 2.0 N
b) W = ½ kx2 = ½ (20 N/m)(-0.10 m)2 = 0.10 J

61. B. Work Done by a Variable Force: Check for Understanding4 3 2 1

62. B. Work Done by a Variable Force

63. Homework for Work Done by a Variable Force
See handout

64. 5.5 The Conservation of Energy

65. Warmup: Road Hazard

67. 5.5 The Conservation of Energy

68. 5.5 The Conservation of Energy
Remember, mechanical energy is kinetic or potential.
If there is a nonconservative force such as friction doing work on the system, the total mechanical energy of the system is not conserved, but total overall energy is conserved.

69. 5.5 The Conservation of Energy
Example 5.8: A 70 kg skier starts from rest on the top of a 25 m high slope. What is the speed of the skier on reaching the bottom of the slope? (Neglect friction)

70. 5.5 The Conservation of Energy
If there is a nonconservative force doing work in a system, the total mechanical energy of the system is not conserved.
However, the total energy (not mechanical!) of the system is still conserved.
Some of the total energy is used to overcome the work done by the nonconservative force.
The difference in mechanical energy is equal to the work done by the nonconservative force, that is
Wnc = E – Eo = E

71. 5.5 The Conservation of Energy
Example 5.10: In Example 5.8, if the work done by the kinetic friction force is -6.0 x 103 J (the work done by kinetic friction force is negative because the angle between the friction force and the displacement is 180˚). What is the speed of the skier at the bottom of the slope?

72. 5.5 The Conservation of Energy
Example 5.9: A 1500 kg car moving at 25 m/s hits an initially uncompressed horizontal spring with spring constant 2.0 x 106 N/m. What is the maximum compression of the spring? (Neglect the mass of the spring.)

73. 5.5 The Conservation of Energy: Check for Understanding

74. 5.5 The Conservation of Energy: Check for Understanding

75. 5.5 The Conservation of Energy: Check for Understanding

76. 5.5 The Conservation of Energy: Check for Understanding

77. 5.5 The Conservation of Energy: Check for Understanding

78. 5.5 The Conservation of Energy: Check for Understanding

79. 5.5 The Conservation of Energy: Check for Understanding

80. 5.5 The Conservation of Energy: Check for Understanding

81. Explanation of the Math for Pulley Problem:
5.5 The Conservation of Energy
Explanation of the Math for Pulley Problem:
Uo = K Just block A: K = ½ m vA2 Both blocks together: K = ½ 3m vAB2 Since the change in energy is the same for both cases, set them equal to each other: ½ 3m vAB2 = ½ m vA2 vAB2 = vA2/ 3 vAB = vA / √3

82. Homework for Section 5.5
See handout

83. Chapter 5 Formulas K = ½ mv2 kinetic energy
Ug = mgh gravitational potential energy
W = F r cos  work
Pave = W power as defined by work over time t
P = Fv cos  power if the force is making an angle  with v
Fs = -kx Hooke’s Law; relates spring force with spring constant and change in length
Us = ½ kx2 elastic potential energy