1. Chapter 5
Section 1 Work
Preview
Objectives
Definition of Work

2. Chapter 5
Section 1 Work
Objectives
Recognize the difference between the scientific and ordinary definitions of work.
Define work by relating it to force and displacement.
Identify where work is being performed in a variety of situations.
Calculate the net work done when many forces are applied to an object.

3. Chapter 5 Definition of Work
Section 1 Work
Definition of Work
Work is done on an object when a force causes a displacement of the object.
No displacement = no work done
Work is done only when components of a force are parallel to a displacement.
In other words, if an object moves in the +x direction but the force is applied in the +y direction, NO work is done. (ex: waitress carrying a tray)

4. Chapter 5
Section 1 Work
Definition of Work

5. Work has SI units of Joules (J) Work is scalar
1 J = 1 Nm
Work is scalar
+W means force and displacement are in the same direction
-W means force and displacement are in opposite directions (ex: friction)

6. Sign Conventions for Work
Chapter 5
Section 1 Work
Sign Conventions for Work
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Visual Concept

7. Examples
A tugboat pulls a ship with a constant net horizontal force of 5.00 x 103 N and causes the ship to move through a harbor. How much work is done on the ship it if moves a distance of 3.00 km?
A weightlifter lifts a set of weights a vertical distance of 2.00 m. If a constant net force of 350 Nis exerted on the weights, what is the net work done on the weights?

8. Examples
A shopper in a supermarket pushes a cart with a force of 35 N directed at an angle of 25⁰ downward from the horizontal. Find the work done by the shopper on the cart as the shopper moves along a 50.0 m length of aisle.
If 2.0 J of work is done in raising a 180 g apple, how far is it lifted?

9. Chapter 5 Preview Objectives Kinetic Energy Sample Problem
Section 2 Energy
Preview
Objectives
Kinetic Energy
Sample Problem

10. Chapter 5 Objectives Identify several forms of energy.
Section 2 Energy
Objectives
Identify several forms of energy.
Calculate kinetic energy for an object.
Apply the work–kinetic energy theorem to solve problems.
Distinguish between kinetic and potential energy.
Classify different types of potential energy.
Calculate the potential energy associated with an object’s position.

11. Chapter 5 Kinetic Energy Kinetic Energy
Section 2 Energy
Kinetic Energy
Kinetic Energy
The energy of an object that is due to the object’s motion is called kinetic energy
Units of energy are Joules (J)
Kinetic energy depends on speed and mass.

12. Chapter 5 Kinetic Energy Section 2 Energy
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Visual Concept

13. Examples
Calculate the speed of an 8.0 x 104 kg airliner with a kinetic energy of 1.1 x 109 J.
What is the speed of a kg baseball if its kinetic energy is 109 J?
Two bullets have masses of 3.0 g and 6.0 g, respectively. Both are fired with a speed of 40.0 m/s. Which bullet has more kinetic energy? What is the ratio of their kinetic energies?

14. Examples
Two 3.0 g bullets are fired with speeds of 40.0 m/s and 80.0 m/s, respectively. What are their kinetic energies? Which bullet has more kinetic energy? What is the ratio of their kinetic energies?
A car has a kinetic energy of 4.32 x 105 J when traveling at a speed of 23 m/s. What is its mass?

15. Kinetic Energy, continued
Chapter 5
Section 2 Energy
Kinetic Energy, continued
Work-Kinetic Energy Theorem
The NET work done by all the forces acting on an object is equal to the change in the object’s kinetic energy.
Wnet = ∆KE
net work = change in kinetic energy

16. Work-Kinetic Energy Theorem
Chapter 5
Section 2 Energy
Work-Kinetic Energy Theorem
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Visual Concept

17. Chapter 5 Sample Problem Work-Kinetic Energy Theorem
Section 2 Energy
Sample Problem
Work-Kinetic Energy Theorem
On a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10?

18. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Work-Kinetic Energy Theorem
1. Define
Given:
m = 10.0 kg
vi = 2.2 m/s
vf = 0 m/s
µk = 0.10
Unknown:
d = ?

19. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Work-Kinetic Energy Theorem
2. Plan
Choose an equation or situation: This problem can be solved using the definition of work and the work-kinetic energy theorem.
Wnet = Fnetdcosq
The net work done on the sled is provided by the force of kinetic friction.
Wnet = Fkdcosq = µkmgdcosq

20. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Work-Kinetic Energy Theorem
2. Plan, continued
The force of kinetic friction is in the direction opposite d, q = 180°. Because the sled comes to rest, the final kinetic energy is zero.
Wnet = ∆KE = KEf - KEi = –(1/2)mvi2
Use the work-kinetic energy theorem, and solve for d.

21. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Work-Kinetic Energy Theorem
3. Calculate
Substitute values into the equation:

22. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Work-Kinetic Energy Theorem
4. Evaluate
According to Newton’s second law, the acceleration of the sled is about -1 m/s2 and the time it takes the sled to stop is about 2 s. Thus, the distance the sled traveled in the given amount of time should be less than the distance it would have traveled in the absence of friction.
2.5 m < (2.2 m/s)(2 s) = 4.4 m

23. Examples
A student wearing frictionless in-line skates on a horizontal surface is pushed by a friend with a constant force of 45 N. How far must the student be pushed, starting from rest, so that their final kinetic energy is 352 J?
A 2.1 x 103 kg car starts from rest at the top of a driveway that is sloped at an angle of 20.0⁰ with the horizontal. An average friction force of 4.0 x 103 N impedes the car’s motion so that the car’s speed at the bottom of the driveway is 3.8 m/s. What is the length of the driveway?

24. Examples
A 2.0 x 103 kg car accelerates from rest under the actions of 2 forces. One is a forward force of 1140 N provided by the traction between the wheels and the road. The other is a 950 N resistive force due to various frictional forces. Use the work-kinetic energy theorem to determine how far the car must travel for its speed to reach 2.0 m/s.
A 75 kg bobsled is pushed along a horizontal surface by two athletes. After the bobsled is pushed a distance of 4.5 m starting from rest, its speed is 6.0 m/s. Find the magnitude of the net force on the bobsled.

25. gravitational PE = mass  free-fall acceleration  height
Chapter 5
Section 2 Energy
Potential Energy
Potential Energy is the energy associated with an object because of the position, shape, or condition of the object.
Gravitational potential energy is the potential energy stored in the gravitational fields of interacting bodies.
Gravitational potential energy depends on height from a zero level.
PEg = mgh
gravitational PE = mass  free-fall acceleration  height

26. Chapter 5 Potential Energy Section 2 Energy
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Visual Concept

27. Potential Energy, continued
Chapter 5
Section 2 Energy
Potential Energy, continued
Elastic potential energy is the energy available for use when a deformed elastic object returns to its original configuration.
The symbol k is called the spring constant, a parameter that measures the spring’s resistance to being compressed or stretched.

28. Elastic Potential Energy
Chapter 5
Section 2 Energy
Elastic Potential Energy

29. Chapter 5 Spring Constant Section 2 Energy
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Visual Concept

30. Chapter 5 Sample Problem Potential Energy
Section 2 Energy
Sample Problem
Potential Energy
A 70.0 kg stuntman is attached to a bungee cord with an unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, what is the total potential energy relative to the water when the man stops falling?

31. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Potential Energy
1. Define
Given:m = 70.0 kg
k = 71.8 N/m
g = 9.81 m/s2
h = 50.0 m – 44.0 m = 6.0 m
x = 44.0 m – 15.0 m = 29.0 m
PE = 0 J at river level
Unknown: PEtot = ?

32. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Potential Energy
2. Plan
Choose an equation or situation: The zero level for gravitational potential energy is chosen to be at the surface of the water. The total potential energy is the sum of the gravitational and elastic potential energy.

33. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Potential Energy
3. Calculate
Substitute the values into the equations and solve:

34. Sample Problem, continued
Chapter 5
Section 2 Energy
Sample Problem, continued
Potential Energy
4. Evaluate
One way to evaluate the answer is to make an order-of-magnitude estimate. The gravitational potential energy is on the order of 102 kg  10 m/s2  10 m = 104 J. The elastic potential energy is on the order of 1  102 N/m  102 m2 = 104 J. Thus, the total potential energy should be on the order of 2  104 J. This number is close to the actual answer.

35. Examples
A spring with a force constant of 5.2 N/m has a relaxed length of 2.45 m. When a mass is attached to the end of the spring and allowed to come to rest, the vertical length of the spring is 3.57 m. Calculate the elastic potential energy stored in the spring.
The staples inside a stapler are kept in place by a spring with a relaxed length of m. If the spring constant is 51.0 N/m, how much elastic potential energy is stored in the spring when its length is m?

36. Examples
A 40.0 kg child is in a swing that is attached to ropes 2.00 m long. Find the gravitational potential energy associated with the child relative to the child’s lowest position under the following conditions:
When the ropes are horizontal
When the ropes make a 30 degree angle with the vertical
At the bottom of the circular arc

37. Chapter 5 Preview Objectives Conserved Quantities Mechanical Energy
Section 3 Conservation of Energy
Chapter 5
Preview
Objectives
Conserved Quantities
Mechanical Energy
Sample Problem

38. Section 3 Conservation of Energy
Chapter 5
Objectives
Identify situations in which conservation of mechanical energy is valid.
Recognize the forms that conserved energy can take.
Solve problems using conservation of mechanical energy.

39. Chapter 5 Conserved Quantities
Section 3 Conservation of Energy
Chapter 5
Conserved Quantities
When we say that something is conserved, we mean that it remains constant.
It may change forms but the total amount is always the same
For example, an ice cube has a mass of 11 g. It begins melting. Now it has a mass of 8 grams but there is 3 g of water. The overall mass stays the same but the form changes.

40. Chapter 5 Mechanical Energy
Section 3 Conservation of Energy
Chapter 5
Mechanical Energy
Mechanical energy is the sum of kinetic energy and all forms of potential energy associated with an object or group of objects.
ME = KE + ∑PE
Mechanical energy is often conserved.
MEi = MEf
initial mechanical energy = final mechanical energy (in the absence of friction)

41. Conservation of Energy
If initial mechanical energy equals final mechanical energy then…
What energy transfers are being made when…
A person swings on a swing?
A rollercoaster rolls?
A clock pendulum swings?

42. Conservation of Mechanical Energy
Section 3 Conservation of Energy
Chapter 5
Conservation of Mechanical Energy
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Visual Concept

43. Chapter 5 Sample Problem Conservation of Mechanical Energy
Section 3 Conservation of Energy
Chapter 5
Sample Problem
Conservation of Mechanical Energy
Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00 m. What is her speed at the bottom of the slide? Assume she has a mass of 25.0 kg.

44. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
1. Define
Given:
h = hi = 3.00 m
m = 25.0 kg
vi = 0.0 m/s
hf = 0 m
Unknown:
vf = ?

45. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
2. Plan
Choose an equation or situation: The slide is frictionless, so mechanical energy is conserved. Kinetic energy and gravitational potential energy are the only forms of energy present.

46. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
2. Plan, continued
The zero level chosen for gravitational potential energy is the bottom of the slide. Because the child ends at the zero level, the final gravitational potential energy is zero.
PEg,f = 0

47. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
2. Plan, continued
The initial gravitational potential energy at the top of the slide is
PEg,i = mghi = mgh
Because the child starts at rest, the initial kinetic energy at the top is zero.
KEi = 0
Therefore, the final kinetic energy is as follows:

48. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
3. Calculate
Substitute values into the equations:
PEg,i = (25.0 kg)(9.81 m/s2)(3.00 m) = 736 J
KEf = (1/2)(25.0 kg)vf2
Now use the calculated quantities to evaluate the final velocity.
MEi = MEf
PEi + KEi = PEf + KEf
736 J + 0 J = 0 J + (0.500)(25.0 kg)vf2
vf = 7.67 m/s

49. Sample Problem, continued
Section 3 Conservation of Energy
Chapter 5
Sample Problem, continued
Conservation of Mechanical Energy
4. Evaluate
The expression for the square of the final speed can be written as follows:
Notice that the masses cancel, so the final speed does not depend on the mass of the child. This result makes sense because the acceleration of an object due to gravity does not depend on the mass of the object.

50. Mechanical Energy, continued
Section 3 Conservation of Energy
Chapter 5
Mechanical Energy, continued
Mechanical Energy is not conserved in the presence of friction.
As a sanding block slides on a piece of wood, energy (in the form of heat) is dissipated into the block and surface.

51. Examples
A bird is flying with a speed of 18.0 m/s over water when it accidentally drops a 2.00 kg fish. If the altitude of the bird is 5.40 m and friction is disregarded, what is the speed of the fish when it hits the water?
A 755 N diver drops from a board 10.0 m above the water’s surface. Find the diver’s speed 5.0 m above the water’s surface. Then find the diver’s speed just before striking the water.

52. Examples
If the diver in item 2 leaves the board with an initial upward speed of 2.00 m/s, find the diver’s speed when striking the water.
An Olympic runner leaps over a hurdle. If the runner’s initial vertical speed is 2.2 m/s, how much will the runner’s center of mass be raised during the jump?
A pendulum bob is released form some initial height such that the speed of the bob at the bottom of the swing is 1.9 m/s. What is the initial height of the bob?

53. Chapter 5
Section 4 Power
Preview
Objectives
Rate of Energy Transfer

54. Chapter 5 Objectives Relate the concepts of energy, time, and power.
Section 4 Power
Objectives
Relate the concepts of energy, time, and power.
Calculate power in two different ways.
Explain the effect of machines on work and power.

55. Rate of Energy Transfer
Chapter 5
Section 4 Power
Rate of Energy Transfer
Power is a quantity that measures the rate at which work is done or energy is transformed.
P = W/∆t
power = work ÷ time interval
An alternate equation for power in terms of force and speed is
P = Fv
power = force  speed

56. Chapter 5 Power Section 4 Power
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Visual Concept

57. Examples
A 1.0 x 103 kg elevator carries a maximum load of kg. A constant frictional force of 4.0 x 103 N retards the elevator’s motion upward. What minimum power, in kilowatts, must the motor deliver to lift the fully loaded elevator at a constant speed of 3.00 m/s.
A car with a mass of 1.5 x 103 kg starts from rest and accelerates to a speed of 18 m/s in 12 s. Assume that the force of resistance remains constant at 400 N during this time. What is the average power developed by the car’s engine?

58. Examples
A rain cloud contains 2.66 x 107 kg of water vapor. How long would it take for 2.00 kW pump to raise the same amount of water to the cloud’s altitude , 2.00 km?
How long does it take a 19 kW steam engines to do 6.8 x 107 J of work?
A 1.50 x 103 kg car accelerates uniformly from rest to 10.0 m/s in 3.00 s
What is the work done on the car in this time interval?
What is the power delivered by the engine in this time interval?