1. Chapter 6
Work and Energy

2. Introduction
Newton’s Second Law leads to definitions of work and energy
The concept of energy can be applied to individual particles or a system of particles
The total energy of an isolated system remains constant in time
Known as the principle of conservation of energy
Plays an important role in many fields
Introduction

3. Force, Displacement, and Work
The connection between force and energy is work
Work depends on the force, the displacement and the direction between them
Work in physics has a more specific meaning than in everyday usage
Section 6.1

4. Work
Experiments have verified that although various forces produce different times and distances, the product of the force and the distance remains the same
To accelerate an object to a specific velocity, you can exert a large force over a short distance or a small force over a long distance
Section 6.1

5. Work, cont. The product of F Δx is called work
For one-dimensional motion, W = F Δx
In two- or three-dimensions, you must take the vector nature of the force and displacement into account: W = F (Δr)cos θ
θ is the angle between the force and the displacement
Section 6.1

6. More About Work Units Work is a scalar
Newton x meter = Joule
N . m = J
Work is a scalar
Although the force and displacement are both vectors
Work can be positive or negative
These are not directions
Section 6.1

7. Work and Directions
The term F cos θ is equal to the component of the force along the direction of the displacement
When the component of the force is parallel to the displacement, the work is positive
When the component of the force is antiparallel to the displacement, the work is negative
When the component of the force is perpendicular to the displacement, the work is zero
Section 6.1

8. Relationships Among Work, Force and Displacement
Work is done by a force acting on an object
The work depends on the force acting on the object and on the object’s displacement
The value of W depends on the direction of the force relative to the object’s displacement
W may be positive, negative, or zero, depending on the angle θ between the force and the displacement
If the displacement is zero (the object does not move), then W = 0, even though the force may be very large
Section 6.1

9. Work, Physics Definition
The term work is used in everyday language
Its definition differs from the physics definition
If work is positive, the object will speed up
If work is negative, the object will slow down
Section 6.1

10. What Does the Work?
When an agent applies a force to an object and does an amount of work W on that object, the object will do an amount of work equal to –W back on the agent
The forces form a Newton’s Third Law action-reaction pair
Multiple agents
When multiple agents act on an object, you can calculate the work done by each separate agent
Section 6.1

11. Graphical Analysis of Work
So far, have assumed the force is constant
Look at a plot of force as a function of the displacement
When the force is constant, the graph is a straight line
The work is equal to the area under the plot
Section 6.1

12. Graphical Analysis of Work, cont.
The force doesn’t have to be constant
For each small displacement, Δx, you can calculate the work and then add those results to find the total work
The work is equal to the area under the curve
The area can be estimated by dividing the area in a series of rectangles
Section 6.1

13. Kinetic Energy
Find the work done on an object as it moves from the initial position xi to the final position xf
W = m a Δx
The acceleration can be expressed in terms of velocities
Combining: W = ½ m vf² - ½ m vi²
The quantity ½ m v² is called the kinetic energy
It is the energy due to the motion of the object
Section 6.2

14. Work and Kinetic Energy
The kinetic energy of an object can be changed by doing work on the object
W = ΔKE
This is called the Work-Energy theorem
The units of work and energy are the same
Joules, J
Another useful unit of energy is the calorie
1 cal = J
Section 6.2

15. Work and Amplifying Force
Suppose the person lifts his end of the rope through a distance L
The pulley will move through a distance of L/2
W on crate = (2T)(L/2) = TL
W on rope = TL
Work done on the rope is equal to the work done on the crate
Section 6.2

16. Work and Amplifying Force, cont.
The work done by the person is effectively “transferred” to the crate
Forces can be amplified, but work cannot be increased in this way
The force is amplified, but not the work
The associated displacement is decreased
The work-energy theorem suggests work can be converted to energy, but since work cannot be amplified the exchange will not increase the amount of energy available
The result that work cannot be amplified is a consequence of the principle of conservation of energy
Section 6.2

17. Potential Energy
When an object of mass m follows any path that moves through a vertical distance h, the work done by the gravitational force is always equal to mgh
W = mgh
An object near the Earth’s surface has a potential energy (PE) that depends only on the object’s height, h
The PE is actually a property of the Earth-object system
Section 6.3

18. Potential Energy, cont.
The work done by the gravitational force as the object moves from its initial position to its final position is independent of the path taken
The potential energy is related to the work done by the force on the object as the object moves from one location to another
Section 6.3

19. Potential Energy, final
Relation between work and potential energy
ΔPE = PEf – PEi = - W
Since W is a scalar, potential energy is also a scalar
The potential energy of an object when it is at a height y is PE = m g y
Applies only to objects near the Earth’s surface
Potential energy is stored energy
The energy can be recovered by letting the object fall back down to its initial height, gaining kinetic energy
Section 6.3

20. Conservative Forces
Conservative forces are forces that are associated with a potential energy function
Potential energy can be associated with forces other than gravity
The forces can be used to store energy as potential energy
Forces that do not have potential energy functions associated with them are called nonconservative forces
Section 6.3

21. Potential Energy and Conservative Forces, Summary
Potential energy is a result of the force(s) that act on an object
Since the forces come from the interaction between two objects, PE is a property of the objects (the system) involved in the force
Potential energy is energy that an object or system has by virtue of its position
Potential energy is stored energy
It can be converted to kinetic energy
Section 6.3

22. Potential Energy and Conservative Forces, Summary
Potential energy is a scalar
Its value can be positive, negative, or zero
It does not have a direction
Forces can be associated with a potential energy function
The work done is independent of the path taken
These forces are called conservative forces
Section 6.3

23. Potential Energy and Conservative Forces, Summary
Some forces are non-conservative forces
Examples: air drag and friction
Do not have potential energy functions
Cannot be used to store energy
Work depends on the path taken by the object of interest

24. Adding Potential Energy to the Work-Energy Theorem
In the work-energy theorem (W = ΔKE), W is the work done by all the forces acting on the object of interest
Some of those forces can be associated with a potential energy
Assume all the work is done by conservative forces
Gravity would be an example
W = - ΔPE = ΔKE
KEi + PEi = KEf + PEf
Applies to all situations in which all the forces are conservative forces
Section 6.3

25. Mechanical Energy
The sum of the potential and kinetic energies is called the mechanical energy
Since the sum of the mechanical energy at the initial location is equal to the sum of the mechanical energy at the final location, the energy is conserved
Conservation of Mechanical Energy
KEi + PEi = KEf + PEf
The results apply when many forces are involved as long as they are all conservative forces
A very powerful tool for understanding, analyzing, and predicting motion
Section 6.3

26. Conservation of Energy, Example
The snowboarder is sliding down a frictionless hill
Gravity and the normal forces are the only forces acting on the board
The normal is perpendicular to the object and so does no work on the snowboarder
Section 6.3

27. Conservation of Energy, Example, cont.
The only force that does work is gravity and it is a conservative force
Conservation of Mechanical Energy can be applied
Let the initial point be the top of the hill and the final point be the bottom of the hill
KEi + PEi = KEf + PEf → ½ m vi² + m g yi = ½ m vf² + m g yf
With the origin at the bottom of the hill, yi = h and yf = 0
Solve for the unknown
In this case, vf = ?
The final velocity depends on the height of the hill, not the angle
Section 6.3

28. Charting the Energy
A convenient way of illustrating conservation of energy is with a bar chart
The kinetic and potential energies of the snowboarder are shown
The sum of the energies is the same at the start and end
The potential energy at the top of the hill is transformed into kinetic energy at the bottom of the hill
Section 6.3

29. Conservation of Energy or Motion Equations?
In the snowboarder example, the hill in A is a straight incline and motion equations could be used to solve for the final velocity
In B, though, the complicated hill is more realistic
Since the slope isn’t a constant, the acceleration is not a constant
Motion equations could not be used
Easily solved using energy conservation
The shape of the hill has no effect on the final velocity
Section 6.3

30. Problem Solving Strategy
Recognize the principle
Find the object or system whose mechanical energy is conserved
Sketch the problem
Show the initial and final states of the object
Also include a coordinate system with an origin
Needed to measure the potential energy
Identify the relationships
Find expressions for the initial and final kinetic and potential energies
One or more of these may contain unknown quantities
Section 6.3

31. Problem Solving Strategy, cont.
Solve
Equate the initial mechanical energy to the final mechanical energy
Solve for the unknown quantities
Check
Consider what the answer means
Check that the answer makes sense
Reminder
This approach can only be used when the mechanical energy is conserved
Section 6.3

32. Conservation of Energy and Projectile Motion
Example, the ball is thrown straight upward and returns to its starting point
The potential energy varies parabolically with time
The kinetic energy varies as an inverted parabola
The total energy remains constant
Section 6.3

33. Changes in Potential Energy
The figure shows two possible choices for an origin in the problem
The change in potential energy is the same in both cases
It is the change in potential energy that is important
The change in potential energy does not depend on the choice of the origin
Section 6.3

34. Other Potential Energy Functions
There are potential energy functions associated with other forces
Examples include
Newton’s Law of Gravitation
Springs
Section 6.4

35. Gravitational Potential Energy Extended
A more general case of a potential energy function associated with gravity can be based on Newton’s Law of Gravitation
Remember, the equation for the force of gravity is
The negative sign indicates an attractive force
The gravitational potential energy of two objects separated by a distance r is
The negative sign means the potential energy is lowered as the objects are brought closer together
Section 6.4

36. Gravitational Potential Energy Extended, cont.
The change in potential energy is
An example would be the spacecraft: its potential energy changes as its separation from the Earth changes
Section 6.4

37. Escape Speed
The escape speed of a satellite is the speed needed for it to escape from the Earth’s gravitational pull
Measure distances from the center of the Earth
rf = ∞
Apply Conservation of Mechanical Energy
Section 6.4

38. Escape Speed, cont. Applying conservation of mechanical energy:
The initial distance is the radius of the Earth
The final distance is ∞, so PEf = 0
The final speed is 0
Section 6.4

39. Escape Speed, final For the Earth, vi = 1.1 x 104 m/s ~ 24,000 mph
For objects fired from different planets (or the Moon), vi depends only on the mass and radius of the planet
In reality, air drag would have to be taken into account when there is an atmosphere
Section 6.4

40. Which Gravitational Potential Energy?
PEgrav = m g y
The potential energy associated with the Earth’s gravitational force for objects near its surface
PEgrav = - G m1 m2 / r
The potential energy associated with the gravitational force between any two objects separated by a distance r
This is required for problems involving motion in the solar system

41. Springs and Elastic Potential Energy
There is a potential energy associated with springs and other elastic objects
When there is no force applied to its end, the spring is relaxed
Not stretched or compressed
Section 6.4

42. Springs and Elastic Potential Energy, cont.
Assume you exert a force to stretch the spring (B)
The spring itself exerts a force that opposes the stretching
You could also compress the spring (C)
The spring again exerts a force back in the opposite direction
Section 6.4

43. Hooke’s Law The force exerted by the spring has the form
Fspring = - k x
x is the amount the end of the spring is displaced from its equilibrium position
x = 0 at equilibrium
k is called the spring constant
Units are N/m
This is known as Hooke’s Law
Applies to objects other than the spring
Section 6.4

44. Hooke’s Law, cont.
Hooke’s Law is not a “law” of physics in the same sense as Newton’s Laws
It is an empirical relationship that experiments show works well for springs and some other objects
The force is a conservative force
A potential energy function can be associated with the force
Section 6.4

45. Potential Energy Stored in a Spring
Since the force is not constant, the work is found by looking at the area under the curve of the force-displacement curve
Area of triangles is ½ F x
But F = - k x
W = - ½ k x2
The negative sign confirms the force and displacement are in opposite directions
Section 6.4

46. Potential Energy Stored in a Spring, Summary
From the work, an expression for the potential energy can be found: PEspring = ½ k x2
The force exerted by the spring always opposes the displacement
So Fspring can be either positive or negative
Depends on if the spring is stretched or compressed
The potential energy is 0 when the spring is in its relaxed state
The spring potential energy always increases as a spring is either stretched or compressed
Section 6.4

47. Potential Energy and Force in a Spring: Summary
Section 6.4

48. Potential Energy with Multiple Forces
Conservation of Energy states KEi + PEi = KEf + PEf
Several different forces can contribute to the potential energy term
For example, there might be gravity and a spring acting on an object: PEtotal = PEgrav + PEspring
PEtotal = m g h + ½ k x2
Section 6.4

49. Elastic Forces and Holding Objects
When you hold an object, the work done it on is zero
Displacement is zero, so work is zero
However, the muscles deform
Muscle fibers are slipping
Motor-like molecules in your muscles are moving and do work
There is chemical energy expended in your muscles
Section 6.4

50. Non-conservative Forces
The work done by conservative forces is independent of the path
The work done by non-conservative forces does depend on the path taken
Non-conservative forces cannot be associated with a potential energy
Friction is an example of a non-conservative force
Section 6.5

51. Friction Example
The work done by friction in moving the block along path B is larger than if it moved along path A
Other non-conservative forces have the same property
Section 6.5

52. Non-conservative Forces and the Work-Energy Theorem
The work in the work-energy theorem was the total work
This work can be due to several different forces
Wtotal = Wcon + Wnoncon
ΔPE = -Wcon
Then, the work-energy theorem can be restated as KEi + PEi + Wnoncon = KEf + PEf
This is the general work-energy theorem with nonconservative forces
The final mechanical energy is equal to the initial mechanical energy plus the work done by any nonconservative forces that act on the object
Section 6.5

53. Conservation of Energy, Revisited
Suppose a system of particles or objects exerts forces on one another as they move about
These forces may be conservative or non-conservative
Assume no forces from outside the system act on the system
So total energy will be conserved
The mechanical energy may be converted to another form of energy such as heat, electrical, chemical, etc.
If some agent from outside exerts a force on one of the particles within the system, the associated amount of work will change the total energy of the system
The same amount of energy must be removed from the agent, so total energy is conserved
Section 6.5

54. Friction Details
For the sliding block shown, the energy associated with Wnoncon goes into heating up the surface of the block and the surface of the floor
This increase is associated with more movement of the atoms
Total energy is still conserved
It is not possible to return all the energy of the atoms back to the block’s motion
Section 6.6

55. Power
Time enters into the ideas of work and energy through the concept of power
The average power is defined as the rate at which the work is being done
Unit is watts (W)
1 W = 1 J/s
Sometimes expressed as horsepower
1 hp = W
Also applies to chemical and electrical processes and devices
Section 6.7

56. Power and Velocity
Power can also be expressed in terms of the velocity at which an object is moving
This also applies for instantaneous power and velocity: P = F v
For a given power,
The motor can exert a large force while moving slowly
The motor can exert a small force while moving quickly
Section 6.7

57. Efficiency
The efficiency, ε, of a system can be used to find the maximum allowable force consistent with Newton’s Laws and conservation of mechanical energy
The efficiency can not be greater than 1
Section 6.7

58. Molecular Motor Example
Myosin can be modeled as a motor
It moves along long filaments of actin molecules
Energy source is chemical
The maximum force is 10 pN
Assumes an efficiency of 1
Section 6.8