Work and Energy

1
Chapter 6

• Work and Energy

2
Introduction

• Newtons 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 ?x is called work
• For one-dimensional motion, W F ?x
• In two- or three-dimensions, you must take the
  vector nature of the force and displacement into
  account W F (?r)cos ?
• ? is the angle between the force and the
  displacement

Section 6.1
6
More About Work

• Units
• 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 ? 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 objects displacement
• The value of W depends on the direction of the
  force relative to the objects displacement
• W may be positive, negative, or zero, depending
  on the angle ? 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 Newtons 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 doesnt have to be constant
• For each small displacement, ?x, 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 ?x
• 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 ?KE
• 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 4.186 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 Earths surface has a
  potential energy (PE) that depends only on the
  objects 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
• ?PE 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 Earths 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 ?KE), 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 - ?PE ?KE
• 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 isnt 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
• Newtons 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
  Newtons 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 Earths
  gravitational pull
• Measure distances from the center of the Earth
• rf 8
• 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 8, 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 Earths
  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
Hookes 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 Hookes Law
• Applies to objects other than the spring

Section 6.4
44
Hookes Law, cont.

• Hookes Law is not a law of physics in the same
  sense as Newtons 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
• ?PE -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 blocks 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 745.7 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, e, of a system can be used to
  find the maximum allowable force consistent with
  Newtons 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