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Title: Review of Lecture 7


1
Review of Lecture 7
  • Energy
  • Work
  • Work Kinetic Energy
  • Work Done by a Gravitational Force
  • Work Done by a Spring Force
  • Work Done by an Applied Force
  • Work Energy Theorem
  • Power

2
Potential Energy
  • Last week we defined something called kinetic
    energy the energy associated with the motion of
    an object
  • We will now define a 2nd kind of energy called
    potential energy
  • This is the energy associated with a change in
    the configuration of a system
  • OK but what the heck does that mean???

3
Potential Energy
  • Potential energy is perhaps best explained by a
    couple examples
  • If you lift a weight (for example, groceries from
    a bag on the floor to a shelf in the pantry), the
    groceries are now farther away from the center of
    the earth you changed the configuration of the
    grocery-earth system
  • In other words, you changed the gravitational
    potential energy of the grocery-earth system

4
Potential Energy
  • Here is another example
  • You push the ball down the tube of the projectile
    launcher and hear the ratchet click a couple
    times indicating that the launcher is ready to
    fire the ball
  • In doing so you compressed a spring in the
    launcher and changed its configuration you
    changed the elastic potential energy of the
    spring system

5
Work and Potential Energy
  • Last week we discussed the relationship between
    work and kinetic energy
  • Now that we have defined a new kind of energy, we
    should see how that fits into the picture

6
Work and Potential Energy
  • Last week when we threw the tomato up we noted
    that negative work was being done on the tomato
    which caused it to slow down during its ascent
  • As a result, the kinetic energy of the tomato was
    reduced eventually to zero
  • But where did that energy go???

7
Work and Potential Energy
  • Where it went was into an increase in the
    gravitational potential energy of the tomato
  • This is because the configuration of the
    tomato-earth system was changed (the tomato got
    farther away from the center of the earth)

8
Work and Potential Energy
  • The reverse of course happens when the tomato
    begins to fall again
  • Now the positive work done by the gravitational
    force causes the gravitational potential energy
    to be reduced and the tomatos kinetic energy
    increases

9
Work and Potential Energy
  • From this we can see that for either the rise or
    fall of the tomato, the change ?U in the
    gravitational potential energy is the negative of
    the work done on the tomato by the gravitational
    force
  • In equation form we get

10
Work and Potential Energy
  • This equation applies also to the spring-block
    system shown at the right
  • If we give the block a shove to the right, the
    kinetic energy of the block is transferred into
    elastic potential energy as the spring compresses

11
Work and Potential Energy
  • The work done in compressing the spring is the
    negative of the change in the blocks kinetic
    energy
  • And of course the reverse happens when the spring
    stretches back out potential energy gets
    transformed back into kinetic energy

12
A little formality
  • A system consists of two or more objects
  • A force acts between a particle-like object in
    the system and the rest of the system
  • When the system configuration changes, the force
    does work (W1) on the particle-like object,
    transferring energy between the kinetic energy of
    the object and some other form of energy of the
    system

13
A little formality
  • When the configuration change is reversed, the
    force reverses the energy transfer, doing work W2
    in the process

14
Conservative and Nonconservative Forces
  • If it is true that W1 -W2 in a system, then the
    other form of energy is always a potential energy
    and the force is said to be a conservative force
  • As you can see from the above definition, the
    gravitational and spring forces are conservative
    forces

15
Conservative and Nonconservative Forces
  • Conversely, a system where W1 ? -W2, has forces
    that are nonconservative
  • Examples of nonconservative forces are the
    kinetic frictional force and the drag force of an
    object moving through a fluid
  • Lets look at the kinetic frictional force first

16
Conservative and Nonconservative Forces
  • We know that as we slide a block along the floor,
    the floor and the block have friction
  • The kinetic frictional force does negative work
    on the block (slowing it down) - this negative
    work is transferred into heat (thermal energy)
  • We also know that this energy transfer cant be
    reversed (e.g., warming up the floor-block
    interface wont cause the block to start moving)
  • From this we can conclude that thermal energy
    (heat) is not a potential energy

17
Conservative and Nonconservative Forces
  • When only conservative forces act on a
    particle-like object, we can greatly simplify
    otherwise difficult problems involving the motion
    of objects
  • We will now develop a test for identifying
    conservative forces which will provide us with
    one means of simplifying such problems

18
Path Independence of Conservative Forces
  • The primary test for determining whether a force
    is conservative or not is this
  • Let the force in question act on a particle-like
    object such that it causes the object to move
    from one point to another and then back again to
    the starting point in other words, the object
    makes a round trip along a closed path

19
Path Independence of Conservative Forces
  • The force is conservative if the total energy
    transferred to the object over the course of the
    round trip along this closed path (or any other
    closed path) is zero
  • In other words
  • The net work done by a conservative force on a
    particle around every closed path is zero

20
Path Independence of Conservative Forces
  • We know from experience that the gravitational
    force is conservative
  • The tomato leaves our hand with a kinetic energy
    of

21
Path Independence of Conservative Forces
  • Work equal in amount to the value of K.E. (but
    opposite in sign) slows the tomato to a stop
  • The tomato then accelerates downwards again
  • At the point where it was thrown (lets say y
    0), it reaches the same kinetic energy as it had
    initially

22
Path Independence of Conservative Forces
  • An important result of the closed-path test is
    the following
  • The work done by a conservative force on a
    particle moving between two points does not
    depend on the path taken by the particle
  • Thats precisely what we mean when we say path
    independence

23
Path Independence of Conservative Forces
  • Suppose a particle moves from point a to point b
    along either path 1 or path 2
  • If the force acting on the particle is
    conservative, then the work done on the particle
    is the same
  • In other words Wab,1 Wab,2

24
Path Independence of Conservative Forces
  • This result is very powerful
  • It means that we neednt concern ourselves about
    the details of the path we only need to care
    about the endpoints

25
Path Independence of Conservative Forces
  • So far we have just asserted that Wab,1 Wab,2
    it would be a nice thing if the assertion were
    proved
  • Without worrying about where positive and
    negative work gets done along the paths, lets
    let the work done it getting from a to b along
    path 1 be denoted as Wab,1 and the work done in
    getting from b back to a along path 2 as Wba,2

26
Path Independence of Conservative Forces
  • If the force is conservative, then Wab,1 Wba,2
    0 and thus Wab,1 -Wba,2
  • In other words, the work done on the outward path
    must be the negative of the work done on the
    return path

27
Path Independence of Conservative Forces
  • But we also know that if the force is
    conservative the work done in getting from a to
    b along path 2 must be the negative of the work
    done in getting from b to a along path 2 thus
    Wab,2 -Wba,2

28
Path Independence of Conservative Forces
  • Substituting Wab,2 for -Wba,2 in the initial
    equation of Wab,1 -Wba,2 we then get
    Wab,1 Wab,2which is what we set out to prove

29
Checkpoint 1
  • A single force F does the indicated work on a
    particle moving in the indicated direction along
    each of the paths
  • Is the force conservative?

30
Sample Problem 8-1
  • A 2.0 kg block of slippery cheese slides along a
    frictionless track from point a to point b
  • The total distance traveled along the track is
    2.0 m and the net vertical drop is 0.80 m
  • How much work is done on the cheese by the
    gravitational force during the trip?

31
Sample Problem 8-1
  • We have no detailed information about the path so
    we cant use that to solve the problem (and even
    if we did we wouldnt want to!)
  • But we do have information about the starting and
    ending point of the trip which is all we really
    need

32
Sample Problem 8-1
  • We know that the total work done is the same
    regardless of the path so lets pick an
    alternative path that allows us an easy solution
    to the problem
  • We can do this because the only force we are
    dealing with here is the force of gravity and
    we know that the gravitational force is
    conservative

33
Sample Problem 8-1
  • So instead of using this path

we will use this one
34
Sample Problem 8-1
  • First look at the horizontal segment of the path
  • The work done is

35
Sample Problem 8-1
  • Now lets look at the vertical segment of the
    path
  • The work done is

36
Sample Problem 8-1
  • The total work done is the sum of the horizontal
    and vertical components, thus

37
Determining PotentialEnergy Values
  • Wed like to be able to come up with an equation
    that relates the potential energy of a particle
    to the force acting on it
  • Lets assume that our object is part of a system
    in which a conservative force acts
  • When the force acts on the object, the change ?U
    in the objects potential energy is the negative
    of the work done on the object

38
Determining PotentialEnergy Values
  • In the general case, we can relate the work done
    on an object as
  • Substituting in our earlier relationship for work
    and potential energy we get

39
GravitationalPotential Energy
  • Lets imagine a particle moving along the y axis
    (positive upwards) from point yi to point yf
  • As the particle moves, the gravitational force F
    does work on it we therefore get

40
GravitationalPotential Energy
  • Carrying the integral on we getwhich yields

41
GravitationalPotential Energy
  • If we let the initial value of y 0, then we
    finally get
  • This equation tells us that
  • The gravitational energy associated with a
    particle-earth system depends only on the
    vertical position y (or height) of the particle
    relative to the reference position (y 0)

42
Elastic Potential Energy
  • Now lets do the same analysis for aspring-block
    system (where the spring has a spring constant k)
  • As the block moves from point xi to point xf, the
    spring force F -kx does work on the block

43
Elastic Potential Energy
  • Substituting in kx for the force in our earlier
    equation we get

44
Elastic Potential Energy
  • Which finally results in
  • Again, if we let xi 0, we get

45
Checkpoint 2
  • Given a conservative force F1(x), rank the three
    situations according to the change in potential
    energy during the particles movement from x 0
    to x1, most positive first

46
Sample Problem 8-2
  • A 2.0 kg sloth hangs 5.0 m above the ground
  • What is the gravitational PE of the sloth-earth
    system when measured at (1) the ground, (2) the
    balcony, (3) the sloth, and (4) 1.0 m above the
    limb?

47
Sample Problem 8-2
  • Here we need to use the equationfor the
    different values of yi
  • (1) 98 J(2) 39 J(3) 0 J(4) -20 J

48
Sample Problem 8-2
  • The sloth drops to the ground
  • For each reference point, what is the change in
    PE?

49
Sample Problem 8-2
  • In each case the ?y is the same -5.0 m
  • So the change in PE is

50
Conservation ofMechanical Energy
  • The mechanical energy of a system is simply the
    sum of its potential energy and the kinetic
    energy of the objects within it
  • For the moment, we will assume that all of the
    forces acting on the system are conservative in
    other words, there are no frictional or drag
    forces present

51
Conservation ofMechanical Energy
  • We will also assume that the system is isolated
    meaning that there are no external forces acting
    on it
  • We know that when conservative forces do work
    they act to transfer energy between the kinetic
    energy of objects in the system and the potential
    energy of the system

52
Conservation ofMechanical Energy
  • We know from equation 7-10 that the change in
    kinetic energy is
  • We also know from equation 8-1 that the change in
    potential energy is

53
Conservation ofMechanical Energy
  • We can therefore combine these two equations to
    getwhich tells us that, in an isolated system
    with conservative forces, the kinetic energy
    increases exactly as much as the potential energy
    decreases

54
Conservation ofMechanical Energy
  • Written a little differently, we havewhich
    can be rearranged to bewhere the subscripts
    indicate two different states of the system

55
Conservation ofMechanical Energy
  • But we said from the outset that the sum of the
    kinetic and potential energies was defined to be
    the mechanical energy of the system
  • So we can now see that, for an isolated system
    with only conservative forces, the mechanical
    energy of the system cannot change

56
Conservation ofMechanical Energy
  • Or said slightly differently In an isolated
    system where only conservative forces cause
    change, the kinetic and potential energies can
    change, but their sum, the mechanical energy of
    the system, cannot change

57
Conservation ofMechanical Energy
  • This result is called the principle of
    conservation of mechanical energy
  • It allows us to examine complicated systems
    without having to consider what happens at all
    times (e.g., all of the intermediate states) and
    without having to consider the work done by the
    force(s) involved

58
Conservation ofMechanical Energy
  • A great illustration of the principle of
    conservation of mechanical energy is the pendulum
  • Lets look at Figure 8-7 on page 174

59
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60
Conservation ofMechanical Energy
  • We can see here at state (a) that the pendulum
    bob is at the bottom of its arc of travel
  • As a result, the energy in the system is entirely
    kinetic

61
Conservation ofMechanical Energy
  • Now when we look at state (b) we can see that the
    pendulum bob has passed the bottom of its arc of
    travel and is moving upwards
  • In this case, its velocity will be lower thus
    it has a lower kinetic energy and an increased
    potential energy

62
Conservation ofMechanical Energy
  • Now at state (c) the pendulum bob has reached the
    top of its arc and has stopped momentarily
  • Now its velocity is zero, so it has zero kinetic
    energy on the other hand, since it is at the top
    of its arc (e.g., the highest vertical
    displacement) the pendulum is at its maximum
    potential energy

63
Conservation ofMechanical Energy
  • The process then continues but in the opposite
    direction

64
Conservation ofMechanical Energy
  • Suppose we knew that the kinetic energy at the
    bottom of the arc (point a in Fig. 8-7) was 20 J

65
Conservation ofMechanical Energy
  • Then without any further work we would also know
    that the potential energy at the top of the arc
    (point c in Fig. 8-7) is also 20 J

66
Checkpoint 3
  • Assume a frictionless ramp
  • Rank in order of the kinetic energy at point B,
    greatest first
  • Rank according to the speed of the block at point
    B, greatest first

67
Sample Problem 8-3
  • A child of mass m is released from the top of the
    slide 8.5 m above the water
  • Assuming no friction, what is the speed of the
    child when she reaches the bottom of the slide?

68
Sample Problem 8-5
  • A 61.0 kg bungee-cord jumper is on a bridge 45.0
    m above a river
  • The relaxed length of the cord is 25.0 m
  • Assume the cord obeys Hooks law and has a spring
    constant of 160 N/m
  • What is the height of the jumpers feet above the
    river when she stops assuming she doesnt hit
    the water of course

69
  • Let L 25.0 m (the relaxed length of the cord)
  • Let d the extension of the cord
  • Solve for h the height of her feet above the
    river

70
Potential Energy Curves
  • Suppose that we know the potential energy
    function and want to know the force that
    corresponds
  • For a one-dimensional system, we know

71
Potential Energy Curves
  • Solving for F(x) and passing to the limit of the
    differential we get
  • Now suppose we plot some 1-dimensional potential
    energy function U(x)

72
Potential Energy Curves
  • Suppose the PE function looked like this
  • Lets analyze it and see what we can find out
    about the behavior of the particle

73
Potential Energy Curves
  • We know that we can find the force function by
    simply taking the derivative of the PE function
  • So the force function looks like the lower curve
    which is simply found by taking negative the
    slope of the upper curve

74
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75
Potential Energy Curves(Turning Points)
  • We know thatthus

76
Potential Energy Curves(Turning Points)
  • We also know that Emch is a constant
  • Lets assume that Emch 5.0 J (which is shown in
    Figure 8-10(a))
  • The equation for K(x) tells us how to find its
    value from the PE curve just subtract the value
    of U(x) at the point (x)

77
Potential Energy Curves(Turning Points)
  • Since K can never be negative (because v2 is
    always positive), the particle can never move to
    the left of x1
  • Note that when the particle reaches x1, the force
    on the particle is positive (because the slope
    dU/dx is negative)
  • As a result the particle is driven towards the
    right and the point x1 is called a turning point

78
Potential Energy Curves (Turning Points)
  • Now take a look at Figure 8-10 (c)

79
Potential Energy Curves(Turning Points)
  • Here we have three different values for Emch
    superimposed on the plot
  • Lets examine the behavior of the particle at
    various points along the plot assuming these
    different values of Emch

80
Potential Energy Curves(Turning Points)
Neutral Equilibrium
(Marble on a tabletop)
Turning Points
81
Potential Energy Curves(Turning Points)
Unstable Equilibrium
Turning Points
(Marble on a bowling ball)
82
Potential Energy Curves(Turning Points)
Turning Points
Stable Equilibrium
(Marble in a bowl)
83
Checkpoint 4
  • This is the PE function for a particle in a
    1-dimensional system of motion
  • Rank the regions according to the magnitude of
    the force on the particle, greatest first
  • What is the direction of the force when the
    particle is in region AB?

84
Work Done on a SystemBy An External Force
  • So far we have defined work as the energy
    transferred to or from an object by means of a
    force acting on that object
  • We will now extend the definition to a system of
    particles
  • Work is energy transferred to or from a system by
    an external force acting on that system

85
Work Done on a SystemBy An External Force
  • In (a) energy is transferred to the system thus
    the work done is positive
  • In (b) energy is transferred from the system
    thus the work done is negative

86
Work Done By An External Force (No Friction)
  • If you toss a bowling ball up in the air
    (ignoring air friction) you have obviously done
    some work but what is the system that you did
    the work on?
  • Ask yourself where did the energy change?

87
Work Done By An External Force (No Friction)
  • You clearly changed the KE of the ball
  • And because the separation of the ball and the
    earth increased, you also changed the PE of the
    ball-earth system
  • So the system must include both the ball and
    the earth

88
Work Done By An External Force (No Friction)
  • The work must include both changes in energy,
    thus

89
Work Done By An External Force (Friction Involved)
  • We have a box being propelled by a constant force
    F, with a retarding frictional force fk
  • The box starts with an initial velocity v0, and
    travels distance d ending with a final velocity v

90
Work Done By An External Force (Friction Involved)
  • We begin by applying Newtons 2nd law to get
  • Because the forces are all constant, so is the
    acceleration a as a result we can use the
    equation

91
Work Done By An External Force (Friction Involved)
  • Solving this equation for a and then substituting
    that back into the previous equation and
    rearranging we get
  • But the first two terms on the right are just ?K
    for the block so we get

92
Work Done By An External Force (Friction Involved)
  • As posed, the problem is 1-dimensional
  • But suppose the block were being pulled up a ramp
    in that case there would also be a ?U to
    contend with
  • So to make the solution a little more general we
    will change the ?K term to ?Emec

93
Work Done By An External Force (Friction Involved)
  • We therefore get
  • By experimentation, we find that that as the
    block slides along, the block and the floor get
    warmer due to the friction
  • As you will find out next term, the temperature
    of the block is related to an objects thermal
    energy

94
Work Done By An External Force (Friction Involved)
  • Again by experimentation, we have found that the
    thermal energy ?Eth is
  • So we can rewrite our earlier equation as

95
Work Done By An External Force (Friction Involved)
  • Recall that W Fd, so we finally end up with
  • This is the energy statement for the work done on
    a system by an external force when friction is
    involved

96
Checkpoint 5
  • In 3 trials, a block is pushed along a horizontal
    surface that is not frictionless
  • The magnitude of the force applied and the result
    of the pushing on the blocks speed are shown in
    the table
  • In all three trials the block is pushed the same
    distance d
  • Rank the 3 trials in terms of the change in the
    thermal energy of the block-floor system,
    greatest first

97
Checkpoint 5
98
Conservation of Energy
  • As we have progressed through this lecture, we
    have assumed (correctly) that energy cannot
    magically appear or disappear
  • More formally, we have assumed that energy obeys
    a law called the law of conservation of energy
    which is concerned with the total energy E of a
    system

99
Conservation of Energy
  • This total includes
  • Mechanical energy (kinetic potential),
  • Thermal energy,
  • and any other form of internal energy in addition
    to thermal energy
  • The law states that
  • The total energy E of a system can change only by
    the amounts of energy that are transferred to or
    from the system

100
Conservation of Energy
  • Stated as a formula, we haveremembering
    that
  • Note that this law is not derived from first
    principles, but rather from countless experiments
    made over hundreds of years

101
Conservation of Energy
  • If you have an isolated system, then the law of
    conservation of energy states that
  • The total energy of an isolated system cannot
    change
  • This does not mean that energy transfers are not
    taking place within the isolated system however

102
Conservation of Energy
  • It is not too difficult to come up with an
    isolated system lets take the rock climber on
    page 182 as an example
  • The climber, her gear and the earth can be taken
    as a reasonable approximation of an isolated
    system

103
Conservation of Energy
  • Lets assume that she starts her descent with
    zero kinetic energy
  • During her descent down the rock face, she will
    acquire some kinetic energy (but not too much
    hopefully) and her potential energy (taking the
    valley floor beneath her as the reference point)
    will decrease
  • Lets also assume that she ends her descent with
    zero kinetic energy (which would probably be a
    good thing)

104
Conservation of Energy
  • During her descent she needs to control her speed
    (kinetic energy), so she wraps the rope around a
    carabiner and uses the ensuing friction to
    control her rate of descent
  • Because this is an isolated system, we know that
    the total energy does not change, so we have

105
Conservation of Energy
  • The climbers kinetic energy started and ended at
    zero, so we then know that ?Emch consists
    entirely of the change in potential energy
  • So it should be clear finally that
  • In other words, all of the potential energy is
    transformed into thermal energy

106
Conservation of Energy
  • Note in doing this analysis, we totally ignored
    the details of what happened as the climber
    descended
  • We are able to do this because
  • In an isolated system, we can relate the total
    energy at one instant to the total energy at
    another instant without considering the energies
    at intermediate times

107
Sample Problem 8-8
  • A circus beagle of mass m 6.0 kg runs onto a
    curved ramp with a speed of v0 7.8 m/s at a
    height of y0 8.8 m above the floor
  • It slides along the ramp to the right and
    eventually (momentarily) comes to a stop 11.1 m
    above the floor
  • The ramp is not frictionless
  • What is the increase in thermal energy Eth in the
    beagle-ramp system because of the sliding?

108
Sample Problem 8-8
109
  • Lets start by looking at the forces involved
  • The normal force on the beagle from the ramp does
    no work as it is always perpendicular to the
    beagles displacement
  • The gravitational force is clearly doing work as
    the height of the beagle changes as it slides
    along the ramp
  • And finally, because the ramp is not frictionless
    there is an increase in thermal energy in both
    the beagle and the ramp

110
  • The system includes the ramp, the beagle and the
    earth
  • We can take this system to be isolated (meaning
    that there are no other forces involved)
  • Therefore we know that the total energy of the
    system cannot change

111
  • So from that we know that

112
  • We also know that the change in mechanical energy
    of the system (?Emch) is the sum of the change in
    the kinetic energy and the change in the
    potential energy

113
  • Substituting these back into the previous
    equation we get

114
Power
  • We are now in a position to expand on the
    definition of power with which we ended the last
    lecture
  • At that time I said that it was the rate at which
    work is done by a force
  • In a more general sense, power is the rate at
    which energy is transformed by a force from one
    form to another

115
Power
  • If an amount of energy ?E is transferred in an
    amount of time ?t, then the average power is

116
Power
  • Similarly, the instantaneous power is

117
Next Class
  • Homework Problems Chapter 87, 21, 41, 50, 71,
    106
  • Read sections Chapter 9
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