Review of Lecture 7

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Review of Lecture 7

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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

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