Download Applying Models to Mechanical Phenomena

Chapter 2 Applying Models to Mechanical Phenomena
2
Contents
2-1 Where we are
headed in this
chapter
2-2 Phenomena,
Data Patterns,
Questions
2-3 EnergyInteraction
Model
2-4 Intro SpringMass Oscillator
Model
2-5 The Connection
Between Force
and Potential
Energy
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Applying Models to
Mechanical Phenomena
This is the second chapter that deals directly with the Energy-Interaction Model.
We add all kinds of mechanical interactions to the thermal interactions we treated
in Chapter 1. (Note: the term “mechanical” as in the phase “mechanical
interactions” is typically used to imply everything other than thermal. Since the
Energy-Interaction Model literally applies to every kind of interaction that
scientists have ever encountered, we will be just scratching the surface of the
realm of applications of this powerful model. We will, however, devote some
attention to one area of application that occurs frequently in many phenomena–all
kinds of things vibrate, from atoms and molecules to bridges and skyscrapers; that
is, they move back and forth or oscillate in very predictable ways.
A fundamental understanding of vibrational (or oscillatory as it is often called)
motion is very useful. Therefore, we introduce the Intro Spring-Mass Oscillator
Model in this chapter as an application of the Energy-Interaction Model. The Intro
Spring-Mass Oscillator Model will also play an important role in Chapter 3 when
we develop particle models of matter.
2-6 What energies
are Fundamental
2-1 Where We Are Headed in this Chapter
2-7 Examples of
Particular
Models
In Chapter 2 we continue our focus on interacting systems. We intentionally focus
on the systems as they exist before the interaction and then immediately following
the interaction; we stay away from the details of what happens during the
interaction itself. This is a very general and a very powerful approach. We saw in
Chapter 1 that energy transfers are fundamentally related to interactions. When
one physical system interacts with another, or when parts of the same physical
system interact, we were able to identify energy-systems that either increased or
decreased in energy. If the physical system is closed, i.e., we have included all of
the energy systems that change during the interaction, then energy conservation
tells us that the total of all the increases equals the total of all the decreases. If the
physical system is open, then the net change in all of the energy-systems equals
the net energy added to the physical system from outside that physical system. We
continue this approach in Chapter 2 with non-thermal energy-systems. In your
classroom activities, you will identify several kinds of energy-systems and the
transfers that occur among them. A surprising result for many of us is that when
we consider typical activities such as driving a car or riding a bike, most of the
energies involved ultimately end up being transferred to various thermal systems.
Thermal energy systems seem to have a way of “grabbing and hanging on to”
most of the energy. (This tendency has to do with the vast number of particles
involved in thermal systems. We explicitly discuss this in Chapter 4.)
2-8 Looking Back
and Ahead
Thermal energies are associated with the random or disordered motions of the
particles making up matter. In this chapter we turn our attention to energy-
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Chapter 2
Applying Models to Mechanical Phenomena
systems in which the common motion all of the particles making up the matter is
important. We can often describe this common motion with just two variables: a
position variable (x) and a speed variable (v). These energies (ones that can be
described by position and speed of the entire object) are commonly called
mechanical energies.
As you begin the activities of this chapter in class, you might be tempted to ask,
“How many kinds of energy can there be?” The answer is simple and reassuring:
there are only two fundamental kinds: these are energies that depend on the square
of the speed of a particle or object (kinetic energy, abbreviated KE) and energies
that depend on the positions or configurations of particles or objects (potential
energy, abbreviated PE). All energies, no matter what names we give them, are
either a kinetic energy or a potential energy or some combination of the two. This
is true for the two energies we have discussed up to this point: bond energy and
thermal energy. Bond energy depends on the positions of the atoms making up the
molecules, so it is a potential energy. Thermal energy is a combination of the
kinetic energies of the individual atoms and the potential energies associated with
the motions of the atoms about their equilibrium positions. We will discuss these
relations in much greater detail in Chapter 3.
Where are we going with this energy stuff and why? You will begin to see as you
carry out the activities and work on the assignments in this chapter that using an
Energy-Interaction Model allows us to answer many interesting questions about
sports, bikes, objects falling off buildings, and other common (or not so common)
everyday activities. Mechanical energies involve position and speed variables,
and because transfers of mechanical energy involve work (instead of heat), we
need to understand a little more about work. Work involves the notion of forces
acting through distances, which means we will have to know a little more about
force itself. The payoff is that we will be able to calculate or predict many
unknown distances, speeds, and forces without ever having to know the details of
the interactions. And the beauty of this approach is that it works for all kinds of
physical situations. We don’t have to learn lots of different ways of approaching
questions that depend on the particulars of the situation!
Another significant benefit of starting the way we did in Chapter 1 with thermal
phenomena is that we can also look much more realistically at real-world
phenomena where friction is always present. If you look at a conventional
introductory physics text, you will find that for the first third of the book, it seems
everyone just pretends that there is no friction! It is as if we lived in a distant
galaxy where friction didn’t exist. But that is a pretty idealized galaxy. Because
we are treating energy within a general model that works for mechanical energies
as well as thermal energies, rather than from a purely mechanics approach, we can
deal directly with the transfers of energy to thermal systems and treat a lot of
phenomena much more realistically than we could if we narrowly focused on the
mechanics of frictionless systems.
Chapter 2 Applying Models to Mechanical Phenomena
2-2
Phenomena, Data Patterns, and Kinds of
Questions and Explanations
Phenomena
33
34
Chapter 2
Applying Models to Mechanical Phenomena
Data Patterns
Energy never disappears. It just leaves one energy system and
reappears in some other energy systems. The total energy stays
constant!
Always!
Always!
Always!
Always!
Kinds of Questions and Explanations
How does energy conservation work with mechanical systems?
Why do we get hot when we exercise?
How does an object bouncing on a spring “work?
What is work?
What is a force? Is it a push?
Where does the energy go when this happens?
Chapter 2 Applying Models to Mechanical Phenomena
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2-3 Energy-Interaction Model
(Summary on foldout)
We can apply the Energy-Interaction Model used in Chapter 1 to mechanical
energy-systems as well as to thermal and bond energy-systems. Mechanical
interactions obey the same rules and can be understood in the same way as the
interactions we studied in Chapter 1. Because energy conservation still holds,
regardless of the kind of interaction, energy-system diagrams prove to be just as
useful in describing the interactions that take place as a ball falls as in describing
the interactions taking place in a pot of boiling water. We will also be able to
make numerical predictions of various parameters in mechanical interactions, just
as we were able to do for thermal interactions.
With thermal and bond systems, we could infer changes in the energy of a system
through changes in observable properties, the indicators, such as temperature for
the change of thermal energies and change of mass in phase changes. In the
mechanical systems introduced below, we infer changes in energy by looking at
changes in position and speed. The mechanical energy-systems we describe can
exchange energy with each other or with the thermal and bond systems that we
discussed in Chapter 1. In Chapter 1, we labeled the energy transferred into a
physical system due to a difference in temperature between two objects as heat. In
this chapter, we introduce another type of energy transfer to a physical system,
which is given the label work, W.
New Construct Definitions
(refer back to Chapter 1 for the Construct
Definitions Already Introduced. The only
new one here is Work)
Work
Sub-Constructs Related to Force
Before discussing work explicitly, we will also introduce some ideas about force.
These are ideas that we will come back to thoroughly in Part 2 of the course. For
right now, we just need a few fairly simple ideas related to force. We can simply
think of them as sub-constructs that we will use to make sure we are all on at least
a 10th grade level of understanding of work and force and using the same
vocabulary. This is all we need for now.
Force
Force is the label we give to a fundamental construct that has a prominent
place in our models of the dynamics (the details) of change, which we take
up in Part 2. This construct is closely identified with our everyday
experience with pushes and pulls. Many of the characteristics of pushes
and pulls that we are familiar with also apply to forces. Forces have both a
magnitude and a direction associated with them. We are all familiar with
these ideas. Pushes and pulls have a magnitude (how hard is the push) and
in what direction is the push. The technical term used to indicate that a
concept has both of these properties is to say it has a vector nature or
simply to say it is a vector. We won’t dwell on this in Chapter 2.
Chapter 2
Applying Models to Mechanical Phenomena
or in classroom activities for right now. Both parts, direction and
magnitude, do, however, affect the amount of energy that one physical
system transfers to another physical system. So we need to explore these
ideas a little bit further.
One way to represent vector quantities is to simply draw an arrow pointing
in the direction the force points (think of the direction of a pull or push)
and let the length of the arrow represent the magnitude of the force (how
hard the push is pushing). Shown below are several forces represented in
this way with arrows.
FC
θ
45°
unit length
FA
y
FB
Sub-Constructs Related to Force
36
x
In these examples force FA is almost three units in magnitude and points in
a direction 45˚ below the +x axis, or we could say it points southeast.
Force FB is about four units in magnitude and points in the –y direction, or
south. Force FC is about two units in magnitude and points in a direction
given by the angle θ with respect to the +x axis.
Force is the agent of interactions of TWO OBJECTS
Forces are the agents of non-thermal interactions between two “objects.”
(The quotes around the word “object” signify that it is being used in a
general way that includes matter in the form of gases or liquids as well as
matter in the solid phase. Sometimes it might even be a single atom or
electron. We will omit the quotes from now on, but remember we are
using the word object in this more general sense.) Just as there can’t be an
interaction involving only one object, there can’t be a force involving only
one object.
We normally say, “the force of the (name of an object) on the (name of
another object)” to make clear which two objects are involved in the
interaction and with the force. For example, we could say, “the force of
the bat on the ball” or “the force of the person’s hand on the book.” We
write this in symbols as:
Fbat on ball and Fhand on book,
or in symbolic notation,
FA on B This last notation means object A exerts a force on object B
To realize that this is something you already know and actually does make
sense to you think of the following: Use the word “push” in a sentence that
refers to you actually pushing. What did you have to say? Did you name
an object that you pushed on? Who did the pushing? Can you push
without pushing on something? Remember that sometimes it might be
something pretty squishy, such as air (which is still a “something,” still an
Chapter 2 Applying Models to Mechanical Phenomena
37
object). Can an object be pushed unless there is “a something” pushing on
it? This is what we mean when we say that force is the agent of an
interaction. There simply cannot be an interaction without there being two
objects. And there can’t be a force without there being two objects.
The Two Forces Involved in an Interaction are Opposite and Equal
Now we come to an idea about forces that makes perfectly good sense if
we think about force as the agent of the interaction. But, it is easy to get
our head all messed up if we forget this very basic idea and instead think
of force as the result of the force instead of simply as the agent of the
interaction. Fortunately, we don’t have to worry much about this until we
get to the models in Part 2 of the course.
Sub-Constructs Related to Force
Here is the idea, which does make sense, if we focus on force as described
above. The magnitude of FA on B is the same as the magnitude of FB on A.
The two forces have the same magnitude, but they point in opposite
directions. This important property of forces has been given the label
“Newton’s Third Law” and is written symbolically as
FA on B = – FB on A
which simply says that object A pushes as hard on object B as object B
pushes on object A, but in the opposite direction. There is one interaction,
and the agent of that interaction, the force, has got to be the same in both
directions.
Some forces act only when objects are in contact
Let’s consider further the example of the force of the bat on the ball. Does
the force of the bat continue to act on the ball after the ball is no longer
touching the bat, or does the force act only while the ball and bat are in
contact? Recall that force is the agent of interaction, so another way to ask
the question is whether the ball and bat continue to interact after they are
no longer in contact. The answer is they interact only while in contact.
Thus, the force the bat exerts on the ball (and the force that the ball exerts
on the bat) acts only while the two objects are in contact. Consequently,
we call this force a contact force.
Some forces act even when the objects are not touching. Some act even
when extreme distances separate the objects and these are referred to as
long-range forces. Think about the force of gravity. The Earth pulls down
on all objects near its surface. Note that we have two objects: the Earth
and the object being pulled down. We know from everyday experience
that the gravitational force of the Earth on an object is present whether the
object is touching the Earth or not. Apparently the interaction between the
Earth and a nearby object is unlike the force between the bat and the ball.
The gravitational force is long range. The interaction and the force, the
agent of the interaction, continue to exist even when there is no direct
contact.
The force of gravity between our sun and each the planets in our solar
system does indeed get weaker as the distance between the sun and the
particular planet increases, but it is definitely still there. The gravitational
Chapter 2
Applying Models to Mechanical Phenomena
force is truly long range. There are also non-contact forces that act
between objects that are not touching, but “die off” or “go to zero” as soon
as the objects get more than a little distance apart. Most of the forces that
hold atoms together into molecules and hold the particles of solids and
liquids together are like this. The interaction and the agent of the
interaction, the force, exists, even though the particles are not touching,
but the force decreases to zero in magnitude pretty fast as the particles
begin to separate just a little bit from each other. We will see examples of
this behavior in Chapter 3 when we study the forces between particles.
Sometimes this kind of force is said to be short range.
Balanced and Unbalanced Forces
Sub-Constructs Related to Force
38
There is one last idea that we will need to use, but is also part of our
common experience with pushes and pulls. We can easily imagine two
people pushing on a table from opposite sides. If person A pushes harder
than person B, the table moves. If the two people push the same, that is,
they each exert the same force on the table, but in opposite directions, the
table does not move. In the first case, we say the forces exerted by the two
objects (the two persons) on the table (a third object) were not balanced,
and as a result, the table moved. In the second case, the two objects
exerted balanced forces on the table (the third object) and the table did not
move. We say the forces exerted by the two persons were balanced.
F1 on 3
F2 on 3
Fc on d
object d
Fg on d
object 3
FB on C
FX on P
object C
FA on C
Balanced Forces
object P
FY on P
Unbalanced Forces
Can you give two reasons why the forces acting on object “3” and on
object “C” are balanced and give the two different reasons why the forces
acting on objects “P” and “d” are not balanced?
The reason the idea of balanced forces is useful is that sometimes we
know the force F1 on 3 for example, but we need to know the force F2 on 3 in
order to know the work that object 2 does on object 3. If there is reason to
believe that the forces from object 1 on 3 and object 2 on 3 are balanced,
then we can carry out the analysis and calculate what we want to know.
This will become clear in class when we start working with forces,
springs, pulls, and pushes.
Chapter 2 Applying Models to Mechanical Phenomena
39
Back to the Construct Work
Now we are ready to make sense of the idea of work as the transfer of energy
from one physical system to another physical system or from one object to
another object.
Work, W, is a transfer of energy into or out of one physical system by a force
exerted by another physical system. The change in energy results from an
interaction in which an object moves through a distance parallel to the force
exerted on it. In symbols,
W = Fparallel Δx , or W = F|| Δx,
where the subscript “parallel” means to multiply only the part of the force and Δx
that are parallel to each other. The result is a number without an associated
direction. The figure below shows the work done on a box while pushing it a
distance ∆x.
F
F
final position
initial position
distance moved
∆x
F
Fıı
Work done on box by someone pushing with
force F : W = Fıı ∆x
Like heat, Q, work, W, is a scalar (i.e., has no direction associated with it) and is
measured in the SI unit of energy, joules. A joule is equivalent to a newton·meter,
or J = N·m, since a newton is the SI unit of force. In the example of the box being
pushed across the floor at constant speed, energy is transferred from the person to
the thermal energy-system of the box and floor. Shown below is an open energysystem diagram for this example.
Thermal energy
system of
block and floor
E th
T
work
∆Eth = W
∆Eth = F||∆x
40
Chapter 2
Applying Models to Mechanical Phenomena
Energy is transferred as work to the physical system of block and floor. If the
block moves at constant speed, then the kinetic energy, KE, of the block does not
change and the only change in energy systems is the combined thermal energy
system of block and floor (something gets warmer due to the friction). Energy
was transferred from one physical system, the person doing the pushing, to the
block floor physical system during an interaction that involved a force that acted
through a distance. Work does not have to result in a transfer to a thermal system,
but in this case it did. This example also illustrates that thermal systems can
increase (and decrease as well) due to a transfer of energy as work, and not only
as heat.
Some Useful Forces and Energies to Know About
Force of Gravity
An object near the surface of the Earth is attracted to the Earth with a force
commonly referred to as the force of gravity. This force is proportional to the
object’s mass and fairly constant everywhere on the surface of the Earth. The
constant of proportionality is often referred to as “g,” but properly it should have
a subscript “E” to indicate that it is the earth that is interacting with the object,
and not, e.g., the moon. We will usually include the subscript in this resource
packet, and encourage you to do the same. If the downward force of gravity is
balanced by an upward force acting on the object, then the object’s motion
remains constant (balanced forces). If the object exerting the upward force on the
object is a scale or balance, then it reads the “weight” of the object, which is equal
in this case of balanced forces, to the gravitational force acting down on the
object.
The term weight is often taken to mean the gravitational force. This is OK when
forces are balanced and the object’s motion does not change. If the object’s
motion is changing, the weight (what a scale reads) can be very different from the
gravitational force. (The force of gravity acting on the astronauts in the space
shuttle is only slightly less than the force of gravity acting on them when they are
standing on Earth, yet they are “weightless” in the orbiting shuttle! We study the
interesting state of weightlessness in Part 2 of this course.)
Keeping in mind the reservation mentioned above, we can write:
FEarth on Object = Fgravity = weight = gE m,
where m is the mass of the object and gE = 9.8 N/kg near the surface of the Earth.
With mass expressed in kilograms and gE in units of N/kg, the weight will be in
newtons, N, the SI unit of force. We wrote “gE” with the subscript “E” to
emphasize that the value 9.8 N/kg refers only to the value of g near the surface of
the Earth. The value of g on the moon or on Mars will have a different value.
Chapter 2 Applying Models to Mechanical Phenomena
41
Gravitational Potential Energy
PE grav
Earth/Obj
PE ↑
work
y↑
∆PEgrav= W
A gravitational potential energy-system exists for each pair of objects interacting
by the gravitational force. If we are talking about an object and the Earth, the
energy of this system changes as some other object (perhaps you) does work on
the object by raising it to a higher elevation. We call this a change in the
gravitational potential energy, ∆PEgravity, of the Earth-object system. This change
is simply equal to the work done by the other object and is
W = ΔPEgravity = F|| Δy = weight Δy = m g Δy
ΔPEgravity = m g Δy = mg(yf - yi)
where ∆y is the change in elevation of the object. Notice that ∆PEgravity does not
depend on distance moved parallel to the surface of the Earth, but only on the
change in vertical distance (gravitational force points toward the center of the
Earth). The work done on the Earth-object system by something else moving the
object further from the center of the Earth, is positive, because the force and
change in distance are in the same direction.
Also note that we are imagining that the object is lifted up at a constant speed, so
that there is no change in its motion. Thus, all of the work goes into changing the
gravitational potential energy of the system. We call this a “potential” energy,
because the energy depends only on the relative positions of the object and Earth.
It does not depend on the route taken to get to these positions or on the speeds the
object and Earth might have.
Note that our expression for the gravitational potential energy gives only changes.
Where we put our origin of the coordinate system used to measure “y” does not
matter, since we are always subtracting two elevations. If we get sloppy and say
an object has a gravitational potential energy of so many joules, we mean it has
this amount relative to where we picked the origin of our coordinate system,
which is completely arbitrary.
We will come back to this later, but it is worth noting here that there is a
connection between the force and the change in the potential energy.
Energy of Motion–Kinetic Energy
Now suppose we do work on an object and move it in a horizontal direction,
increasing its speed. Because we have not changed its elevation, we have not
changed its gravitational potential energy. But certainly it now has more energy.
We know this both from experience and from our definition of work. Where is
this energy? It is in the object’s motion. We call this kinetic energy, KE.
If we do work on an object of mass m and change its velocity from vi to vf, then
the change in KE is
W = ∆KE = KEf - KEi = 12 mv 2f − 12 mv i2 Note: it is the difference of the
squares of the speeds that is important in changes in KE.
The next energy system we take up is so important for future work, that we give it
its own model: the Intro
€ Spring-Mass Oscillator Model.
42
Chapter 2
Applying Models to Mechanical Phenomena
2-4
Intro Spring-Mass Oscillator Model
(Summary on foldout)
Construct Definitions and Relationships
Elastic Object
An object, which when deformed (perhaps by squeezing it or stretching it) returns
to its original shape.
Spring
A coiled spiral shaped object that is constructed to be elastic over a large range of
deformation. Some springs can only be stretched, but if the coils are initially
separated, the spring is elastic when both stretched or compressed.
Spring-mass
A spring with an object whose mass is considerably greater than the mass of the
spring attached to one end of the spring.
Hanging spring-mass
A spring-mass with one end solidly attached to a support in a manner that the
spring hangs vertically with the mass at the bottom of the spring.
Before the mass is attached to the spring, the bottom of the spring (the “free
end”) will be at a definite position with respect to the attached end. This is the
equilibrium position of the free end without the mass attached. When the
mass is attached, the free end of the spring will move to a new, lower position.
This is the equilibrium position of the free end of the spring with the mass
attached.
Force (see discussion earlier in this chapter on Force)
Spring force and force constant
The force with which a spring pulls back when stretched (or pushes back
when compressed) is proportional to the amount of stretch from equilibrium,
provided the spring is not stretched too far. (Historically, this linear
proportionality between the force and amount of stretch is referred to as
Hooke’s law behavior.) We write the force with which the spring pulls back
(the restoring force) as
F = -k x,
where k is the “spring constant” or “force constant” (and is dependent on the
stiffness of the particular spring). The minus sign indicates that the force is
opposite in direction to the direction it was stretched or compressed. In order
for the force to have units of newtons, the units of k must be newtons per
meter. The force that you (an external agent) have to exert on the spring to
stretch it a distance x is in the opposite direction to the restoring force and is
equal to +k x.
An important point to notice in the expression for the force of the spring is
that x is measured from the unstretched position of the free end of the spring.
Chapter 2 Applying Models to Mechanical Phenomena
43
(hanging) spring-mass Force and force constant
A spring with a mass hanging on it acts exactly like a spring without a
mass, except that the end of the spring has a different equilibrium
position. There are now two forces acting on the mass: the force from the
spring pulling up or pushing down and the force from the Earth always pulling
down. We will not show this here, but the combination of the two forces is
completely accounted for if we measure all stretches and compressions of
the spring from the equilibrium point of the free end of the spring with
the mass attached.
The force with which a spring-mass pulls back when stretched (or pushes back
when compressed) is proportional to the amount of stretch from the
equilibrium determined with the mass attached (provided the spring is not
stretched too far). We will usually write the force with which the spring-mass
pulls back (the restoring force) using the symbol “y” instead of with an “x”,
but this is just a convention.
F = -k y,
where k is the “spring constant” or “force constant” (dependent on the
stiffness of the particular spring, not on the mass). The minus sign indicates
that the force is opposite in direction to the direction the spring-mass was
stretched or compressed. The force that you (an external agent) have to exert
on the spring-mass to stretch it a distance y is in the opposite direction to the
restoring force and is equal to +k y.
A critically important point to notice in the expression for the force of the
spring-mass is that y is measured from the unstretched position of the free end
of the spring-mass system. By measuring the stretch from this “new”
equilibrium position, the effect of the force of the Earth is automatically taken
into account, so does not have to be added back in. It is as if, we are in far
outer space where there is no force of gravity. The only force is the
“modified’ force of the spring.
Work (see discussion earlier in this chapter on Force)
Work done on a spring-mass system by an External Agent
The work I do when I pull on the mass increases the PE of the mass/spring
system. I can calculate this work: it is simply the product of the force I apply
times the distance through which I move while pulling or pushing. The only tricky
part is that the force is not constant: FIapply = + k y. That is, it is proportional to the
distance I have pushed it or pulled it. So I need to take the average force times the
distance. The force varies from zero to a max of k y . T h e average of this is
1/2 ky. When I multiply this average force times the distance, I get the work I do
on the system:
W = (average force) × (distance) = F|| y
= ( 12 ky)y =
€
€
1
2
ky 2
44
Chapter 2
Applying Models to Mechanical Phenomena
Energies of a Spring-Mass
Potential Energy of a spring-mass
The expression in the preceding paragraph represents the work I did on the
system, so its PE must have changed by that same amount. It doesn’t matter how
the spring got stretched (that is, whether it was extended or compressed). If
stretched by an amount y, it must have a PE relative to the equilibrium position
(with mass attached) of
PE spring-mass = 12 ky 2
The change in the spring-mass potential energy, ∆PE spring-mass can be found
using the Energy-Interaction Model in the standard way. Let’s assume in this
example that I compress
or stretch the spring further from its equilibrium position
€
than when it was in its initial position, yi, to a final position yf. Therefore, I am
doing positive work on the spring-mass system. That is, I am adding energy to the
spring-mass system. (The work I do on the spring-mass system will be a positive
number of joules.) Thus, its energy must increase. If I don’t change its KE, i.e.,
don’t change its velocity (the final velocity is equal to its initial velocity), the only
energy system that changes is the PE spring-mass energy system.
PE mass-
work
spring
E ↑ |y|↑
∆PE spring-mass = W = ∆ 12 ky 2 = (½ ky2)f - (½ ky2)i
Notice the difference between the equation written just above this sentence and
the expression for PE spring-mass written at the end of the previous paragraph. Can
you “tell a story” in your
€ own words what the difference is between these two
equations and why they are truly different?
Kinetic energy of a spring-mass
The kinetic energy of the spring-mass system is the same
as for any mass that has a speed, v.
KE spring-mass = 12 mv 2
where m is the mass of the object hanging on the spring (and part of the spring
mass if it is not negligible compared to the mass of the hanging object) and v is
the speed (the time
€ derivative of the position, y) of the mass.
The change in the KE spring-mass is calculated the same way as the change of any
energy:
∆KE spring-mass = KEf - KEi =
€
1
2
mv 2f − 12 mv i2
Chapter 2 Applying Models to Mechanical Phenomena
45
Spring-Mass Systems: a Universal Motion
A mass attached to a spring will oscillate if it is pulled away from its equilibrium
position and released. Energy is transferred back and forth between the kinetic
energy system of the mass and the potential energy system of the spring. If there
were no friction, the oscillation would continue forever, with energy transferring
back and forth between the kinetic and potential energy systems.
Most physical systems that vibrate back and forth do so like a hanging springmass, particularly when the amplitude of vibration is not too large. Because this
motion is so common, it is worth looking at it a little closer.
The maximum value of the PE of this spring-mass system occurs when the mass
is at its extreme positions and its speed is zero. Conversely, the KE of the springmass system is a maximum when the mass is at its equilibrium position (y = 0).
We know from the last chapter that in the absence of friction, ∆Etot = ∆KE + ∆PE
= 0, or in words, Etot is a constant. Looking at Etot at any particular time in its
cycle of vibration, the energy is still going to be equal to the same total value.
Written symbolically, this becomes:
Etot = PE + KE = constant = PEmax = KEmax
The graph below shows the KE and PE of a spring-mass system as a function of
the position from equilibrium. The kinetic energy as a function of y, KE(y), is just
the difference between the maximum energy, Emax, and PE(y). This is just
another way of expressing conservation of energy for this system. Both PE(y) and
KE(y) are parabolas, centered about y = 0. The shape is due to the dependence on
the square of y in the expression of the PE.
Although we cannot show it now without investigating the time behavior of the
motion of the mass, it turns out that the time-average (that is, the average over
time) of the PE and KE are the same, and are consequently both equal to one-half
the total energy.
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Chapter 2
Applying Models to Mechanical Phenomena
avg KE spring-mass = avg PE spring-mass = ½ Etotal = ½ Emax
The fact that the time-average potential and kinetic energies are the same has a
profound implication for the model of matter that we are about to develop in
Chapter 3. This result is true only for a potential energy that depends on the
square of the variable. It is precisely the fact that the potential energy is quadratic
with respect to position that makes a spring-mass system so special, so universal,
and so important.
The lowest point on the PE(y) curve is frequently called the potential energy
minimum. Why is the value of the position variable for which the potential energy
is a minimum significant? Consider what happens as energy is removed from a
system which is oscillating about the equilibrium value of its position (e.g., a real
spring-mass system, because of friction, gradually transfers mechanical energy to
thermal energy). The amplitude (maximum extent) of the oscillations decreases
until eventually, when all mechanical energy has been transferred to thermal
systems, the system comes to rest at the equilibrium position, the position of the
potential minimum.
Now consider any physical system that oscillates and which will settle down to a
stable position as energy is transferred to thermal systems. That stable equilibrium
position represents the physical state where potential energy is smallest. (All of
the PE and KE have been transferred to thermal systems.) The potential energy
must increase as y increases (either positively or negatively) away from zero, the
equilibrium position. Now nature seems to prefer smooth changes of things like
potential energies. The simplest smooth mathematical function that increases for
both positive and negative y is y2. When we look at real physical systems, it turns
out that sufficiently close to the minimum, the potential energy always “looks
parabolic”! This is a result with far reaching consequences. It implies that any
oscillating system behaves just like our simple spring-mass system, at least for
small amplitudes of oscillation! What’s important is not that the spring-mass
system is so special itself; it is, rather, that the behavior of the spring-mass system
represents a truly universal behavior of any oscillating system! We will use this
property shortly when we model the real atoms of liquids and solids as
“oscillating masses and springs.”
Chapter 2 Applying Models to Mechanical Phenomena
47
2-5 The Connection Between Force and Potential
Energy (This is relationship (8) in the Summary of the intro
Spring-Mass Oscillator Model and relationship (2) in the Intro
Particle Model of Matter)
There is a relationship between a potential energy and the force that is associated
with that potential. This relationship has a useful graphical representation that will
help us better understand the spring-mass potential energy and, in Chapter 3, the
potential energy associated with the bonding between atoms.
The figure shows the potential energy of a spring-mass system. Marked on the
figure are the positions where the force exerted by the spring has the greatest and
the least values. If pulled or pushed to some value of the position y, what do we
know about the restoring force of the spring?
Energy
The force is large and
the force is in the
direction that lowers
the PE; i.e., back to
smaller values of |y|.
The force is large and
the force is in the
direction that lowers
the PE; i.e., back to
smaller values of |y|.
-y
+y
Slope = 0 here tells us that the force = 0 here
Without derivation, we simply state the result that the force exerted by the spring
is equal to the negative of the derivative of the PE with respect to y. Graphically,
the force is the negative of the slope of the PE vs. y curve. Rather than worry
about the negative sign here, however, we simply focus on the fact that the spring
always tries to get back to its equilibrium position. This is what tells us the
direction of the force.
For any potential energy, it is true that the magnitude of the force associated
with that potential energy is equal to the derivative of the potential energy with
respect to the position coordinate associated with the force. Using the symbol r
for generality, we can write this symbolically as:
|F| = d(PE)/dr or |F| = slope of the PE vs. r curve
This is a general result that is true for the force associated with any potential
energyi. Since the value of the slope gives the magnitude of the derivative of a
function, the steeper the slope of a PE curve plotted against its position variable,
the greater the magnitude of the force. The restoring force of the spring (or
anything that oscillates) will be zero when the slope is zero, which occurs at the
equilibrium point, i.e., where the object comes to rest when it stops vibrating.
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Chapter 2
Applying Models to Mechanical Phenomena
2-6
What Energies are Fundamental?
It turns out that gravitational potential and spring potential energies we discussed
above are examples of one of the two fundamental types of energy–the type that
depends on the positions of masses. An energy that depends only on the relative
positions of the particles (objects), and not on their past history (the path they
took) or on their speeds, is called a potential energy. “Positional energy” is a
more descriptive word, but for historical reasons, these are labeled potential
energy.
The second fundamental type of energy depends only on the speeds of particles
(objects). This kind of energy is called energy of motion or kinetic energy.
All of the various kinds of energy fit into one of the two fundamental types. For
example, chemical bonds involve the electric potential energies that depend on the
locations of the electrons of the atoms as well as their kinetic energies as they
whirl around the nucleus. Thermal energy is a combination of the kinetic and
potential energies of individual atoms due to their random motions. Bond energy
is a potential energy due to the force individual atoms exert on each other. The
elastic energy of a spring is a potential energy because it depends only on the
positions of the elements of the spring, not on their speeds. When an object
rotates, it has energy due to its rotational motion—rotational kinetic energy.
One energy is as good as any other—almost. On a microscopic scale, energy is
energy and any kind can be turned into any other kind. But on a macroscopic
scale, where lots of particles (atoms and molecules) are involved, it turns out that
all kinds of energy can be turned into thermal energy, but there are restrictions on
turning thermal energy back into other kinds. We can convert some, but not all
thermal energy to other forms. (We delve into the mysteries of entropy and the
second law of thermodynamics in Chapter 4.)
Chapter 2 Applying Models to Mechanical Phenomena
2-7
49
Examples of Particular Models of Mechanical
Phenomena
Example 1
Suppose we consider a baseball thrown by a pitcher. The pitcher’s hand exerts a
force on the baseball over some distance parallel to the baseball’s velocity. Thus,
the pitcher does a certain amount of work on the ball. Without knowing more
details about how the ball was thrown, all we can say is that the work done by the
pitcher's arm on the baseball equals the total change in energies of all other
energy-systems of the ball. The natural way to model this situation is to treat the
baseball as an open physical system.
work
∆KE + ∆PEgrav + ∆Eth = W
and W = F|| Δx, where ∆x is the distance the pitcher’s hand moved as s/he threw
the ball parallel to the direction the ball traveled. This is about all we can say
without having more information.
Perhaps we want to estimate the average force the pitcher exerts on the ball during
the throw. If the throw is horizontal, there will be no change in gravitational
potential energy. If we assume for now that we are focusing on the speed of the
ball just after it is thrown, then air resistance will
not cause significant energy to be transferred to
KEball
W
thermal energy-systems. Thus, we can simplify the
energy-system diagram to:
E v
↑
initial state: ball at rest
↑
∆KE = W
final state: ball moving horizontally
with velocity v
∆KE = ½ m(v2), W = F|| Δx
since vi = 0, letting vf just be v
Thus, F|| Δx = ½ m(v2)
So, if we knew how fast the ball was thrown (which gives us ∆KE), we can
determine a value for the force exerted on the ball by the pitcher by making a
reasonable estimate for the distance ∆x.
50
Chapter 2
Applying Models to Mechanical Phenomena
Example 2
Instead of being interested in the force the pitcher exerts on the ball when s/he
throws it, we might be interested in how much energy the pitcher uses to make a
large number of throws. In this case the natural approach is to include the pitcher
in the energy-system diagram and not focus on the work done by the pitcher.
We create a closed physical system consisting of the pitcher and the ball.
before: ball at rest
after: ball with velocity v
pitcher
energy
E⇓
∆Epitcher
KE ball
E⇑
v⇑
+ ∆KEball = 0
∆KEball = ½ m(v2)
so,
∆Epitcher = - ½ m(v2) for each pitch.
Note the sign of the change in energy of the pitcher. Does this make sense? You
should be able to calculate how many food (big C) calories the pitcher burns after
throwing say 100 pitches with the ball averaging 50 miles per hour as it leaves the
pitcher’s hand. Is this a good way to stay slim and still eat large hamburgers with
fries?
Chapter 2 Applying Models to Mechanical Phenomena
2-8
51
Looking Back and Ahead
We now have quite a few different forms of energy–translational kinetic as well
as rotational, thermal, bond, gravitational potential and spring (or elastic)
potential energy. A given physical situation could involve any number of these.
Energy can be transferred among physical systems either as heat (when there is a
temperature difference between two objects) or as work (when one object exerts a
force (on another object) that acts through a distance). There are yet other forms
of energy that involve electrical interactions, magnetic interactions, and a more
general form of the gravitational energy.
By treating thermal interactions and mechanical interactions on an equal footing,
we can approach realistic situations without having to automatically assume
friction or air resistance is negligible. By now you should be very comfortable
with the energy-interaction model. When we encounter new “kinds” of energy, it
won’t be a “big deal.” We simply add them to our repertoire of energy-systems
that might change in any particular interaction.
Now we are in a position to delve into particle models of matter. Our goal is to be
able to understand, in a general or universal way, as many of the properties of
matter as we can. As we do this, we will also make a much more direct
connection to thermodynamic concepts you have worked with in chemistry
courses.
Even as we extend and perfect our energy-interaction model, we recognize that
many questions are beyond its reach. For example, our before-and-after approach
can’t tell us, “How long did it take an object to fall?” Questions like this involve
the dynamics (the details) of interactions. We will spend more time in Part 2 of
the course and accompanying course understanding the dynamics of rigid objects.
This is fundamentally the relation of force to motion known as Newton’s 2nd law.
Using Newton’s laws and kinematics to describe the details of interactions, we
can answer questions that are unanswerable using the before-and-after approach.
But for right now, we stick to an energy approach and avoid, as much as possible,
the details of interactions.
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Chapter 2
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Endnotes
i
Everything we say here about the relation of force to potential energy is strictly
true when the force depends on only one spatial dimension. That is, we consider
the spring to move in only one direction in space. It the force depends on
movement in two or three dimensions, then technically we say that force is the
negative of the gradient of the potential. This is analogous to releasing a ball on a
smooth hillside; the ball starts to roll in the direction of the steepest slope of the
hill. In two or three dimensions, the force is the derivative of the potential in the
direction of “steepest slope.”