Chapter 7 Momentum Conservation: Overview
53
Chapter 7 Momentum Conservation
Overview
We began this course by focusing on the idea of physical systems, energy systems, and
transfers of energy between different physical systems. In earlier chapters, we concentrated on
applying an approach to understanding our physical universe that emphasized the results of
interactions. The question we tried to answer was what happened to a physical system from a
time before to a time after the system interacted with other systems. We tried to avoid needing to
understand the details of the interaction. We discovered that changes of energy of a physical
system is a very useful measure of the interaction, not the only measure, but certainly a very
useful measure. We couldn’t completely avoid the details of interactions, however. We saw that
force, the agent of interaction, was involved in the amount of energy transferred during an
interaction. Specifically, the differential amount of energy transferred as work, dW, is equal to
the product of the parallel component of an externally applied force and the distance moved:
dW = F||dx. We wrote a conservation of energy expression (a more general form of the 1st law of
thermodynamics that allows for all kinds of energy changes) to express how the energy of a
system changes in response to energy inputs in the form of heat or work: dE = dQ + dW. We
saw how we could apply this energy formalism to more traditional thermodynamic systems
(gases, heat engines) as well as to mechanical systems. We also developed a simple particulate
model of matter in earlier chapters that involved modeling the bonds between atoms and
molecules as analogous to masses hanging on springs, the masses being in continuous random
oscillation. This simple model allowed us to explain and predict many of the thermal properties
of matter in its various states. Again, we avoided the details of oscillations and focused only on
changes in energies.
In this chapter, we continue our focus on the results of interactions. We are still trying to
address what happens to a physical system from a time before to a time after the system
interacted with other systems. We will analyze two new physical quantities (momentum and
angular momentum) that round out our understanding of the results of an interaction. We still
cannot completely avoid the details of interactions. We will see that force, the agent of
interaction, is also involved when either momentum or angular momentum is transferred during
an interaction.
xf
Instead of calculating an energy transfer called work, W = ∫ F|| dx (or, in differential form,
xi
dW = F||dx), we calculate a momentum transfer called impulse, J = ∫ F dt from t1 to t2 (or, in
differential form, dJ = F dt). We will write a conservation of momentum expression, ∆p = J, to
express how the momentum of a system changes in response to momentum inputs in the form of
impulse. We will see both similarities and differences with energy. One difference is that both
momentum and impulse, unlike energy and work, are vector quantities. A physical quantity is a
vector if it has both a magnitude and a direction associated with it. We indicate vectors by either
making the symbol bold, or using arrows over the symbols. One interesting aspect of forces for
us right now is the relationship of forces to the motion of material objects. It is traditional to say
that these relationships are governed by Newton's three laws. However, there are many features
of forces, some rather subtle, that we need to wrestle a bit with before we can appreciate and use
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Chapter 7 Momentum Conservation: Overview
Newton's Laws to answer interesting questions regarding so much of our everyday experience in
the physical world. So initially we avoid the details of the motion during the interaction and
focus only on changes in momentum. Then, in Chapter 8: The Relation of Force to Motion, we
will explicitly use the time dependence of the impulse to find the detailed time dependence of the
motion, rather than just comparing the end result of changes between two points in time.
The first model/approach in this chapter, Momentum Conservation, gets us into the
meaning of momentum and how changes in momentum are related to forces. We will solidify a
lot of learning regarding forces that was introduced in Chapter 6.
Then in the second
model/approach of this chapter, Angular Momentum Conservation, we explore the fascinating
world of rotating objects, from molecules to galaxies. We extend the ideas/constructs of force,
impulse, and momentum to their analogous rotational or angular counterparts: torque, angular
impulse, and angular momentum. You will have ample opportunity to sharpen your vector
manipulation skills that were introduced in Chapter 6.
Chapter 7 Momentum Conservation: Linear Momentum Model
55
(Linear) Momentum Conservation Model
(Summary on foldout)
Overview of the Model
As you begin this chapter, listening in lecture and working in class, it must seem, at least at
first, that you are being introduced to a lot of new concepts. The representation of the motion of
an object and the forces acting on an object are necessary ideas to understand before we can fully
understand this new conserved quantity called momentum. One goal of this (and the next)
chapter is to understand the effects of forces on motion and we begin to do this in this chapter
through a discussion of momentum and transfers of momentum.
We introduce two concepts which are completely new: momentum and impulse. However,
we are taking great pains to help you see how these concepts play roles very similarly to energy
and work. So, yes, you have to memorize that momentum, p, is the product of mass and velocity
(p = mv). And you have to be careful to not forget that momentum has vector properties, just as
velocity does. But, impulse is not an isolated construct you file away in your brain somewhere.
Rather, you should really strive to understand impulse in analogy to work. A transfer of energy
as work changes the energy of a physical system. Similarly, a transfer of momentum as an
impulse changes the momentum of a system. Energy is conserved. Momentum is also
conserved. Of course, there are differences between momentum and energy and between
impulse and work. As you work in class, as you study this text, as you work the FNT’s, and as
you interact with other students and with instructors as you mentally struggle with this material,
try to understand these new concepts in relation to what you already know, rather than as simply
some more isolated facts that you memorize.
Review and Extension of the “Before and After” Interaction Approach
In Chapters 1 and 2 we focused on changes in the energy of a physical system. Energy has
meaning for one particle or 1023 particles, for objects as small as the nucleus of an atom and as
large as a galaxy. It really is a universal concept that applies to any physical system. It turns out
that there are two other concepts that are like energy in that they are universally applicable, are
transferred among systems as a result of interactions, and the amount transferred gives very
useful information that does not depend on the details of the interaction. These are the concepts
of momentum and angular momentum.
Integrating “the Agent” of Interactions
We have called force the “agent of interactions”. Interactions occur between objects as they
exert forces on each other. Objects experience changes in energy when other objects exert forces
on them and do work on them. We recall that the amount of energy change caused by a force is
the integral of the force over a distance. This integral is called the work done on a system.
The only component that contributes to the work, however, is the component of the force that
is parallel to the motion. We usually indicated this component with the symbol F||:
xf
W=
∫ F||(x) dx = Ef - Ei = ∆ E
xi
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Chapter 7 Momentum Conservation: Linear Momentum Model
Or, if F is constant, or we define an average force Favg, we can write
W =Favg||∆x = Ef - Ei = ∆E
In other words, the parallel component of force integrated over the path of the motion is the
work, and this work equals the amount of energy transferred to the system due to the application
of the force by an object outside the system.
A similar integral of the force is equal to the change in momentum of the system. But
instead of integrating over distance, we now integrate over time. This integral is called the
impulse of the force, F. We represent the impulse with the symbol J.
tf
J=
∫ F(t) dt = pf - pi = ∆p
ti
Or, if F is constant, or we define an average force, Favg, then
J = Favg ∆t = pf - pi = ∆p
Impulse is a vector quantity and causes a change in a vector property of a system:
specifically, a change in the linear momentum, ∆p. The change in momentum, is of course,
independent of what Galilean reference frame we choose to measure the momenta in.
Note on units: Force has SI units of newtons, of course. Impulse must therefore have units
of newton seconds, N s. Momentum, the product of mass and velocity, must have SI units of
kilogram meter per second, kg m/s. Since these two quantities are equated, these units must be
equivalent, as you can show using the relation N = kg m/s2.
Linear Momentum
The linear momentum of an object is simply the product of the object’s mass and velocity:
p = mv
Linear momentum incorporates the notion of inertia, expressed as mass, as well as the speed and
direction of motion. In some ways it is similar to kinetic energy, 12 mv2, but an obvious
difference is that momentum has a direction; it is described as a vector. (Often, the word
“momentum” is used without the modifier “linear,” when talking about linear momentum. Later,
however, the modifier “angular” is always used when talking about angular momentum.)
Temporary restriction to non-rotating objects and center of mass
Until we consider rotation of objects in the next model/approach, Angular Momentum
Conservation, we will consider phenomena in which extended objects act only like point
particles. A useful construct that will become much more meaningful when we consider
rotation, is center of mass. Right now we can simply consider that any extended object acts like
a single particle whose mass is equal to the mass of the object, located at the special point, the
center of mass.
The Construct of Net Force and net Impulse
Be sure to review the discussion of Net Force in Chapter 6. The central point is that the
effect of all forces acting on an object can be represented by a single vector construct called the
Chapter 7 Momentum Conservation: Linear Momentum Model
57
unbalanced force or the net force, Σ F. When we use the concept of impulse, it is sometimes
useful to consider the impulse, J, due to a particular force; we would write this as:
JA = FA∆t
(The subscript “A” tells us that the impulse is due to particular force
exerted by the object “A”)
However, the power of the construct of impulse comes into play when we consider the impulse
of the net force; we write this as
Jnet = J = Σ F∆t = ∆p (J without a subscript usually means the impulse due to the net
force.)
In words, the net impulse is equal to the change in the linear momentum of the system. We
explore this relationship further below.
Momentum of a System of Particles
The momentum of a single object is simply the product of its mass and velocity. Suppose we
define a physical system that contains several particles which move with different velocities.
The total linear momentum of this physical system is the vector sum of the individual linear
momenta.
psystem = p1+p2+p3+...=∑pi
If the particles in our system interact with each other, they exert forces on each other, and
there will be an impulse associated with each of these forces. Newton’s 3rd law tells us,
however, that the impulse that particle a, for example, exerts on particle b is equal in magnitude
and opposite in direction to the impulse exerted by particle b on particle a. And using the
relation that the impulse is equal to a change in momentum of a particle, we see that the change
in momenta of particles a and b due to their interaction will be equal in magnitude, but opposite
in direction.
Generalizing the above argument to interactions between any of the particles within the
system, we see that if the momentum of one particle changes a certain amount, another particle’s
momentum changes the same amount in the opposite direction. Thus, when we sum over all the
momenta of the system, the total momentum of the system does not change in response to
interactions among the particles within the system.
However, if the particles of our system interact with particles (objects) outside the system,
then the total momentum of the system might change. The figure below shows some of the
forces that might be acting on the particles of the system. Some, labeled int (for internal) don’t
change the total momentum of the system. The forces labeled ext (for external) do change the
momentum of the system.
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Chapter 7 Momentum Conservation: Linear Momentum Model
Fext on c
c
Fint b on c
d
Fint c on b
b
Fint a on b
a Fint b on a
Of the various impulses shown in the figure, only the impulse caused by Fext on c causes a change
in momentum of the system of particles.
Statement of Conservation of Momentum
So, for a system of particles (objects) it is useful to write the impulse/momentum relation in a
way that emphasizes the external interaction:
Net Impulseext = Jext = ∫ ΣFext(t) dt = pf - pi = ∆psystem
A system acted on by external forces undergoes a change in total linear momentum equal to the
net impulse (total impulse) of the external forces.
We can rephrase the relationship stated above as a conservation
momentum of a system of particles.
←
← for
principle
←
the total
If the net external impulse acting on a system is zero, then there is no
change in the total linear momentum of that system; otherwise, the
change in momentum is equal to the net external impulse.
This statement is an expression of Conservation of Linear Momentum. The total linear
momentum of a system of objects remains constant as long as there is no net impulse due to
forces that arise from interactions with objects outside the system. It does not matter that the
objects of the system interact with each other and exert impulses on each other. These internal
impulses cause changes in the individual momenta of the objects, but not the sum or total
momentum of the system of objects.
We can rephrase this discussion in terms of open and closed systems:
1) Closed system - A closed system does not interact with its environment so there is no net
external impulse. The total momentum of a closed system is conserved. That is, the total
momentum of the system remains constant.
2) Open system - An open system interacts with its environment, so that it can exchange
both energy and momentum with the environment. For an open system the change in the
total momentum is equal to the net impulse added from the environment–from objects
outside the system.
Chapter 7 Momentum Conservation: Linear Momentum Model
59
Applications of Momentum Conservation
As an example of the use of the application of momentum conservation, consider the
collision of two automobiles. The total momentum of the system of two autos is the sum of the
individual momenta before the collision. If the forces exerted by the road in the horizontal
direction are small compared to the forces exerted by the autos on each other (usually a very
good approximation), then the momentum of the system of two autos is conserved. That is, it is
the same immediately after the collision as it was before. Why? Because the forces the two cars
exert on each other are internal forces and don’t contribute an external impulse. Note that this
would not be true if a car hits the proverbial brick wall, since we would typically take the system
to be the car, and the brick wall would exert a net external impulse to the system. The diagram
shows a not quite head-on collision between car (a) moving to the right with a faster and/or
heavier car moving to the left:
Before
p
p
ai
bi
b
a
p
tot i
After
p
pa
f
p
tot i
p
tot f
a
b
p
tot f
bf
Regardless of what happens during the collision, as long as the road exerts a negligible
impulse during the collision (compared to the impulses exerted by the colliding autos on each
other), the total momentum of the two autos immediately before the collision equals the total
momentum immediately after the collision. Note that in the example shown in the figure, the
total momentum before and after the collision is shown in the right part of the diagram, and the
vectors are equal. They are equal in spite of the fact that after the collision, the autos bounce off
at an angle wrt (with respect to) the original direction of motion. The components of momenta in
the perpendicular direction cancel each other out, since there was no momentum in the
perpendicular direction before the collision. But before we explore momentum transfers more
closely, we want to examine collisions in general. We especially want to bring energy
conservation as well as momentum conservation into the analysis, so we can use these powerful
conservation laws together. We will see that together, these conservation laws enable us to
answer many (if not most) questions that arise in collisions, whether they be collisions of cars or
galaxies or the elementary particles physicists study in the collisions in particle accelerators.
And we can do this without having to know any details of the actual forces that act during the
collision or the details of how the motion actually changed during the collision. That is, we do it
with a “before and after” approach, not a detailed analysis of the forces and motion approach,
which we will take up later in this chapter. With the combination of energy and momentum
conservation, we have an extremely powerful and general method of analyzing many physical
phenomena. There are, however, some important questions that can’t be answered without using
a detailed analysis of forces and motion.
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Chapter 7 Momentum Conservation: Linear Momentum Model
Collisions: Momentum & Energy Conservation
We just saw that if the external forces are negligible in a collision, the total momentum is
conserved. What about energy? If, during a collision, kinetic energy is not converted to thermal
energy or into deforming the objects (bond energy), then kinetic energy is also conserved. That
is, the kinetic energy of the system before the collision will equal the kinetic energy after the
collision. When two cars crash into each other, kinetic energy is usually not conserved (unless
they are “bumper cars” with heavy spring bumpers that convert kinetic energy to spring (elastic)
potential energy and then back to kinetic energy). In the example shown in the previous figure,
the kinetic energy just before the collision is much greater than the kinetic energy immediately
after the collision. We can see this, by recognizing that kinetic energy is proportional to the
square of the magnitude of the momentum vector (length of the vector). The magnitudes of the
two arrows representing initial momenta are much longer than the magnitudes of the two arrows
representing final momenta. Much of the kinetic energy must have gone into deformation of the
cars and into thermal energy during the collision. If the collision is between two protons or two
billiard balls, kinetic energy might be exactly or almost conserved. Collisions in which kinetic
energy is conserved are called elastic collisions.
Elastic Collisions
In an elastic collision between two objects (particles)
both the momentum and kinetic energy are conserved.
That is, the values of the total momentum and kinetic
energy of the system before the collision are equal to the
values they have after the collision. This gives us an
equation relating the squares of the speeds of the objects
from KE conservation, and a vector equation (one
equation for each dimension) relating the velocity
components from linear momentum conservation.
KEi = KEf
and
ptoti = ptotf
As an example, consider the one-dimensional
collision of two identical billiard balls. Suppose the first
ball is at rest and is hit “head on” by a second ball which
has velocity v. What are the velocities of the two balls
after the collision?
Both momentum and energy
conservation hold so both equations above must be
satisfied. Writing them out in detail we have:
1
2
mv2i2 =
1
2
mv2f2+
1
2
Elastic Collision
For Equal Mass Objects
Before
p1 = 0
i
p
2i
2
1
After
p
p2 = 0
f
1f
1
2
p = p
1f
2i
mv1f2
m v2i = m v2f + m v1f
The only solution of these two simultaneous equations is v1f = v2i and v2f = 0
That is, the second ball comes to rest and the first moves off with the same velocity the second
ball had initially. This is illustrated in the accompanying figure.
If the masses are not equal, both objects will have non-zero velocity after the collision.
When the collision involves motion in more than one dimension, we can write a momentum
conservation equation for each component of the total momentum. The algebra might get a little
messy, but the idea is pretty straightforward.
Chapter 7 Momentum Conservation: Linear Momentum Model
61
Inelastic Collisions
In an inelastic collision between two objects kinetic energy is not conserved, so we can not
equate initial and final kinetic energies. However, an interesting special case occurs when the
collision is “completely inelastic” so that the objects stick together. Then they both have the
same final velocity after the collision.
Momentum Conservation Model Summary
One way of summarizing the main ideas in a model/approach is to list the (1) constructs, i.e.,
the “things” or ideas that are “used” in the model, (2) the relationships–in mathematical or
sentence form–that connect the constructs in meaningful ways, and (3) the ways of representing
the relationships. During your study of the model/approach you should have developed a good
understanding of the meaning of each of the constructs. Some of these constructs probably start
out as nothing but memorized definitions, but eventually take on a deeper meaning. The
relationships might also start out as nothing more meaningful than a simple equation relating
some of the constructs, e.g., J = Σ F ∆t = ∆p. By the time you finish this part of the course,
however, you should understand this particular relationship, for example, as expressing one of
the most fundamental, universal, and widely applicable principles in all of physics. Developing
a deep and rich understanding of the relationships in a model/approach comes slowly. It is
absolutely not something you can memorize. This understanding comes only with repeated
mental effort over a period of time. A good test you can use to see if you are “getting it” is
whether you can tell a full story about each of the relationships. It is the meaning behind the
equations, behind the simple sentence relationships, that is important for you to acquire. With
this kind of understanding, you can apply a model/approach to the analysis of phenomena you
have not thought about before. You can reason with the model.
Listed here are the major, most important constructs, relationships, and representations
of the momentum conservation model.
Constructs
Velocity, v
Momentum, p
Net Force, ΣF
Impulse, J
Newton’s 3rd law
Conservation of momentum
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Chapter 7 Momentum Conservation: Linear Momentum Model
Relationships
The velocity is the time derivative of the displacement: v =
dr
Δr
or v average =
dt
Δt
The linear momentum of an object measured in some coordinate system is simply the product
of the object’s mass and velocity:
€
€
p = mv
The linear momentum of a system of particles is the vector sum of the individual momenta:
psystem = ∑pi
The net force acting on an object (physical system) is the vector sum of all forces acting on
that object (physical system) due to the interactions with other objects (physical systems).
ΣFA = FB on A + FC on A + FD on A + …
The impulse of the total (or net) external force acting on a system equals the product of the
average force and the time interval during which the force acted.
Net Impulseext = J = ΣFavg ext∆t = ∫ ΣFext(t) dt
The force (impulse) exerted by object A on object B is equal and opposite to the force
(impulse) exerted by object B on object A.
FA on B = – FB on A and JA on B = – JB on A
Conservation of Linear Momentum
If the net external impulse acting on a system is zero, then there is no change in the total
linear momentum of that system; otherwise, the change in momentum is equal to the net
external impulse.
Net Impulseext = J = ∫ ΣFext(t) dt = pf - pi = ∆psystem
Collisions
The momentum of the system of objects (particles) remains constant if the external impulses
are negligible. This is true whether the collision is elastic or inelastic
ptoti = ptotf
If a collision is elastic, then none of the mechanical energy is transferred to bond or thermal
energies and both the total mechanical energy (all kinetic and elastic energies) and the
momentum remain constant.
(mechanical energy)i = (mechanical energy)f and ptoti = ptotf
Representations
Graphical representation of all vector quantities and (vector relationships) as arrows whose
length is proportional to the magnitude of the vector and whose direction is in the direction of the
vector quantity.
Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the
symbol.
Component algebraic equations, one equation for each of the three independent directions.
Chapter 7 Momentum Conservation: Linear Momentum Model
63
A useful way to organize and use the representations of the various quantities that occur in
phenomena involving momentum, change in momentum, and impulse and forces is a momentum
chart. The momentum chart, like an energy-system diagram, helps us keep track of what we
know about the interaction, as well as helping us see what we don’t know.
The boxes are to be filled in with scaled arrows representing the various momenta and changes in
momenta.
Closed System
Typically used for collisions/interactions involving two or more objects.
Closed
pi
pf
System
∆p
Object 1
Object 2
Total
System
0
For total system: ∆p = 0
For each object: pi + ∆p = pf
(written as component equations, if useful)
Write expressions for each momentum vector, such as p = mv
Open System
Typically used when the phenomenon involves a net impulse acting on the system.
Open
pi
pf
System
∆p
Total
System
For total system: ∆p = J
pi + ∆p = pf
(and for component equations, if useful)
Write expressions for each momentum vector, such as p = mv
Below the momentum chart draw a force diagram for the object. The net force gives the
direction of the impulse and ∆p.
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Chapter 7 Momentum Conservation: Angular Momentum Model
Angular Momentum Conservation Model
(Summary on foldout #7 at back of text)
Overview
The ideas we have developed for linear momentum and impulse apply to rotational motion as
well. But first, we will need to develop the rotational analogs of the various variable and
constructs we have been using. Force, momentum, velocity, impulse all have rotational analogs.
The concept that impulse equals change in linear momentum has its analog in rotational motion
as does the principle of conservation of momentum.
In the last model, we focused both on the properties of forces and the momentum transfers
governing the connection of force to motion. We found that forces can be rather tricky to deal
with, and we, hopefully, began to appreciate the usefulness of being very precise about technical
terminology as it relates to force and motion and to the usefulness of representations such as
momentum charts and force diagrams.
Now we extend the formalism to enable us to analyze and make sense of the motion of
extended objects that can rotate as well as translate. We also introduce the last conserved
quantity that we will work with, angular momentum (which could also be called rotational
momentum). We will introduce a couple of additional concepts: torque and rotational inertia as
well as to ways to describe rotational motion. We will then be in a position to answer detailed
questions and make specific predictions about the magnitudes of individual forces and the
changes in motion caused by the applied forces in a wide variety of situations.
Angular momentum is analogous to momentum (translational or linear momentum) even
though they are quite different physical quantities. For instance, we found in the last model that
the momentum of an object is conserved if there is no net external force acting on it. In this
model we will find that the rotational analogue of force is called torque and that the angular
momentum of an object is conserved if there is no net external torque acting on it (even if there
is a net force). Similarly, a transfer of angular momentum is called angular impulse. Remember
from earlier chapters that work is the integral of the applied force over the distance the system
moves. In this model we broaden our idea of work a little by including the energy transferred if
a torque is applied over the angle that the system rotates.
However, translational or linear momentum (usually just called momentum) and angular
momentum are clearly very different physical quantities and you will have to work hard and be
careful at keeping them separate in your thinking. The difference is obvious when you see a
physical situation but, when discussing abstract ideas without a physical picture in mind, it is
easy to confuse the two quantities. For instance, a ball may be spinning (i.e. have angular
momentum) and flying through the air in a straight line (i.e. have momentum). Or, it may be
spinning at any speed (have any angular momentum) and not be flying through the air. Or, it
may be flying through the air but not spinning at all. So, you see that the amount of angular
momentum the ball has is completely independent of its momentum. The moral of this little
story is the same as with all physics problems: try to keep a concrete physical picture in your
head as you learn new abstract ideas.
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Chapter 7 Momentum Conservation: Angular Momentum Model
The Center of Mass Idea
You may have realized by now that modeling objects as point particles is a rather drastic
oversimplification, but often very useful. When does the extended geometry of a non-point
object become important? Focusing on just one point of an object can describe perfectly
adequately the translational motion of that object, but it does not tell us anything about the
object’s rotation. Whether an object rotates or not, depends on where forces are applied to the
object. We will not derive or prove the general result, described in the following paragraphs,
that we use to handle this situation: combined translation and rotation of rigid objects. We will
simply state it.
It turns out that we can consider all of the forces acting on the object as if they acted at one
point, the center of mass, as far as translation is concerned. That is, if we are concerned only
about an object’s translation, it doesn’t matter where the forces act on the object. We can
consider them all to act at a single point! This is truly a great simplification. We have been
using this result throughout this course without making a “big deal” about it. The special point
where we consider the forces to act is called the center of mass. It is the same as the center of
gravity (where you can support the object and it won’t rotate) as long as the gravitational force is
uniform. Near the surface of the Earth, for all objects of ordinary size, the gravitational force can
certainly be considered uniform, so for all problems we consider, the center of mass and center
of gravity are the same point.
Now, what about rotations? To take into account the effect of applied forces on the rotation
of an object, we have to know where the forces are applied. We use a new construct, the torque,
τ , which takes into account the magnitude and direction of the applied force as well as its
distance from the point or axis about which the object rotates. If objects are constrained to rotate
about a particular axis, such as a wheel mounted to an axle, the torques are typically computed
about that axis. If there is no constraint, torques should be computed about the center of mass,
the point about which the object will rotate.
In order to properly discuss the rotational analog of momentum, we need to develop a
consistent way to describe rotational motion. We find an analogous set of rotational motion
variables to translational motion variables. We will introduce these motional variables by
looking at both the circular motion of a point object and the rotational motion of an extended
object (an extended object has size, so it is not a point object). By dividing up the general
motion of a rigid object into translation plus rotation, we can separately discuss the momentum
(actually the translational momentum) and the angular momentum.
The Detailed Description of Rotational Motion—Rotational Kinematics
We begin by developing some useful relationships to
describe the motion of a point object. Rather than using
rectangular coordinates to describe the position P of a particle
moving in a circle, we find it convenient to use polar
coordinates, r and θ. The coordinate r is the distance of the
point from an axis of rotation (the origin ϑ ); θ is the angular
displacement from an arbitrarily chosen axis that defines zero.
As in the figure, θ is frequently measured from the positive x
axis, but it could be measured from any reference line.
P
y
ϑ
r
θ
x
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Chapter 7 Momentum Conservation: Angular Momentum Model
When θ changes by an amount ∆θ, the particle moves an
amount ∆s along the circumference of the circle defined by the
radius r. The arc length, ∆s is simply the product of r and ∆θ:
Pf
∆s
r
The instantaneous velocity of the point P is always
tangential to the curve at that point. If we differentiate the
displacement with respect to time to get the tangential velocity
of this object, we get an expression that depends only on the
time derivative of θ:
ds
dθ
.
= vtangential = r
dt
dt
Pi
∆θ
∆s = r∆θ
r
ϑ
The time rate of change of the angular position, θ, is called the angular velocity or rotational
velocity and is usually represented by the Greek letter ω ("omega").
dθ
.
ω=
dt
The rotational velocity and tangential velocity are related by: v tangential = rω .
The Units of θ and ω.
The units of θ and ω are respectively an angle unit and an angle unit divided by time. We
can use any units we want and that are useful for a particular application for θ and ω. Typical
units are degrees, degrees/second; revolutions, revolutions/second or rpm or revolutions/hour,
etc. The "natural" units, are, however, radians and radians per second. We must use radians and
radians per second when we use the relations connecting v to ω, etc. Note that a “radian” is a
rather “funny” kind of unit. For instance, radians multiplied by meters is just meters, not
radian·meters. It is a useful word to put into sentences to tell us we are talking about angular
motion (and to make phrases “sound right”), but it does not behave like a “real” unit such as
meter or second.
Note that so far we have been discussing a point object constrained to move in a circle. We
can also describe the kinematics of any extended object (e.g. a baseball bat) that is rotating about
a fixed origin (where we grip it) by θ and ω, as long as we define the polar coordinates about the
fixed axis of rotation. Actually, we can use this same approach for objects that are rotating as
well as moving translationally, if we define the polar coordinates about the “center of mass.”
The Directions of θ and ω.
Just as the translational variables position, Direction of θ and ω is given by the Right-Hand-Rule
r, and velocity, v, have both direction and
magnitude, so do the angular variables θ and
ω. It is useful to treat these variables as
vectors, θ and ω. What direction do these
variables point? The only unique direction in
space associated with a rotation is along the
rr
axis of rotation. So, if the axis of rotation
gives the direction, we need only specify
which way along the axis corresponds to a rotation
direction
67
Chapter 7 Momentum Conservation: Angular Momentum Model
particular direction of rotation. By convention, the direction is specified by the “right-handrule.” If you curl the fingers of your right hand in the direction of positive θ or the direction
rotation is occurring, your thumb points in the direction (along the axis of rotation) of θ or ω.
We will see several more examples of the right-hand-rule (RHR).
When forces act on extended objects, they not only cause the object to change its
translational motion, but can also cause it to change its rotational motion. That is, these forces
can cause an angular acceleration as well as a translational acceleration. It turns out that it is not
just the magnitude and direction of the force that is important in
causing angular accelerations, but also where the force is applied on
F
an extended object. Torque is the construct that incorporates both
r
the vector force as well as where it is applied to an object.
ϑ
The Rotational Analogs of Force, Momentum, Mass,
and Impulse
The Torque Construct; the rotational analog to force
Consider a force F exerted tangentially on the rim of a wheel or
disk. The rim is at a distance r from the axis of rotation. We can
formally define torque, represented by the Greek letter τ, in terms of
the force F and the distance r:
F
Fradial
Ftangential
r
ϑ
τ = r⊥F = rFtangential
where r is sometimes referred to as the moment arm of this applied
force—the further away from the axis a particular force is applied,
the more torque is exerted, producing more change in rotational motion. Torque can be thought
of as the “turning effectiveness of a force” or “rotational force.”
What happens if the applied force is not purely tangential, as in the second figure? This force
can be broken down into its tangential and radial components, F tangential and F radial. Note that the
radial component F radial of this force has no effect on the rotational motion of this disk! So, for
any general force exerted a distance r from a rotation axis, it is only the tangential component of
this force ( F tangential) that will affect rotational motion. The tangential component of the force can
always be found with the appropriate trig function. If θ is the angle between the applied force, F,
and r, the tangential component is Fsinθ .
Torque, along with other angular variables, has vector properties. If
we imagine the torque causing the object to rotate about an axis
perpendicular to the plane defined by the force and the moment arm, r,
we can use the same right-hand-rule introduced for finding the direction
of θ and ω to find the direction of the torque, τ . If you curl the fingers
of your right hand in the direction of rotation that the torque would
cause, then your thumb points in the direction of the torque.
"direction" of τ
F
r
ϑ
68
Chapter 7 Momentum Conservation: Angular Momentum Model
The Angular Momentum Construct; the rotational analog to momentum
It is useful to consider the angular momentum of both a point object as
well as the angular momentum of extended objects. In either case, we
need to be clear about the axis (or point) about which we are calculating
the angular momentum.
A particle with momentum p, located at position r from some point in
space has angular momentum L about that point with a magnitude given
by
L
p
r
L = r⊥p = rptangential
Note that the angular momentum is related to the linear momentum the same way as torque is
related to force. Both L and τ depend on the choice of the point in space to which they are
referenced. Like torque, angular momentum is a vector. Its direction is perpendicular to both r
and p and is given by the RHR.
If a system has many parts, its total angular momentum is the
Direction of θ,
vector sum of the angular momenta of all the parts:
and
L = L1 + L2 + L3 ... = Σ Li
ω
L
A rigid object with rotational inertia I about some particular
axis has an angular momentum about that same axis given by
L=Iω
The direction of L is parallel to the direction of ω. These
directions are shown in the figure.
rr
rotation
Rotational Inertia; the rotational analog to mass
direction
Recall that for translational motion an object with a large
amount of inertia has a greater momentum than an object with a small amount of inertia, both
moving at the same speed. Mass, m, is the measure of inertia in translational motion.
The rotational motion analogy to inertia is rotational inertia (or rotational mass), or in very
technical language, moment of inertia. With a given net torque, Στ,, different objects will
experience different rotational accelerations.
The rotational inertia of an object does not depend solely on the amount of mass in the
object, but on how this mass is distributed about the axis of rotation.
For the simplest case of a point mass m moving in a circle of radius r, its rotational inertia is
given by:
I = mr2 .
This definition allows us to calculate the rotational inertia of any object, provided we know
the position r of every portion of its mass as measured perpendicularly with respect to the axis of
rotation:
I = m1r12 + m2 r22 + m3 r32 + … = ∑ mi ri2 .
i
€
Chapter 7 Momentum Conservation: Angular Momentum Model
69
This looks a lot like calculus (which it is in the limit of infinitesimally small mass increments.)
The table below gives the rotational inertia of several simple geometric shapes, as calculated in
the limit of infinitesimal increments of mass using this equation.
Object
Rotational
inertia
point mass m moving in radius
r
I = mr2
r
thin ring of mass m, radius r
rotating about center
I = mr2
r
thin rod of mass m, length L
rotating about one end
perpendicular to the rod
thin rod of mass m, length L
rotating about the center
perpendicular to the rod
I=
1 2
mL
3
L
I=
1
mL2
12
L
disk of mass m, radius r,
rotating about an axis
perpendicular to disk through
the center
sphere of mass m, radius r,
rotating about an axis through
the center
thin hollow spherical shell of
mass m, radius r, about an axis
through the center
I=
1 2
mr
2
r
I=
2 2
mr
5
r
I=
2 2
mr
3
r
As seen from the formulas in the table, objects with the same mass can have very different
rotational inertias, depending on how the mass is distributed with respect to the axis of rotation.
70
Chapter 7 Momentum Conservation: Angular Momentum Model
Also, it is possible for an object to change its rotational inertia (e.g., a gymnast tucking in or
extending arms and legs), which can lead to dramatic results as net torques are applied.
The rotational inertia of a composite object is the sum of the rotational inertias of each
component, all calculated about the same axis.
Itotal = I1 + I2 + I3 + 
.
So for a ring and a disk stacked upon each other and rotating about the symmetry axis of both,
the total rotational inertia is: Itotal = Iring + Idisk .
€
The SI units of rotational inertia are kg·m2.
Angular Impulse; the rotational analog to impulse
The angular analog to impulse, is angular impulse:
AngJext =∫ τ ext(t) dt
or, if the torque is constant with time, or we define an average torque, τ avg
AngJext = τ avg ∆t
A Statement of Angular Momentum Conservation:
AngJext =∫ τ ext(t) dt = ∆Lsystem
or
AngJext = τ ext ∆t = ∆Lsystem
If the net external angular impulse acting on a system is zero, then there is no change
in the total angular momentum of that system; otherwise, the change in angular
momentum is equal to the net external angular impulse.
Other Angular Counterparts
Work
We are familiar with the concept of work as a way that the energy of a system is changed. In
terms of force and distance, work is:
W = ∫ F||dx
where the parallel symbol reminds us that it is only the components of force and displacement in
the same direction that contribute to the integral.
A similar expression holds for the work done by a torque which acts through an angle:
W = ∫ τ||dθ
The energy of a particular system can be changed by the process of a force exerted by an
outside object doing work on an object in the system and/or by a torque exerted by an outside
Chapter 7 Momentum Conservation: Angular Momentum Model
71
object doing work on an object within the system. In either case, the work can be positive
(increases the energy of the system) or negative (decreases the energy of the system). If the
force or torque is constant (or we assume an average force or torque), the integral is immediately
performed and we have
W = F||∆x
and
W = τ||∆θ
Energy Systems
The total energy of a system is the sum of all of the various energy systems, which can
include both translational and rotational energy systems. During collisions among parts of a
physical system, energy can be transferred among these separate systems. We have previously
mentioned rotational kinetic energy. Another energy system with a rotational counterpart is
elastic or spring potential energy. The elastic potential energy of a system described by a spring
constant k is:
PEelastic = 1/2 kx2
Similarly, the elastic potential energy of a rotating system which has a linear restoring force is
given by the expression:
PEelastic = 1/2 kθ2
The Rate of Energy Transfer: Power
We previously discussed power as the time derivative of energy transfer, or the rate at which
the energy of a system changes. This applies, of course, to any type of energy system. Recall
that the SI unit of power is the watt (W) which is equal to a joule per second.
In mechanical systems, in which energy is transferred as work, it is often useful to consider
the rate of energy transfer, power, associated with a particular force. Since the energy
transferred is the work done by the force, the power associated with that force is the time
derivative of the work. If the force is constant in time, then:
P = dW/dt = d(F||avgx)/dt = F||avg dx/dt = F||avg v
Thus, the power is simply the applied force times the velocity of the object the force is acting on.
The rotational counterpart is:
P = τ||ω
Putting it all together
The chart on the next page shows all of the linear motion and dynamic variables along with
their rotational counterparts. Keep this chart out and handy for ready reference to help you from
getting “lost” in all the symbols. You should make sure that you recognize the meaning behind
the symbols when you see on of these relationships. (Note: acceleration, angular acceleration,
and Newton’s second law are treated in detail in the next chapter, but are shown here for
completeness and convenience)
72
Chapter 7 Momentum Conservation: Angular Momentum Model
Summary Listing Fundamental Concepts Used in Mechanics
Emphasizing Translational and Rotational Counterparts
Category
Concept
Translation
Rotation
Relation
kinematic
variables
position
x
θ
θ = arclength /r
velocity
v = dx/dt
ω = dθ/dt
ω = vt/r
acceleration
a = dv/dt
α = dω/dt
α = at/r
force/torque
F
τ
τ = r⊥F = rFtang
inertia
momentum
m
p = mv
I
I = Σ mr2
L = Iω
L = r⊥p = rptang
Elastic Energy
1/2 kx2
1/2 kθ2
Kinetic
Energy
1/2 mv2
1/2 Iω2
Work
System
Energy
W = ∫ F||dx
fundamental
dynamic
variables
Energy
All Kinetic and Potential Energies
plus Thermal
Energy
Conservation
P = dE/dt =τ||ω
Momentum
p = mv
L = Iω
Impulse
Momentum
Conservation
J = ∫ Fdt
Newton’s
1st law
Newton’s
Laws
∆Esystem = Wall + Q
P = dE/dt = F||v
Power
Momentum
W = ∫ τ||dθ
Newton’s
2nd law
Newton’s
3rd law
∆p = Jext
system
if ΣF = 0, then
∆v = 0, ∆p = 0
ΣF = ma
or, ΣF = dp/dt
F1on2 = -F2on1
J1on2 = -J2on1
AngJ = ∫ τ dt
∆L = angJext
system
if Σ τ = 0, then
∆L = 0
Σ τ = Iα
or, Σ τ = dL/dt
L = r⊥p = rptang
Chapter 7 Momentum Conservation: Angular Momentum Model
73
Interesting Effects Involving Angular Momentum
There are two fascinating aspects of angular motion that don’t exist for linear motion in quite
the same way. The first is that the rotational inertia is readily changed, as for example, when a
skater extends or pulls in a leg. The second is related to the fact that both p and L are vector
quantities and can change in direction without changing in magnitude.
When an ice skater begins to spin with a leg extended, there is only a small torque exerted on
the skater by the ice. Thus, angular momentum diminishes rather slowly (she can spin for a long
time). Now, if she pulls in her leg, her rotational inertia is reduced considerably, and her
rotational velocity (spin velocity) increases considerably. This is most easily seen by writing the
angular momentum as L = Iω and noting that if L remains almost constant, then the product Iω
must remain constant.
Another fascinating, and rather startling situation, is the change in direction of the angular
momentum of a spinning object when it is acted upon by a torque that is not along the direction
of the angular momentum vector itself. This is the weird behavior exhibited by a spinning top or
gyroscope.
The figure shows a bicycle wheel supported by a rope at the left end of a
short axle. The second figure is the extended force diagram. The torque
caused by the force of the Earth acting down at the center of gravity of the
wheel produces a torque that is perpendicular to this force and the axle; it
points into the figure. If the wheel is not spinning, it just falls, rotating about
the pivot point, because this is the only point of support. However, when the
Side
wheel is spinning with angular momentum L0, the situation is much view
different.
The third figure is a top view, showing the original angular momentum
points into page
vector, Li, the new angular momentum vector, Lf and the torque, τ . The
torque acts for a time ∆t . We use the angular impulse equation to give the
F E on wheel
ϑ
change in angular momentum,
τ ∆t = ∆L = Lf - Li
or Lf = Li + τ ∆t
That is, the direction of the initial angular momentum, Li, is changed by
Top View
the presence of the angular impulse, and is moved to the direction shown
by Lf. But if L is in a new direction, then the orientation of the wheel
must have changed, because L is due to the spinning wheel and points
τ ∆t
along ω. This turning motion of the orientation of the wheel is called
precession. Instead of falling, the wheel precesses. Of course, once the
angular momentum (and the wheel) point in a new direction, the torque
comes into play again, causing the wheel to precess still farther. In this
fashion, the wheel is caused to precess in a horizontal circle about the L f = L i + τ ∆t
pivot point.
Precession is analogous to the situation of a ball being twirled around in a circle on the end of
a string. Why doesn’t the tension in the string pull the ball in toward the center of the circle?
The answer is that it does, but the large tangential velocity also moves the ball in a direction
tangent to the circle. The net result is that the ball travels in a circular path. If there were no
large tangential velocity, the ball would indeed be pulled directly toward the center of the circle
due to the tension in the string. A similar thing happens with the bike wheel. The torque causes
74
Chapter 7 Momentum Conservation: Angular Momentum Model
a change in the direction of the large angular momentum of the spinning wheel. If the wheel did
not have this large angular momentum, the torque would cause the wheel to tip over, or “fall
down.”
Angular Momentum Conservation Model Summary
Just as we did for linear momentum conservation, we will summarize the main ideas of the
angular momentum conservation model/approach by listing the (1) constructs, i.e., the “things”
or ideas that are get “used” in the model, (2) the relationships–in mathematical or sentence form–
that connect the constructs in meaningful ways, and (3) the ways of representing the
relationships.
Developing a deep and rich understanding of the relationships in a
model/approach comes slowly. It is absolutely not something you can memorize. This
understanding comes only with repeated hard mental effort over a period of time. A good test
you can use to see if you are “getting it” is whether you can tell a full story about each of the
relationships. It is the meaning behind the equations, behind the simple sentence relationships,
that is important for you to acquire. With this kind of understanding, you can apply a
model/approach to the analysis of phenomena you have not thought about before. You can
reason with the model.
Listed here are the major, most important constructs, relationships, and representations
of the angular momentum conservation model.
(Be sure to refer back to the chart on page 34 showing the analogous linear and angular
constructs and relationships.)
Constructs
Angular Velocity, ω
Rotational Inertia, I
Angular Momentum, L
Net Torque, Σ τ
Angular Impulse, angJ
Newton’s 3rd law
Conservation of angular momentum
Relationships
The angular velocity is the time derivative of the displacement:
ω=
€
dθ
dt
or ω average =
Δθ
Δt
The angular momentum of an object measured about some fixed axes is simply the product of
the object’s rotational inertia and angular velocity:
€
L = Iω
The angular impulse of the total (or net) external torque acting on an object equals the
product of the average torque and the time interval during which the torque acted.
Net Angular Impulseext = angJ = Σ τ avg ext∆t = ∫ Σ τ ext(t) dt
The directions of torque, impulse, angular velocity, and angular momentum as determined by
the right-hand rule
Chapter 7 Momentum Conservation: Angular Momentum Model
75
The torque (angular impulse) exerted by object A on object B is equal and opposite to the
torque (angular impulse) exerted by object B on object A.
τ A on B = – τ B on A and angJA on B = – angJB on A
Conservation of Angular Momentum
If the net external angular impulse acting on a system is zero, then there is no change in
the total angular momentum of that system; otherwise, the change in angular momentum
is equal to the net external angular impulse.
Net Angular Impulseext = angJ = ∫ Σ τ ext(t) dt = Lf - Li = ∆Lsystem
Representations
Graphical representation of all vector quantities and (vector relationships) as arrows whose
length is proportional to the magnitude of the vector and whose direction is in the direction of the
vector quantity.
Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the
symbol.
Component algebraic equations, one equation for each of the three independent directions.
A useful way to organize and use the representations of the various quantities that occur in
phenomena involving angular momentum, change in angular momentum, and angular impulse
and torques is an angular momentum chart, which is totally analogous to the linear momentum
chart. The angular momentum chart helps us keep track of what we know about the interaction,
as well as helping us see what we don’t know.
The boxes are to be filled in with scaled arrows representing the various angular momenta and
changes in angular momenta.
Closed System
Open System
Typically used for interactions
involving two or more objects.
Closed
Li
Lf
System
∆L
Typically used when the phenomenon
involves a net angular impulse acting on the
system.
Open
System
Li
∆L
Lf
Object 1
Total
System
Object 2
Total
System
0
For total system: ∆L = angJ
Li + ∆L = Lf
(and for component equations, if useful)
For total system: ∆L = 0
For each object: Li + ∆L = Lf
(written as component equations,
if useful)
Write expressions for each momentum
vector, such as L = Iω
If appropriate, show the extended force
diagram that determines angJ = Σ τ ∆t
Write expressions for each
momentum vector, such as L = Iω
Make sketches if required to show
directions of various vectors
Make sketches if required to show directions
of various vectors and the application of the
right-hand rule.
76
Chapter 7 Momentum Conservation: Wrap-up
Wrap-up
We have now developed approaches/models to enable us to use the three fundamental
conservation laws of all of science: energy, momentum and angular momentum. The “before
and after the interaction” approach, which now includes momentum and angular momentum, as
well as energy, is extremely general and universally applicable. It is the foundation of
equilibrium thermodynamics. It allows us to get answers to most questions we ask regarding the
behavior of interacting systems, as long as we don’t need the time dependence of the dynamical
variables.
What are the limitations to the approaches we have developed these past two chapters? We
have mentioned some of these before, but it is good to emphasize them again. We know from
our prior studies in chemistry and from some of what we have done in this course, that strange
things begin to happen when the systems we are studying get very small, the size of molecules
and atoms. Energies become quantized. Atoms and molecules can absorb and emit only certain
amounts of energy, not a continuous range. We saw how specific heat modes became frozen out
at low temperatures in solids. Things also get weird when speeds become large. In this case,
large means moving at speeds that begin to approach the speed of light—3 x 108 m/s. Both at
very small scales and when things go fast, our approach breaks down and must be replaced by
more complicated theories. But the primary variables in both quantum mechanics and in special
relativity turn out to be energy, momentum, and angular momentum. There is something very
special about these constructs. They apparently represent some of the most basic aspects of the
universe. The fundamental ideas of conservation of energy, momentum, and angular momentum
carry through all of the models we use to describe our universe.
The concepts of energy, momentum, and angular momentum (and the conservation of energy
and momentum) remain as we delve into the details of the microscopic and the realm of very
high speeds, but we do have to make changes in our understanding of these constructs. Energy,
momentum and angular momentum take on discreet values; i.e., they become quantized. When
we go to high speeds momentum and energy become intertwined. Even the separate idea of
mass conservation gets pulled into a unified mass-energy conservation principle. We will
explore the quantum world a little further in Part 3, but you will need to explore the fascinating
world of special and general relativity on your own or in more advanced courses.