We hold these truths to be self evident...
You may have detected a cause and effect relationship among
the elements of motion we have been studying under the heading of
kinematics . Consider a
particle at rest in our reference frame. It can't undergo a
displacement, that is change position, without getting a velocity.
It can't change its velocity from zero without getting an
acceleration. Acceleration seems to be at the root of motion as
far as kinematics is concerned. Now we need to peel back the next
layer and look at the cause of acceleration. That takes us out of
the realm of kinematics into the next level of mechanics, called
A fundamental concept in dynamics is the dynamical system. Since
dynamics is the study of motion and forces , things that move or things
under the influence of forces or both are called dynamical
systems. For example a satellite in orbit around the Earth is a
dynamical system. So also might a beating heart be considered a
So where did the word "system" come from? What makes
an object a system? In general a system is considered to be a
thing composed of more than one part. Now I do not want to make
too big a deal about this one word "system" because I
am not sure all that much thought went into choosing this word
back in the early days, but in fact we cannot study the motion
of a single isolated point. Think about it. Things only move
relative to other things, like an observer for example. So if
there is motion there is a system. Likewise no force exists for a
single isolated particle so again the word system seems to
By applying the laws of nature to a dynamical system, we may
determine its future behavior. That is really what science is all
about. Predicting the future. We are in this business so as to
predict the future early enough and accurately enough to profit
from our knowledge.
The technique of predicting the future of a dynamical system
by application of the laws of nature which govern its change as
time passes is called "mathematical modeling". The way
it works is this. We discover somehow the laws relating the
physical quantities of the dynamical system to time. We express
those laws in mathematical terms. The resulting equations may
involve the derivative of some of the variables with respect to
time. Then we solve the equations using the things we know to
calculate the things we don't know.
This is called modeling because the mathematical functions
derived from the laws of nature display the behavior of the
observable quantities of the actual system. That means that we
can avoid the time and expense involved in building an actual
model of the dynamical system, assuming that it is even possible
to build such a model. We will not, in this course be building
mathematical models like that in the diagram on the left. It just
illustrates one of the possibilities.
Newton's first law of motion, sometimes called the law of
inertia, was actually adopted by Newton from the work of Galileo.
It states that a particle's velocity will not change unless a
force is applied to the particle. This means that if a
"body" (a collection of particles having appreciable
mass) is at rest it remains at rest unless a force is applied. If
a body is moving with some velocity, it will continue to move
with the same velocity unless a force is applied. The first law
is really just a qualitative statement about the persistence of
Newton's second law gets into a quantitative relationship
between forces and motion. For this we need to revisit the rate
of change idea. We defined velocity as the rate of change of
position, a small displacement divided by the corresponding
change in time, as v =
Dr / Dt. The velocity
vector itself may change over time so it too has a rate of
change. That rate of change of velocity is called
"acceleration". The acceleration of a body is the
change in velocity divided by the corresponding change in time,
as a = Dv / Dt. We have no single word analogous to
displacement, describing the change in velocity.
The second law states that the acceleration of a body is
proportional to the force on it. This is consistent with our
experience that the harder we push on a moveable body, the
quicker its speed changes. The second law goes on to state that
the constant of proportionality between the force and the
acceleration is the "mass" of the body. In the form of
an equation the second law reads F=m*a, where
F is the force vector, m is the scalar mass, and
a is the acceleration vector. The mass may be considered the
property of a body that determines its resistance to changing
We are using the kilogram as the unit of mass, as stated earlier.
The unit of length is the meter, so displacement is in meters.
Velocity is calculated as displacement divided by time so its
units are meters per second. Acceleration is calculated as
velocity divided by time so its units are meters per second per
second, or meters per second squared. The force sufficient to
accelerate one kilogram by one meter per second squared is called
one Newton, in honor of the old gentleman himself, seen at the
You may have observed that the first law is contained in the
second as the special case where force and therefore acceleration
Newton's third law addresses the nature of forces. The
implicit assumption is that a force is simply a manifestation of
the interaction between a pair of bodies. You might say there can
not be a pushee without a pusher. The third law states that the
force resulting from the interaction of two bodies acts with
equal magnitude on both of them and in opposite directions. For
every action, there is an equal and opposite reaction.
These three laws of nature credited to Newton are not all
there are but they are enough to allow us to get started in
building and analyzing mathematical models of some dynamical
Suppose for example that I toss a ball straight up from the
surface of the Earth. Once it leaves my hand let's assume the
only force acting on it is gravity. The
Vertical Ball Toss
display shows that situation. I could not
resist adding a bit of reality to this system by making the ball
bouncy. Notice that it looses some height on every bounce. Later
we will relate that to some properties of the ball and the
surface on which it is bouncing.
Suppose in tossing the ball I fail to toss it straight up.
That means that the initial velocity vector has not only a
vertical component but also a horizontal one. In our discussion
of vectors we showed how a vector may be resolved into
perpendicular vector components.
Now that we have introduced force into the
picture, I am going to go over again here an important idea about
the independence of motion in different dimensions. One of the
things implicit in Newton's second law is that the change in
velocity, acceleration and force all point in the same direction.
This is evident from the fact that multiplying a vector by a
scalar does not change its direction. Acceleration is force
divided by the scalar, mass, so acceleration is
"collinear" with force, meaning that the vectors lie in
the same direction. The change in velocity is acceleration
multiplied by the scalar, time, so the change in velocity is
collinear with the force. Notice that it is only the change in
velocity which lies in line with the force, not necessarily the
Since force and acceleration are collinear, a force vector
which is vertical can have no effect on horizontal motion.
Likewise a horizontal force cannot affect vertical motion. This
means that the motion of a particle in two or three dimensions
can be studied as two or three independent one-dimensional
problems where the dimensions are chosen along axes that are
mutually perpendicular. For example in our thrown ball situation
one of the axes would be chosen vertical and the other horizontal
with distance increasing in the direction of the horizontal
component of the initial velocity.
Next we apply Newton's laws to each of the independent
motions separately. In the vertical direction the ball moves
exactly as when tossed straight up. In the horizontal direction
gravity has no effect on the ball's motion since it acts
perpendicular to that path. It is Newton's first law which
applies in this direction. Since the ball experiences no
horizontal force it moves with uniform velocity horizontally
until it returns to the ground. The
Combined Ball Toss
display shows the effect of having both
horizontal and vertical components of initial velocity.
These ball toss displays present the output of the model in
the most physical but least useful manner. What you see is an
animation of the motion of a ball tossed with a certain initial
velocity. Except for the ability to precisely set the initial
velocity, this model brings us no closer to predicting the future
than actually tossing a ball would do. To get a better grip on
the data generated by the model, we could make a videotape of
the moving ball, run it one frame at a time and measure the
ball's position in each frame. Then we could tabulate the
results and plot the table on a graph. In fact that is the way
dynamics data was handled in the old days.
Now that mathematical modeling on a
computer is possible, we can go directly to a more useful
presentation. As a first approach we can capture the motion by
recording the ball's flight, plotting its
so that we can see where it has been as well as where it is. That is the sort of
display we saw in the Projectile
Motion example. A tossed ball and an artillery shell should
have some things in common.
A still more useful representation of the model output would
allow you to answer directly questions like, "What is the
maximum height reached by a projectile?", "At what time
is the maximum height reached?", "When does the
projectile hit the ground... At what point?" and so on. This
information is contained in plots of vertical and horizontal
position vs. time as illustrated on the next display. On such
plots you can just place the cursor to get the required value. If
more precision is required than you can get from the graph, you
will need a table of values. For that you may use our
There is a sort of a recipe for solving problems involving
Newton's laws as applied to individual
Remember we are still
dealing with particles, even though we have moved on from
kinematics. However messy a problem may seem to be, frequently we
can boil it down to the movement of one or more particles under
the influence of various forces. The key to solving these
problems is to use a trick called the "
diagram" to keep track of the forces applied.
Even if there are multiple particles involved, isolate them
one at a time and carefully identify the forces which apply to
the one on which you are working. Then choose the orientation of
your reference frame so as to simplify your work as much as
possible. Next draw a diagram showing the particle with each
force vector attached to it. Then resolve each of the force
vectors into its components along the reference frame dimensions.
Add up the forces in the x, y, and z(if applicable) directions.
Then to get the acceleration along each dimension divide the
appropriate force by the mass. Once the accelerations are known,
apply the rules we developed in kinematics, or use calculus, or
do mathematical modeling on a computer, to get the velocities and
positions as functions of time.
Play around with the
Free Body Diagram Tool
to get some idea what it can do for you. Detailed
instruction are included with the display.
So far we have exercised Newton's first and second law.
The third law deals with action and reaction. What this law
really means is that forces always occur in pairs. In Newtonian
terms there is no such thing as an isolated force. The action and
reaction force pair also always acts on different bodies. If they
acted on the same body, there would be no motion because the net
force on everything would be zero. For any body you can imagine
which is subject to some force, there is another body subject to
the reactive force. This business of action and reaction does not
imply a cause and effect. Either force could be considered the
action force, the other is then the reaction.
Correctly identifying the action/reaction
force pairs and associating each force with the correct object is
sometimes a stumbling block. Imagine a book lying on a table
standing on the surface of the Earth. There is a table-book
force, a book-table force, a table-Earth force, an Earth-table
force, an Earth-book force and a book-Earth force. The force of
the Earth on the book and the equal but opposite force of the
table on the book are not an action/reaction pair. They both act
on the same body. The force of the Earth on the book and the
force of the book on the table are not an action/reaction pair.
They are both in the same direction. The force of the Earth on
the book and the force of the Earth on the table are not an
action/reaction pair. They involve three different bodies.
Consider the ball toss situation, If I toss a ball straight up
with a certain force, my hand exerts the force on the ball and
the ball exerts an equal and opposite force on my hand. The force
on my hand from the ball is transmitted through the structure of
my body to my feet, which are planted firmly on a very large ball
called the Earth. So the ball is accelerated in one direction and
the person-Earth body is accelerated in the opposite direction.
The reason we don't notice the movement of the Earth is the
huge difference in mass between it and the ball.
Let's consider a situation where the reaction force
produces a more noticeable effect. A friend of mine once tried to
use a six-gauge goose gun to propel a modified harpoon at giant
blue fin tuna. A few moments of quiet reflection on Newton's
laws could have saved him considerable discomfort. The goose gun
had a mass of about 8KG. The harpoon had a mass of about 2KG. The
burning of the charge of gun powder, about a quarter of a cup of
it, produced equal force on the projectile and the gun. The
acceleration produced by a force is inversely proportional to the
mass of the body, so the velocity of the gun toward the shooter
only built up one fourth as fast as the velocity of the harpoon
toward the fish. Even so the collision between the gun and the
shooter was hard to ignore.
Perhaps a more common example of the third law is the collision
between two billiard balls. During the time the balls are in
contact, they are subject to equal and opposite forces. The
situation in this little two body system from shortly before to
shortly after the collision is shown in the Billiards display.
In the Billiards display, the operation of Newton's third
law is seen in the action and reaction of the two balls. Perhaps
later we will expand the capability of this display to see some
more interesting effects. You will notice that the magnitude of
the average force between the two balls during their contact time
is displayed at the end of the run. The directions of the action
and the red and black vectors show reaction forces,
which are not to scale by the way. The assumptions used in
calculating this average force is that the balls each have a mass
of 0.25kg and they remain in contact for 1 millisecond. We will
deal with this sort of
impulse force in more detail later.
In the next section of the course, I will go into some of the
details of setting up and solving the kind of equations which
make mathematical modeling possible.
Are there any questions?