Download earlier section

Newtonian Mechanics in one dimension
As explained in the last section, we consider a
body which can move freely along a horizontal
track fixed to the surface of the earth. The track
is rigid and straight, and has tick marks one
meter apart.
−4 −3 −2
−1
0
1
2
3
4
We also have a uniformly-running clock, with
which we can measure time in seconds. Using
the clock and the tick marks on the track, we can
determine the velocity of the body. For present
purposes, all we really need is to be able to say
whether the velocity is constant or not.
The principle of inertia
Suppose we start the body with some initial
velocity. Then eventually it slows down and
stops due to friction. However, we can repeat the
experiment using ever more elaborate methods of
reducing friction. We find that it becomes harder
and harder to detect the decrease in velocity.
By suitably refining the apparatus, it is possible
to come arbitrarily close to the ideal situation in
which the velocity of the body remains constant.
According to Newton, this is no accident, but
rather a fundamental law of nature.
This law is often called Newton's first law, but a
more descriptive name is
The principle of inertia:
• (tentative version only!) In the ideal case in
which a body is not acted upon by any external
influence, its velocity will remain constant.
Inertia is the name given to this tendency of a
body to remain in a state of uniform motion
unless acted on by an external influence.
Two remarks are needed, in order to make the
above statement of the principle of inertia
precise. First, the body must have constant mass.
The meaning of "constant mass" should be intuitively clear, although we have not yet actually
defined mass. Second, it is important to specify
what we are measuring the velocity with respect
to. The principle of inertia is not valid, for example, if we measure the velocity of our freelymoving body with respect to a track which is
accelerating - speeding up or slowing down -
with respect to the surface of the earth.
To see this, suppose the body and track are
initially at rest with respect to the surface of the
earth. Then the track begins to speed up to the
right. As seen from the earth, the body remains
at rest - moves with a constant velocity of zero:
(This is what happens when a waiter quickly
pulls a tablecloth out from under some dishes.)
From the point of view of the track, however, the
body begins to speed up to the left:
Its velocity with respect to the track is therefore
not constant.
Similarly, if the clock does not run uniformly,
then the velocity of the body with respect to the
fixed track will not appear to be constant, even if
no external influence acts on the body.
The track, together with the clock, is our frame of
reference. If the track is not accelerating and the
clock runs uniformly, then the principle of inertia
is valid and the frame is an inertial frame. If not,
then the frame is noninertial. To be accurate, the
last part of the above statement of the principle of
inertia should read ...its velocity with respect to
an inertial frame will remain constant.
A final comment is in order. In our example of a
track on the earth, we have ignored two effects:
the rotation of the earth, and tidal forces. These
cause the surface of the earth to be a noninertial
frame. We will deal with these phenomena in
later chapters. For present purposes, it is
sufficient to know that their effects can be made
negligibly small in the experiment described.
Newton's second law
Now let's see what the effect of an "external
influence" is. We will join the body to a spring
whose other end is attached to the track, and then
stretch the spring.
If we hold the body at rest and then release it, it
accelerates to the right - its velocity increases.
We say that a force has been applied to the body.
Force:
• Anything which causes a body having constant
mass to accelerate relative to an inertial frame is
called a force.
Recall that we have units for time and distance seconds and meters, respectively. Notice that the
above definition does not tell us what units to use
for force, or how to measure its value in those
units. This definition is qualitative, rather than
quantitative. To make it quantitative, we need to
give a precise definition of mass.
Suppose we now join a second, identical body to
the first one. We stretch the spring by the same
amount as before, and observe the motion of the
combined bodies. We look for a relation between
the motion of the original body and that of the
combined bodies.
Of all the properties of the motion, one in particular is related very simply. We find that the
acceleration is exactly half as large as before.
(We are talking here about the initial acceleration, when the springs are still both stretched
by the same amounts.)
Notice that it didn't have to be this way. The acceleration could have been some other fraction of
its original value. Or the property which behaves
simply could have been the rate of change of the
acceleration. There are many such possibilities.
But the motion happens to be governed by a law
of nature which says that the acceleration is half
as large for the combined bodies, for a given
force.
Now, the size of the initial force applied to the
combined bodies is the same as that applied to
the single body. The force is something external
to the body, its value being completely
determined by the internal properties of the
spring and by how far the spring was stretched.
And the spring was stretched by the same initial
amount in both cases.
What has changed is the amount of matter contained within the body - this has doubled. The
word for the "amount of matter", used in this
sense, is mass. The mass of the new body is
twice that of the old single body.
If we made another new body out of three of the
original ones, the mass would be three times as
large. The measured initial acceleration would
be one-third as large as the original. Thus, the acceleration produced by a given force is inversely
proportional to the mass.
What is the unit of mass? All we can do is pick
some reference object and say that it has one unit
of mass. Then the masses of all other bodies are
fixed in terms of this unit. We simply compare
their accelerations under the influence of the
same force. By convention, we choose the mass
unit to be the kilogram.
Note that this procedure is no different from the
choice of second and meter as the units of time
and distance. The choice is completely arbitrary.
(In this course, we will not be concerned with the
actual standards which define the second, meter
and kilogram.)
Newton's second law:
Now let's consider what happens when we double
the force, but leave the mass the same. We do
this by using two identical springs instead of one,
and stretching them by the same amount as
before.
• In an inertial frame,
We compare the motion with the original, and
find that the initial acceleration is doubled.
Again, this did not have to be so, but nature says
it is. For three springs, the acceleration is tripled,
and so on. The acceleration is directly proportional to the force, when the mass is constant.
If the acceleration is inversely proportional to the
mass when the force is constant, and directly
proportional to the force when the mass is
constant, then the mathematical relation between
them is
acceleration = (a number) H force ÷ mass.
We fix the number by choosing the size of the
unit of force. We say that a force of one unit
acting on a mass of one unit produces an
acceleration of one unit. Then the number is 1.
It is conventional to multiply both sides of the
above relation by the mass, and express the
relation symbolically in the form of
F = ma
for a body having constant mass.
The unit of force is symbolized by the letter N,
and is called the
Newton:
1 N = 1 kg m s–2
(kg stands for kilogram, m for meter and s for
second. The unit of acceleration is m s–2 - read
"meters per second per second".)
As we have defined it, mass is a measure of a
body's resistance to being accelerated. The larger
the mass of a body, the less it will be accelerated
by a given force. It has "more inertia", so to
speak. For this reason, it is appropriate to call
mass defined in this way inertial mass.
When we later study gravity, we will see that
there is another kind of mass, the gravitational
mass, defined completely differently. The distinction between the two definitions of mass is
tremendously important, as is the experimental
fact that they are one and the same!
Re-stated using the concept of force, the
principle of inertia reads
The principle of inertia:
that a force acts on the body. But this force is
fictitious - it is entirely an effect of the acceleration of the reference frame.
• In the ideal case in which a body having
constant mass is not acted upon by any external
force, its velocity with respect to an inertial
frame will remain constant.
Despite the fact that such forces are fictitious,
they are extremely important and we will have
occasion to discuss them in detail later on.
It is interesting to look at the relation between
Newton's second law and the principle of inertia.
The principle of inertia deals with the case in
which the force on the body is zero, while
Newton's second law tells what happens when
the force is arbitrary. So the principle of inertia
is just a particular case of Newton's second law the case where the force is zero.
We should check that this is reflected in the
mathematics. Beginning with F=ma, we set F to
zero. The mass is not zero, so the acceleration a
must be zero. But zero acceleration is the same
as constant velocity. This shows that Newton's
second law reduces to the principle of inertia
when the external force is zero.
We note that Newton's second law is not valid in
a noninertial frame. We return to our earlier
example in which the track accelerates but the
body remains at rest with respect to the earth.
From the point of view of the track, the body
accelerates. A person attached to the track might
then apply Newton's second law and conclude
Several forces
More than one distinct force can act on a body.
For example, a body could have several springs
attached to it, and there could be friction present,
and so on. In this case, it is found experimentally
that the body’s acceleration is the same as that
which would result from the application of a
single force which is the sum of the individual
forces. In symbols,
ma = F net = 3 F i ,
i
where the index i labels the individual forces.
The sum of all the individual forces is sometimes
called the “net” force. In this course, we will
often write “F=ma” with the understanding that
F means the net force if more than one force is
acting.
Newton's second law in action
In many cases, the nature of the net force acting
on a body is known. It might depend on time,
position, velocity, or some combination of these,
but its dependence is known from experiment. In
such cases, Newton's law becomes an equation of
motion which we can solve.
How is this equation obtained? We begin with
Newton’s law, F=ma, where the (net) force is a
function of x, v, and t:
F(x,v,t) .
Using the definition of the velocity and
acceleration as the first and second derivatives of
x(t), respectively, we get the equation of motion
äå
d 2 x(t)
dx(t) ëìì
å
m
=
F
x
(
t
)
,
,t ì .
å
dt í
dt2
ã
This is a differential equation for the function
x(t). That is, it is an equation whose solution is a
whole function of time, not just an algebraic
number.
The solution allows us to predict the position of
the body at any time, as long as we know its
initial position and velocity. This predictive
quality is the main power of Newton’s law.
Don’t worry if this looks a bit abstract right now.
In the next few sections, we will see how it
works in several concrete cases.
Here is a directory of the cases we will consider:
• constant force, with and without damping
• force proportional to distance (the harmonic
oscillator), with and without damping
• (advanced topic) vertical motion near earth,
taking into account variation of gravity with
height.