Download Newtonian Mechanics

Newtonian Mechanics
We are now going to proceed with Newtonian
mechanics in three dimensions. The treatment
will parallel that given in the earlier section on
Newtonian mechanics in one dimension, and will
be more brief.
An inertial frame typically consists of a uniformly-running clock and a set of identical measuring rods with uniform tick marks, mounted at
right angles to one another. The rods are not
accelerating in any way (in particular, not
rotating).
In more than one dimension, position, velocity,
and acceleration are vectors: xP, vP, and a
P . If the
position is a function of time, then the velocity
and acceleration are
vP =
z
2
dxP
dvP
d xP
and Pa =
=
.
dt
dt
dt2
The first step is to state the principle of inertia
and use it to define what we mean by an inertial
frame.
The principle of inertia
• If a body having constant mass is not acted
upon by any external influence, its velocity will
remain constant.
Inertial frames
• An inertial frame is one in which the principle
of inertia holds.
ä
y
x
We will choose to measure time in seconds and
distance in meters.
Now that we have defined inertial frames, and
know how to tell when no external influence is
acting, we can proceed to cases in which an
external influence does act.
of the same external influence. By convention,
we choose the mass unit to be the kilogram.
Mass
The next step is to define (inertial) mass. We do
this by first figuring out a way to apply the same
external influence to different bodies in an
inertial frame. For example, we stretch a spring
by a fixed amount, and attach composite bodies
consisting of varying numbers of identical bodies
to the spring.
fixed
In this case, we know that the external influence
doesn’t depend on the nature of the body on
which it acts. It is determined solely by the
distance the spring is stretched.
We find that the acceleration of a composite
body is inversely proportional to the number of
its constituents - that is, to the m a s s of the
composite body.
We 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
Newton's second law
We then figure out how to vary the external
influence, but leave the mass the same. For
example, we do this by using several identical
springs instead of one, and stretching them by the
same amount as before.
fixed
We find that the acceleration is proportional to
the number of springs - that is, to the force
exerted by the bunch of springs.
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 takes the form of
Newton's second law:
Newton's second law in action
• In an inertial frame,
P = m aP
F
for a body having constant mass.
Notice that this is a vector equation. Both the
force and the acceleration have a direction associated with them, and Newton’s second law says
that their directions are the same.
In cases in which more than one distinct force
acts on a single body, 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,
P = 3F
P ,
ma
P=F
net
i
i
where the index i labels the individual forces.
The sum of all the individual forces is sometimes
called the “net” force.
In many cases, the nature of the 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 a set of
three equations of motion which we can solve.
There is one equation for each dimension.
Example: motion under a constant force
As our first example, we consider the motion of a
P - one
body under the action of a constant force F
which does not vary over time, is the same no
matter where the body is, and does not depend on
the body's velocity.
We begin with Newton’s law. Using the definition of the acceleration as the first derivative
of the velocity, we find the following three
equations of motion:
P = m dvP(t) .
F
dt
Rearranging and integrating once, we find
vP(t) =
P
F
t + a constant vector,
m
where the constant vector does not depend on
time. The constant must be fixed by the value of
the velocity at some particular time. For
simplicity, let's suppose that this time is t=0, and
that the velocity at that time is vP0 . Substituting
t=0 into both sides, we find that the value of the
constant vector is vP0 . The velocity is therefore
given by
P
dxP(t)
F
=
t + vP0 .
dt
m
We then integrate again, and find
xP(t) =
P 2
1 F
t + Pv0 t + another constant vector.
2 m
The new constant vector is fixed by the value of
the position at another particular time, which we
may again take to be t =0. If the position then is
xP0 , then the value of the new constant is xP0 . The
final answer for the position as a function of time
is thus given by
xP(t) =
P 2
F
t + Pv0 t + xP0
2m
.
This formula allows us to find the position at any
time, as long as we know the values of the initial
position and velocity vectors. The reason we
have to know two vectors is because Newton's
law gives rise to a second-order vector differential equation. That is, the highest derivative
which appears is the second derivative.
You might find it useful to compare this analysis
with our previous treatment of motion under a
constant force in one dimension. You will find
that they are remarkably similar.