Environmental Engineering Reference
In-Depth Information
in size in proportion to each of the masses. Notice that if the distance
r
is fixed so
that it is equal to the radius of the Earth,
R
E
, then Eq. (2.18) simplifies to
F
=−
mg
e
r
,
(2.19)
where
GM
R
E
g
=
and
M
is the mass of the Earth. In this way we can view our earlier expression,
F
mg
as a special case and we have the bonus of relating
g
to the mass and
radius of the Earth once we know
G
.
=
2.2.1 Newton's Third Law
So far we have not worried too much about just how forces act on extended
bodies. What about all of the forces internal to the body? They certainly do not
appear to play a role in the motion of the body as a whole so it seems they must
cancel each other out somehow. Similarly, when we speak of the acceleration of an
extended body, it is not immediately clear whether we are speaking about all parts
of the body or perhaps one special point within it. The example of a spinning ball
thrown through the air illustrates the point because different parts of the ball clearly
accelerate differently (remember that rotation is associated with acceleration). In
this section we shall make progress towards resolving these matters by considering
the behaviour of extended bodies, although we shall have to wait until Chapter 10
before we finally solve the problem of a spinning object thrown through the air.
As a bonus, we shall also solve another problem that we have left hanging in the
air - just how do we define an inertial frame? That is a serious problem because
we have shown that non-inertial frames are characterized by the fact that isolated
particles accelerate. But there is a nasty loophole since we can presumably never
be sure that the acceleration has not arisen as the result of a force and that the
particle is not actually isolated.
Progress in addressing these matters can be made once we have a grasp of
Newton's Third Law, which expresses the empirical fact that real forces are found
in pairs. Applied to particles it asserts that:
If particle
B
exerts a force on particle
A
given by
F
AB
,then
A
will exert a force
on
B
(
F
BA
) such that
F
BA
=−
F
AB
.
In other words the force exerted on particle
A
by particle
B
is equal in magnitude,
but opposite in direction, to the force exerted on particle
B
by particle
A
.
Immediately we see that the Third Law gives us a mechanism for distinguishing
between acceleration caused by a real force, and that which is the result of choosing
a non-inertial frame of reference. Let us consider particle
A
. If this particle is
observed to be accelerating, then either there is a force acting on it, or the observer
is using a non-inertial frame. The observer may then look for another particle that
is responsible for the force. If that particle (
B
) can be identified, it must be subject
