Environmental Engineering Reference
In-Depth Information
in that state. Observer A is moving, relative to a frame of reference in which B is
at rest, with velocity
v
and acceleration
a
. A will therefore measure the particle to
be moving with velocity
a
. So what does A conclude? The
particle is accelerating, so there must be a force acting on it (according to the First
Law). However, as the particle is isolated, by definition it cannot be subject to any
forces. This contradiction is a direct result of observing the particle from an accel-
erating frame of reference. If
a
−
v
and acceleration
−
v
,
but with zero acceleration. Since this is uniform motion, the First Law still holds
for the isolated particle. Thus there are two classes of frames of reference, those
in which
a
=
0
then A observes the particle with velocity
−
0
,
called non-inertial frames of reference. The First Law thus becomes essentially a
statement upon the existence of inertial frames of reference:
=
0
, called inertial frames of reference, and those for which
a
=
There exist inertial frames of reference, with respect to which an isolated particle
moves in a straight line of constant velocity (including zero).
This reformulation of the first law supposes that we can find an isolated particle.
Clearly there is no real object so alone in the Universe that it is devoid of all
forces; the very act of observing something involves an interaction at some level,
even if it is only the force involved in reflecting light. So how do we ever find, in
practice, a good inertial frame? From a practical point of view we must find ways
of isolating a particle other than by removing it to a remote region of the Universe.
This involves using our knowledge of forces to arrange things in such a way that
there is no
net
force on a body. An air-hockey table is just such a construction: air
is blown through tiny holes to create a force on the puck that cancels the effect of
the Earth's gravity. In addition, supporting the puck on a layer of air means that
frictional forces are greatly reduced for most laboratory experiments. With the table
adjusted properly, a puck will glide at nearly constant velocity across the table, with
only a small change in speed. So does the air-hockey table define an inertial frame
of reference? Approximately, yes, but at some level of precision the effects of the
Earth's rotation will become apparent. As was shown in Section 1.3.4, an object
moving in a circle at constant angular speed is accelerating towards the centre of
the circle. Thus any laboratory fixed to the surface of the Earth is accelerating
and therefore constitutes a non-inertial frame of reference. Similarly the rotation
of our neighbouring stars about the galactic centre means that even the “fixed”
stars cannot be counted upon to define a perfect inertial frame. We cannot take the
principle of inertia as a statement that can be verified experimentally in isolation
of the rest of mechanics. By itself, the First Law may be thought of as a statement
of faith in the existence of inertial frames of reference. In practice it matters little
that we cannot find perfect inertial frames. Approximate ones are good enough for
the development of classical mechanics and experiment confirms the results to a
sufficiently high degree of accuracy under a wide range of conditions.
Now that we have hammered out the definition of an inertial frame we are in
a position to clarify what we mean by 'no motion' in our force experiment of
Figure 2.1. We define a particle to be in static equilibrium when it is acted on by
forces and yet is at rest in some inertial frame. This is equivalent to saying that the
particle must be moving with constant velocity when measured in any inertial frame.
