Environmental Engineering Reference
In-Depth Information
8
Non-inertial Frames
To this point, our attention has focused mainly on physics as viewed from inertial
frames of reference. Inertial frames have the substantial advantage that Newton's
laws hold within them and that Einstein's Special Relativity is formulated using
them. For example, bodies not acted upon by some external force travel in straight
lines (or remain at rest) and acceleration arises as a result of the action of a force.
However, it is not always advantageous to work in an inertial frame. For example,
a natural frame to choose when describing physics on the surface of the Earth
would be a frame at rest relative to the Earth. Any such frame is not inertial
because the Earth is spinning on its axis (and rotating in orbit about the Sun). In
this chapter, our goal is to understand the implications of working in non-inertial
frames of reference. As we shall see, Newton's laws can be rescued provided we
are prepared to introduce the idea of fictitious forces. In order not to complicate
matters too much we shall assume that all speeds are sufficiently small so that we
can ignore the effects of relativity. We will in fact return to consider relativistic
effects in accelerating frames of reference towards the end of the topic.
8.1 LINEARLY ACCELERATING FRAMES
Let us start with the simplest type of acceleration, namely acceleration in a
straight line. In Figure 8.1 we show two frames of reference. It looks rather similar
to the pictures in the last chapter on Special Relativity except that now the frame
S
is accelerating uniformly relative to
S
. Ignoring the relativistic effects, if a particle
is located at position
x
(t)
in
S
then its co-ordinates in
S
are given by
x
(t)
=
x
(t)
−
X
(t),
(8.1)
where
X
(t)
is the position of the origin
O
relative to the origin
O
. Differentiating
twice gives us a relationship between the acceleration of the particle as it would
