Environmental Engineering Reference
In-Depth Information
z
z'
S
S'
A
inertial frame
non-inertial frame
x
x'
O
O'
Two frames of reference S and S which are accelerating relative to each other.
Figure 8.1
be determined in the two frames of reference, i.e.
a (t)
=
a (t)
A (t),
(8.2)
where A (t) is the acceleration of S relative to S (the double arrow in the figure is
intended to denote acceleration).
Now we know that Newton's Second Law holds in the inertial frame and hence
the acceleration of the particle in S is related to an applied force F via
F
=
m a
(8.3)
(assuming a particle of fixed mass m ). We can use Eq. (8.2) to re-write this
equation as
m( a +
F
=
A ).
(8.4)
This can obviously be re-cast into the form
F =
m a
(8.5)
provided F =
m A . Thus from the point of view of an observer at rest in S
the particle moves around as though it is acted upon not only by the real force F
but also by a fictitious force
F
F fict =−
m A .
(8.6)
A very simple and familiar illustration of such a fictitious force occurs if one
considers a ball on the floor of an accelerating car. As the car accelerates forwards
so the ball rolls towards the back of the car. From the viewpoint of someone
sitting in the car it is as if the ball is being pushed along. Of course there is no
physical force acting upon the ball: viewed from the point of view of a person
standing watching the car accelerate past, in the absence of any friction the ball
would remain at rest whilst the car accelerates. The very same fictitious force
is responsible for pressing the driver of the car back into their seat as the car
accelerates.
Another example is provided if we consider the case of a freely falling lift (or
elevator). As the lift accelerates downwards it can be used to define a non-inertial
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