Environmental Engineering Reference
In-Depth Information
frame of reference. Any objects within the lift will experience not only their own
weight but also an upwards fictitious force of magnitude
mg
where
g
is the accel-
eration of the lift (i.e. the acceleration due to gravity). But this is none other than
the weight of the object. Hence an unfortunate passenger within the lift will feel
weightless as they plummet towards the ground. We invoked the lift for dramatic
effect but it should be clear that a person falling freely towards the ground will
feel weightless (in the absence of any air resistance). This is a very interesting
and intriguing result and in fact provides us with our first hint towards Einstein's
theory of gravitation, also known as his General Theory of Relativity. It is worth
our spending a moment or two to consider just why the weightlessness of free fall
is a such remarkable phenomenon.
Take a look at Eq. (8.6). The mass which appears in this equation and which
determines the magnitude of the fictitious force is just the mass which appears in
Newton's Second Law. Now, the exact cancellation of the weight of a body in
free fall only occurs if this mass is identically equal to the mass which appears in
the law of gravitation and which defines the weight
mg
of the body. This may not
at first strike us as a remarkable result but we really ought to be very impressed
that the inertial mass which appears in Newton's Second Law is, as far as we can
tell, identical to the mass which appears in the law of gravity. After all, these are
two totally independent laws of physics. The significance of this equivalence of
inertial and gravitational mass can be glimpsed if we return to the example of a
lift in free fall within a uniform gravitational field and realise that physics within
the lift is totally indistinguishable from physics in a lift floating in the zero gravity
environment of outer space. The suggestion is that uniform gravitational fields can
be eliminated if we work in freely falling frames of reference. As we shall see
in Chapter 14.2, Einstein took this idea to its logical conclusion and succeeded
in eliminating the force of gravity altogether in exchange for a description of the
world in terms of an infinity of carefully chosen freely falling frames of reference
which, as we shall later show, is equivalent to a curved spacetime.
8.2 ROTATING FRAMES
Sitting on a merry-go-round, one is in a rotating frame of reference. In order to
remain at rest in that frame we feel a fictitious force called the centrifugal force
which pushes outwards and balances the real centripetal force pulling us towards
the centre. The centrifugal force is not the only fictitious force associated with
a rotating frame. You may even have noticed the other force if you have ever
attempted to play a game of 'catch' whilst riding on a merry-go-round: it is called
the Coriolis force and it arises when objects are in motion in a rotating frame of
reference. In this section we shall derive mathematical expressions to quantify the
role of the centrifugal and Coriolis forces.
Let us consider a set of co-ordinate axes which rotate with an angular velocity
ω
about some axis
1
and we shall place the origin somewhere on the axis of rotation.
For definiteness, you might think of such a set of axes fixed to the Earth, with the
origin located at the centre of the Earth. We denote the basis unit vectors in the
1
See Section 4.3 for the definition of
ω
.
