Environmental Engineering Reference
In-Depth Information
Example 1.3.1
Calculate the acceleration of a particle with the time-dependent
position vector given by
1
2
at
2
j
.
r
(t)
=
A
sin
(ωt)
i
+
Solution 1.3.1
Differentiation once gives
v
(t)
=
Aω
cos
(ωt)
i
+
at
j
and again to obtain
Aω
2
sin
(ωt)
i
=−
+
a
(t)
a
j
.
1.3.1 Frames of reference
To describe the motion of a particle we need a position vector and a point of
origin. We have seen that a position vector may be represented by components in a
given co-ordinate system. The question immediately arises as to how one chooses
a co-ordinate system to best suit a given physical situation.
For example, consider a cabin attendant who pushes a trolley along the aisle of
an aircraft in flight. For a passenger on the aircraft a natural co-ordinate system
to use would be one fixed to the aircraft, perhaps a Cartesian system with one
axis pointed along the aisle. On the other hand, an observer on the ground might
prefer a co-ordinate system fixed to the Earth. The reason why the observers tend
to choose different co-ordinate systems is that each observer is surrounded by a
different collection of objects that appear to be stationary. The passenger on the
aircraft regards the structure of the aircraft as fixed whereas the observer on the
ground regards objects on the Earth as stationary. We say that the passenger and
the observer on the ground have different frames of reference.
A frame of reference is an abstraction of a rigid structure. We might think of a
collection of particles whose relative positions do not change with time. However,
it is not necessary for the particles to actually exist in order to define a frame
of reference, we simply understand that the particles
could
exist in some sort of
static arrangement that defines the frame of reference. Within a particular frame of
reference there is always an infinite choice of co-ordinate systems. For example,
if the observer on the ground chooses Cartesian co-ordinates, there are an infi-
nite number of ways in which the axes may be oriented. Alternatively, latitude,
longitude and distance from the centre of the Earth may be chosen as the three
co-ordinates, with an arbitrary choice of where the meridian lines lie. The choice
of co-ordinate system implies a particular frame of reference, but we can discuss
frames of reference without commitment to a particular co-ordinate system.
1.3.2 Relative motion
In describing the motion of two particles it is often advantageous to use relative
position and velocity vectors. The relative position vector
R
ab
(t)
r
a
(t)
is the displacement from the position of particle
a
to that of particle
b
and it is,
=
r
b
(t)
−
