Environmental Engineering Reference
In-Depth Information
Example 2.2.3
Calculate the position of the centre of mass of a uniform thin rod
of length l and linear mass density ρ(x)
d
m
d
x
=
=
κx where x is the distance from
one end of the rod.
Solution 2.2.3
We consider the rod as being made up of many tiny pieces, each of
length
d
x and mass
d
m
=
ρ(x)
d
x (see Figure 2.9). The position of the centre of
mass is given by
M
i
M
i
1
1
R
=
m
i
x
i
i
=
ρ(x
i
)x
i
d
x
i
and in the limit that
d
x
→
0
the sum can be replaced by an integral, i.e.
l
l
1
M
1
M
κx
2
d
x
i
,
R
=
xρ(x)
d
x
i
=
0
0
l
0
κx
d
x
1
2
κl
2
and
i
is the unit vector in the direction of increasing
where M
=
=
x. Thus
2
3
l
i
,
R
=
i.e. the centre of mass lies two-thirds of the way along the rod, on the high-density
side of the geometric centre.
d
x
x
Figure 2.9 Slicing a rod into pieces in order to compute the centre of mass of a thin rod
of non-uniform density.
2.2.2 Newton's bucket and Mach's principle
Earlier in this chapter we discussed the idea that an isolated particle viewed from
an inertial frame of reference moves with constant velocity. This is the essential
content of the First Law in Newtonian mechanics; it provides a way of select-
ing inertial frames from non-inertial ones. Within an inertial frame, accelerations
are the result of pairs of forces operating between particles. We have shown that
all inertial frames move at constant relative velocities and are thus led to the
idea of an infinite set of inertial frames, which are all equally valid for doing
physics. The concept of the inertial frame thus underpins classical mechanics, but
there is no deeper explanation given as to
why
these particular frames of refer-
ence are so special in our Universe. In this section we will try to probe a little
deeper.
In classical mechanics there are no absolute velocities and only relative velocities
have any meaning. The same thing cannot be said for accelerations. Acceleration
