Environmental Engineering Reference
In-Depth Information
where
A
is the acceleration of the centre of mass. Eq. (2.28) represents an extremely
important simplification of the dynamics of complex systems. In a macroscopic
object there are of the order of 10
23
particles. If these particles all interact with
each other, there will be of the order of 10
46
forces to consider; clearly, solving
the motion for so many particles is an impossible task. The beauty of Eq. (2.28)
is that irrespective of the details of the internal forces, the motion of the centre of
mass is governed only by external forces.
Example 2.2.2
Show that the centre of mass of an extended body falls with a uni-
form acceleration g near the Earth's surface. Neglect air resistance.
Solution 2.2.2
Each particle within the body experiences an external force m
i
g.
The total force on the body is thus
m
i
g
g
m
i
=
F
=
=
Mg.
The acceleration of the centre of mass is therefore A
=
F/M
=
g, as required.
The Third Law applies equally to extended bodies as it does to particles. If we
consider two bodies,
A
and
B
, then because the internal forces sum to zero in both
bodies, the total force that
A
exerts on
B
is the sum of the forces that the particles
in
A
exert on the particles in
B
. This sum is equal in magnitude but opposite in
direction to the force that the particles in
B
exert on the particles in
A
.Sothe
Third Law can be stated for extended bodies:
If body
B
exerts a force on body
A
given by
F
AB
,then
A
will exert a force on
B
(
F
BA
) such that
F
BA
=−
F
AB
.
It is important to be clear that these two forces act on
different
bodies. Figure 2.8
illustrates this with two bodies connected by a massless spring. The spring in the
figure is a symbolic representation of any real force.
A
B
F
AB
=
m
A
a
A
F
BA
=
m
B
a
B
Figure 2.8 Equal magnitudes but opposite directions for forces acting on mutually inter-
acting bodies A and B.
In many practical situations it is impossible to consider explicitly all the parti-
cles that make up a system to determine the position vector of the centre of mass.
Instead, a macroscopic body can often be approximated as a continuous distri-
bution of matter with a spatially-dependent density function. The calculation of
the centre of mass position then becomes an integral rather than a discrete sum.
Depending on the situation this integral may be either over a line, a surface or a
volume.
