Environmental Engineering Reference
In-Depth Information
is called the gravitational mass. Introducing a constant of proportionality
g
we
write
m
g
.
F
=
(2.5)
As we shall soon see we can choose
g
to be the acceleration of
all
bodies in
free-fall. Since the mass is in static equilibrium, then the two forces that act
upon it must be equal in magnitude and opposite in direction, and we can write
kx
m
g
. Thus the elastic extension of the spring is related to the gravitational
mass of the object, the principle behind some types of weighing scales. We can
do similar experiments with other forces; Millikan's oil-drop experiment uses the
the static equilibrium between the electrostatic, viscous and gravitational forces
on charged oil drops to determine the electrical charge of the electron.
By using static equilibrium in various experiments we can investigate the prop-
erties of different forces, putting them on a common scale. While this procedure
is practicable for many physical systems, it is not always convenient to observe
forces in static equilibrium. Forces more generally affect the motion of things, as
is the case with the gravitational attraction between the Moon and the Earth, or the
force on an electron moving in a magnetic field. It is now time for us to study the
way in which forces affect motion.
=
2.2 FORCE AND MOTION
We return to the experiment with springs depicted in Figure 2.4. Suppose we
were to suddenly sever the three springs on the left-hand side of the mass, what
would we observe? Such an experiment can be easily performed in the lab, and
is depicted in Figure 2.6. Now the mass moves so we write the extension of
the spring as a function of time
x(t)
. From experiment the observed motion is
oscillatory and can be expressed as
2
x(t)
=
A
cos
ωt,
(2.6)
x
(
t
)
Mass
Figure 2.6
Mass on a spring in motion. The mass moves horizontally over a very smooth
surface.
2
Strictly speaking the motion will have some degree of damping and the amplitude of the oscillation
will not be constant. We will assume that the damping is small enough to be ignored.
