Environmental Engineering Reference
In-Depth Information
Mass
Figure 2.1 Identical, equally-stretched springs with a mass in static equilibrium in the
horizontal plane. The mass is supported vertically on a low-friction surface (such as an
air-hockey table). Equally you could imagine the experiment as being performed in outer
space.
force on the mass. Let us further say that the springs are carefully constructed to
be as similar as possible; for the sake of this argument they can be considered to be
identical. To further make sure that the springs behave identically we change the
angles that the springs make with each other and look for a situation in which each
spring is stretched by the same amount. When we perform this experiment in the
lab we observe that the mass is stationary if and only if the angle between any pair
of equally-stretched adjacent springs is 120
◦
. Our definition of force must take the
result of this type of experiment into account. Since we know from experience that
pushing or pulling can produce motion we assert that our experiment with three
springs, in which the mass doesn't move, corresponds to a total force of zero. In
this way we are led to the idea that force must be a vector quantity, which sums to
give zero in our experiment. That the vector should also point along the axis of the
spring can be deduced from a similar experiment constructed with two collinear
springs: there is then no special direction other than the axis of the springs and
any physical property of the system should not break this symmetry, so the force
must point along the length of the spring. Force is thus to be regarded as a vector
quantity representing a push or pull, which: (a) points along the axis of a stretched
spring; (b) is additive when several springs are involved; (c) results in no motion
when that sum is zero. We also know from experience that there are things other
than springs which may push or pull, so we state as part of our definition that any
thing that can potentially replace one of the springs in the above experiment also
exerts a force on the mass.
This sounds pretty close to a good definition of force, albeit in a fairly specific
scenario, but there is a weakness that you may have already noticed. We stated
that the force is zero when the mass is stationary. That is quite reasonable for
experiments performed at rest on the Earth but what happens if we do the exper-
iment in outer space? How do we agree on what frame of reference to use for
