Environmental Engineering Reference
In-Depth Information
Thus we see that although an ideal spring obeys Hooke's Law and therefore
is a harmonic oscillator it is not the only harmonic oscillator. We have just seen
that
any
system close to a point of stable equilibrium also constitutes a harmonic
oscillator and therefore behaves like a spring. The only exception is those systems
for which the second derivative of the potential just happens to vanish at the point
of equilibrium, i.e.
k
=
0.
3.3
COLLISIONS
At the very heart of the Newtonian programme to understand the world is the
idea that particles move around and collide with each other. Collisions will typi-
cally change the energies and momenta of the colliding particles but always such
that the total energy and total momentum remain unchanged. We must be very
careful though in how we use energy conservation because it is possible for some
energy to “leak away”, e.g. the energy carried away by a sound wave when two
objects collide, and it strictly needs to be included when we come to compute
the total energy after the collision. These days, collisions are exploited daily at
the world's particle physics colliders in order to develop an understanding of how
matter behaves at the shortest distances. Although Newton's laws do not apply in
those experiments (we need relativity and the quantum theory instead) it is nev-
ertheless true that energy and momentum remain conserved. It is also true that
the methods used in solving collision problems in Newtonian mechanics are very
similar to those used when it comes to tackling collisions in particle physics and
that is something we will explore in more detail in Section 7.2.
3.3.1
Zero-momentum frames
We will consider the collision between two classical particles that interact by a
force that goes to zero at large distances. This allows us to separate the process
into three stages (see Figure 3.8): (a) the early stage, when the particles are far
enough apart to be each considered isolated; (b) the interaction stage when the
mutual interaction is significant and the particles are accelerating; (c) the late stage
when the particles are once again isolated. The particles have masses
m
1
and
m
2
with initial velocities
v
1
and
v
2
, respectively. In stage (c), after the collision, the
velocities are
v
1
and
v
2
respectively. In constructing Figure 3.8 we have chosen
a frame of reference in which to view the collision. Naturally enough we take
this to be the frame in which the experimenter is stationary. This frame is known
as the
lab frame
. While the lab frame is a familiar frame of reference to use in
thinking about the collision, it turns out not to be the most useful for calculations.
All isolated collisions conserve linear momentum, which is a vector quantity. So if
we want to make things easy for ourselves we could choose a frame of reference
in which the total momentum is zero. We will call this the
zero-momentum frame
.
Let us suppose that we transform to a frame of reference travelling with velocity
V
with respect to the lab frame. How will the particles move in this new frame of
reference? As discussed in Section 1.3.2, we know
5
that
5
These results change in Special Relativity.
