Environmental Engineering Reference
In-Depth Information
they do show that it is possible to obtain gravitational accelerations similar to the
centrifugal and Coriolis accelerations that appear in frames of reference that rotate
relative to the distant stars. That said, the agreement is not exact and depends on
the assumed distribution of matter in the shell. To a large extent, the specialness
of inertial frames is diminished in Einstein's General Theory of Relativity (see
in particular the discussion following Eq. (14.35)) and Mach's concerns become
much less pressing.
2.3 APPLICATIONS OF NEWTON'S LAWS
We have discussed at some length the theoretical content of Newton's laws of
motion and are now in a position to apply them to dynamical problems. While this
is in some cases a straightforward exercise, the versatility of classical mechanics
ensures that there exist a huge range of different types of problems that can be
posed. Different problems often require different approaches. While many people
can happily follow the solution to a given problem, it is often the case that when
facing a fresh problem on their own they cannot see how to start. Problem solving
in dynamics is therefore a skill that needs to be learnt and which improves greatly
with practice. In this section we will look at a technique for solving dynamical
problems based on “free-body” diagrams. We will also show how some problems
can be more easily solved using the principle of momentum conservation. Friction
and viscous forces play an important role in the behaviour of macroscopic systems
and they will also be discussed in the present section.
2.3.1 Free Body Diagrams
Solving problems in dynamics usually involves a sequence of several steps.
While the following programme is not always the most efficient or elegant way to
proceed (the use of conserved quantities often works better) it represents a direct
approach to finding the solution and is a good fall-back position when you cannot
spot a clever trick to use.
The problem-solving recipe is generally as follows:
•
Identify the forces acting on the system
•
Choose a system of co-ordinates (and associated basis vectors) appropriate to
the geometry.
•
Apply the Second Law to the components of the system to get second order
differential equations of motion.
•
Solve the equations of motion together with any constraints or boundary con-
ditions.
To aid in the first step of this procedure it is almost always crucial to draw a
good diagram. We will try to make this diagram as uncluttered as possible, while
including all forces. To this end we draw each body as “free” in the sense that
there are no supports drawn on the diagram (although the forces exerted by them
should certainly be included). In addition we shall represent the position of a body
by the position of its centre of mass. We indicate the forces as arrows and draw
