1549380323-Statistical Mechanics Theory and Molecular Simulation

Newton’s laws of motion 3

dW = -Fcosq dl

dl

F

q

F

dl

θ

Fig. 1.1Example of mechanical work. Here dW=F·dl=Fcosθdl.

system along a particular path. The work dW performed along a short segment dlof
the path is defined to be
dW=F·dl=Fcosθdl. (1.2.5)

The total work done on the object by the force between points A and B along the
path is obtained by integrating over the path from A to B:

WAB(path) =

∫B

A

F·dl. (1.2.6)

In general, the work done on an object by a force depends on the path taken between A
and B. For certain types of forces, calledconservative forces, the work is independent
of the path and only depends on the endpoints of the path. We shalldescribe shortly
how conservative forces are defined.
Note that the definition of work depends on context. Eqn. (1.2.6) specifies the
work donebya forceF. If this force is an intrinsic part of the system, then we refer to
this type of work as work doneby the system. If we wish to calculate the work done
againstsuch a force by some external agent, then this work would be thenegativeof
that obtained using Eqn. (1.2.6), and we refer to this as work doneon the system. An
example is the force exerted by the Earth’s gravitational field on anobject of massm.
If the mass falls under the Earth’s gravitational pull through a distanceh, we can think
of the object and the gravitational force as defining the mechanical system. In this
case, the system does work, and eqn. (1.2.6) would yield a positive value. Conversely,
if we applied eqn. (1.2.6) to the opposite problem of raising the objectto a heighth,
it would yield a negative result. This is simply telling us some external agent must
do workonthe system against the force of gravity in order to raise it to a heighth.
Generally, it is obvious what sign to impart to work, yet the distinctionbetween work
done on and by a system will become important in our discussions of thermodynamics
and classical statistical mechanics in Chapters 2–6.
Given the form of Newton’s second law in eqn. (1.2.4), it can be easily shown that,
in a flat or Euclidean space, Newton’s first law is redundant. According to Newton’s
first law, an object initially at a positionr(0) moving with constant velocityvwill
move along a straight line described by

r(t) =r(0) +vt. (1.2.7)

This is an example of atrajectory, that is, a specification of the object’s position as a
function of time and initial conditions. If no force acts on the object, then, according
to Newton’s second law, its position will be the solution of