Modern Engineering Thermodynamics

10.3 What Are Conservative Forces? 321
10.2.2 Conservative Fields
A vector field is said to be conservative if it has a vanishing line integral around every closed path c in its region
of definition, or
I
c
!
A . dx ! = 0
Since the line integral of a conservative vector field ! A around any closed path is always zero, the value of its
integral between any two arbitrary points ! x 1 and ! x 2 depends only on the end points themselves and is independent
of the path taken between these points (i.e., the integral is a point function). Further, any vector field ! A
that obeys the preceding equation must also obey the equation ! A = −∇ ! P, where ! ∇ is the differential operator 2
(called the del operator) and Pð ! x Þ is called the potential 3 of ! A at ! x .Theterm∇ ! P is called the gradient of P,
and for a point ! x and an arbitrarily chosen reference point ! x 0, we can write
Pð ! x Þ = Pð ! Z
x!
!
x 0Þ − A . dx !
x!
0
The gradient of a scalar function P is usually written as ∇ ! P = A ! , where A ! is called the gradient vector.
10.3 WHAT ARE CONSERVATIVE FORCES?
Any force produced by a reversible process is a point function (i.e., it depends only on the end points of the
process) and is therefore called a conservative force. Thetermconserve means to preserve from loss or decay. The
“conservation” laws of physics state that certain measurable quantities (e.g., mass, energy, momentum) do not
change with time in an isolated system. Similarly, a conservative force is one whose magnitude is not diminished
in time by its own action (i.e., it is nondissipative). Thus, the total work done by a conservative force is
independent of the path producing the displacement and is equal to zero when its path is a closed loop, or
W conserative
force
I
=
!
F . dr ! = 0:
C
Since conservative forces are reversible (i.e., nondissipative), they form a conservative vector field and have an
energy potential (or stored energy), Φ, defined as F ! = −∇ ! Φ. Nonconservative (irreversible) forces (such as friction)
that depend on other forces (such as sliding velocity) are dissipative, and no energy potential can be
defined for them.
OK, BUT WHAT DOES ALL THIS REALLY MEAN?
In simpler terms, this means that we can divide forces into two categories: conservative forces and nonconservative forces. If
the “net” work done by a force acting on a system is always zero, the force is said to be conservative. In other words, if the
work done by a force depends only on the initial and final states of a system and not on the path taken by the force, then
it is a conservative force. Otherwise, it is non-conservative.
Examples of conservative forces:
■
■
■
The force of gravity.
Coulomb’s force in electrostatics.
A completely elastic deformation.
Examples of nonconservative forces:
■
■
■
Friction (both sliding and viscous).
Inelastic (or plastic) deformation.
Electrical resistance.
2 This operator has the following form in Cartesian coordinates
! ∂ðÞ! ∇ ðÞ=
∂x i ∂ðÞ! +
∂y
j ∂ðÞ +
∂z k! :
3 Note that the potential P is not uniquely determined by A ! , since any other potential of the form P′ = P + a constant also satisfies
this equation. Consequently, a conservative force field A ! can always be written as the negative gradient of its potential P.