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CHAPTER 9
CONSERVATIVE FORCES
1.
Introduction.
In Chapter 7 we dealt with forces on a particle that depend on the speed of the particle.
In
Chapter 8 we dealt with forces that depend on the time.
In this chapter, we deal with forces that
depend only on the position of a particle.
Such forces are called
conservative forces
;
While only
conservative forces act, the sum of potential and kinetic energies is conserved.
Conservative forces have a number of properties. One is that the work done by a conservative
force (or, what amounts to the same thing, the
line integral
of a conservative force) as it moves
from one point to another is routeindependent.
The work done depends only on the coordinates
of the beginning and end points, and not on the path taken to get from one to the other.
It
follows from this that the work done by a conservative force, or its line integral, round a closed
path is zero.
(If you are reminded here of the properties of a
function of state
in
thermodynamics, all to the good.)
Another property of a conservative force is that it can be
derived from a potential energy function.
Thus for any conservative force, there exists a scalar
function
V
(
x
,
y
,
z
) such that the force is equal to
−
grad
V
, or
−
∇
V
.
In a onedimensional
situation, a sufficient condition for a force to be conservative is that it is a function of its position
alone.
In two and threedimensional situations, this is a necessary condition, but it is not a
sufficient one.
That a conservative force must be derivable from the gradient of a potential
energy function and that its line integral around a closed path must be zero implies that the
curl
of a conservative force must be zero, and indeed a zero
curl
is a necessary and a sufficient
condition for a force to be conservative.
This is all very well, but suppose you are stuck in the middle of an exam and your mind goes
blank and you can't think what a line integral or a
grad
or a
curl
are, or you never did understand
them in the first place, how can you tell if a force is conservative or not?
Here is a rule of thumb
that will almost never fail you:
If the force is the tension in a stretched elastic string or spring, or
the thrust in a compressed spring, or if the force is gravity or if it is an electrostatic force, the
force is conservative.
If it is not one of these, it is not conservative.
Example
.
A man lifts up a basket of groceries from a table.
Is the force that he exerts a
conservative force?
Answer
:
No, it is not.
The force is not the tension in a string or a spring, nor is it electrostatic.
And, although he may be fighting against gravity, the force that he exerts with his muscles is not
a gravitational force.
Therefore it is not a conservative force.
You see, he may be accelerating
as he moves the basket up, in which case the force that he is exerting is greater than the weight of
the groceries.
If
he is moving at constant speed, the force he exerts is equal to the weight of the
groceries.
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