So far we have looked at the work done by a constant force. In the physical
world, however, this often is not the case. Consider a mass moving back and
forth on a spring. As the spring gets stretched or compressed it exerts more
force on the mass. Thus the force exerted by the spring is dependent on the
position of the particle. We will examine how to calculate work by a position
dependent force, and then go on to give a complete proof of the WorkEnergy
theorem.
Work Done by a Variable Force
Consider a force acting on an object over a certain distance that varies
according to the displacement of the object. Let us call this force F(x), as
it is a function of x. Though this force is variable, we can break the
interval over which it acts into very small intervals, in which the force can be
approximated by a constant force. Let us break the force up into N intervals,
each with length δx. Also let the force in each of those intervals be
denoted by F_{1}, F_{2},…F_{N}. Thus the total work done by the force is
given by:
W = F_{1}δx + F_{2}δx + F_{3}δx + ^{ ... } + F_{N}δx
Thus
W =
F_{n}δx
This sum is merely an approximation of the total work. Its degree of accuracy
depends on how small the intervals
δx are. The smaller they are, the
more divisions of
F arise, and the more accurate our calculation becomes.
Thus to find an exact value, we find the limit of our sum as
δx
approaches zero. Clearly this sum becomes an integral, as this is one of the
most common limits seen in calculus. If the particle travels from
x_{o} to
x_{f} then:
Thus
W = F(x)dx 

We have generated an integral equation that specifies the work done over
a specific distance by a position dependent force. It must be noted that this
equation only holds in the one dimensional case. In other words, this equation
can only be used when the force is always parallel or antiparallel to the
displacement of the particle. The integral is, in effect, quite simple, as we
only have to integrate our force function, and evaluate at the end points of the
particle's journey.
Full Proof of the WorkEnergy Theorem
Though a calculus based proof of the WorkEnergy theorem is not completely
necessary for the comprehension of our material, it allows us to both work with
calculus in a physics context, and to gain a greater understanding of exactly
how the WorkEnergy Theorem works.
Using that equation, the equation we derived for work done by a
variable force, we can manipulate it to yield the workenergy theorem. First we
must manipulate our expression for the force acting on a given object:
F_{net} =
ma =
m =
m =
mv
Now we plug in our expression for force into our work equation:
W_{net} =
F_{net}dx =
mvdx =
mvdv
Integrating from v_{o}
to v_{f}
:
W_{net} =
mvdv =
mv_{f}^{2} 
mv_{o}^{2}
This result is precisely the WorkEnergy theorem. Since we have proven it with
calculus, this theorem holds for constant and nonconstant forces alike. As
such, it is a powerful and universal equation which, in conjunction with our
study of energy in the next
topic, will yield powerful
results.