Calculus
Based Section: Variable Forces
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 Work-Energy 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 =
|
|
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 Work-Energy Theorem
Though a calculus based
proof of the Work-Energy 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 Work-Energy
Theorem works.
Using that equation, the
equation we derived for work done by a variable force, we can manipulate it to
yield the work-energy 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 =
mv
dx =
mvdv
Integrating from v o to v f :
W net =
mvdv =
mv f 2 -
mv o 2
This result is precisely
the Work-Energy 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.
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