Work
This is the final application of integral that we’ll be looking at in this course. In this
section we will be looking at the amount of work that is done by a force in moving an
object.
In a first course in Physics you typically look at the work that a constant force,
F
, does
when moving an object over a distance of
d
. In these cases the work is,
However, most forces are not constant and will depend upon where exactly the force
is acting. So, let’s suppose that the force at any
x
is given by
F(x)
. Then the work
done by the force in moving an object from
to
is given
by,
To see a justification of this formula see the
Proof of Various Integral
Properties
section of the Extras chapter.
Notice that if the force is *constant we get the correct formula for a constant force.
where
ba
is simply the distance moved, or
d
.
So, let’s take a look at a couple of examples of nonconstant forces.
Example 1
A spring has a natural length of 20 cm. A 40 N force is required to stretch (and hold
the spring) to a length of 30 cm. How much work is done in stretching the spring from 35 cm to
38 cm?
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This example will require Hooke’s Law to determine the force. Hooke’s Law tells us that the
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 Fall '08
 sc
 Force, Work

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