# Work Done by a Varying Force

##

An Important Limitation:

As mentioned in the fine print, all of the methods so far for
calculating work depend on the fact that the force doing the work is
constant. After all, if the force doing the work is not constant,
what value are you going to use in the calculation? Yet, many common
situations involve non-constant forces, such as the force
exerted to stretch a spring.

What to do? Depending on what you know about the force (and your
mathematical sophistication), there are a few things to try that
could give you, at the minimum, a decent approximation of the work
done by the force...

##

Method 1: Average Force

Suppose a force is not constant as it does work, but you know (or
can estimate) the **average value of the force,
F**_{ave}, over the distance, x, in which it does
work. In this case, the work done by the force is:

** **

**Work = F**_{ave}x - if the
direction of F_{ave} is in the direction of motion

or:

**Work = -F**_{ave}x - if the
direction of F_{ave} is opposite the direction of motion

or, for the mathematically inclined:

##

For Example:

Suppose
that it takes a force of 40 Newtons to hold a spring when it is
stretched 1 meter from its normal rest position. How much work was
done in stretching the spring this distance?

### Solution:

The force required to hold the spring varies from 0 Newtons at its
rest position to 40 Newtons at the 1 meter position. You might be
tempted to guess that the average force that you exert as you stretch
the spring would be 20 Newtons (the *average* of 0 Newtons and
40 Newtons). The work done to stretch the spring would be:

Work = F_{ave}x = (20 Newtons)(1 meter) = 20
Joules

This would be a good guess!

last update May 15, 1998 by JL Stanbrough