We already defined work in one dimension
is the product of force times displacement
for a constant force, but now let's look at a case
where we're applying a force, f, that is a function of x.
So, our component, f of x, is a function
of x in the i hat direction.
And one of the simplest examples of a force like this
is what we call the spring force.
Now, when we apply this force to our object--
and let's look at an object, i, i hat direction,
we'll have an origin.
And this is our plus-x coordinate system.
When we apply force-- what we want to look at
is because the force is a function of position-- then
we want to look at the displacement
over a small amount.
So let's call this the point xj.
And out here, let's refer to this as xj plus 1.
And what we've done is we're going
to ask how much work is done when the force is displaced
from here to there.
And that's what we'll call delta x.
So now our displacement is a small displacement x of j
plus 1 minus x of j.
And our work for this small displacement-- and that's
why we'll indicate it with a delta-- is
equal to the force, which is a function of x,
times this displacement.
And so we get f of x times x of j plus 1 minus x of j.
And what we have to indicate here
is because the force is varying, we're
looking at just this displacement here.
Let's refer to this force as in the j-th part.
The total work is just the sum of all these scalar quantities.
Remember, although force is a vector and displacement
is a vector, the product of these two quantities
is a scalar.
And so, if we want the total work,
we have to sum from j goes from 1 to n of this quantity
f of j dot-- and I'll put a little j there
to indicate that-- delta xj.
What does this sum mean?
Well imagine that we're making a series of displacements
all the way out to a final position,
x final, and we divided this interval into n pieces.
And so this represents the little bit
of work done for all of these displacements.
And that is what we define to be the work
for a non-constant force.
The issue here is about how fine we cut this interval in.
We made n individual pieces.
But if we want to ask ourselves, what
is the limit as n goes to infinity,
then that's what we now need to consider.
So what we're doing is we're making
smaller and smaller and smaller little displacements.
And we're taking this sum-- j goes from 1 to n
of x of xj times delta xj.
And this limit of a sum is by definition
the integral of the force with respect to dx.
So what we end up with is our work is the integral of f of x.
It's a function and I'm going to have an integration variable, x
prime of dx prime, where x prime is
going from our initial position to our final position.
And this is now our definition of work
which generalizes a constant force to a non-constant force.
Again, let's try to look at some type
of geometric interpretation.
So if we plotted f of x versus x.
And now let's consider a case where
we have some arbitrary force.
So I'm going to just draw the force as if it were arbitrarily
increasing as a function of x.
And here is our x initial.
And here's our x final.
And again, we would like to make a geometric interpretation.
Let's consider xj.
And I'll make this very big.
xj plus 1 for the sake of visualization.
And this value here is f of xj.
And so what we see now is that little bit of work,
like we had for a constant force,
can be approximated as the area underneath the curve for just
this small interval between xj and xj plus 1.
And if we make this sum, what we're doing
is we're just taking a series of approximations
to the area under these curves.
Now you can see, graphically, that there
is very little error when the function was nearly constant.
But in this position, where the function is growing,
this represents our little error.
However, in the limit, as n goes to infinity,
we can make that error go vanishingly small.
So once again, we see, as a geometric interpretation,
that this is the area under the curve of the force
versus position between the points
initial xi and initial-- and the final, x
final, where the particle is being
displaced from an initial position
to some final position.
And here we can call our initial position anywhere we want.
If I write that, this x initial, to the final position.
And that's our generalization of the definition of work.