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## Work and energy

# Work as area under curve

## Video transcript

- [Instructor] There's a
hamburger sitting right in front of you, but you don't want it. Maybe it's because you're a vegetarian, maybe you're already full,
maybe you found someone that needs it more than you do,
so you're gonna push this hamburger to the right,
with a force of four newtons and you're gonna do this
and you're gonna push this a distance of five meters to the right. So a question we get asked would be how much work did we do in passing this hamburger to the person to our right? Now since this is a pretty
simple problem we can just plug straight into the work formula,
but I'm not gonna do that. I'm gonna show you an alternate
way to think about this, because what we learn from this alternate approach is gonna help us in more challenging, complicated work examples. So I ask you to imagine this. Instead of just plugging straight in, consider the fact that
we exerted a force of four newtons for the entire five meters that we pushed this
hamburger to the right, so if we were to plot what the force was on our hamburger as a
function of its position, it would look something like this. So if it started at zero it
moved five meters to the right and we exerted a constant
force of four newtons, That's why this is a horizontal line. It's horizontal, that means
it was a constant amount of force, and the reason
I'm doing this is because there's gonna be a geometrical
significance to this work. Check it out. We can just use the work formula. We know work is the force
in the direction of motion times the displacement or you
could say the entire force times displacement times cosine theta. If you don't like this cosine
theta, this just makes sure that you're singling out
the force in the direction of motion and only that
component of force that's directed in the direction of
motion but for our hamburger example, the force already
was in the direction of motion so we don't really need this. In other words the angle
here would be zero. Whenever you take cosine
of zero, you just get one and one times anything is
just that thing so whenever your force is already
in the direction of the motion of the object you
don't really need that cosine theta and we don't need it here. So I'm gonna say that the
force here was four newtons so we can calculate four
newtons was the force for the entire displacement of five meters and we get the work that
we did on the hamburger was positive 20 joules of work. Now if you're clever you might be like wait, four newtons times
five meters, look it, that's just the area of this rectangle. So notice that this line
is a straight line in this force graph just forms a rectangle in here and all we did, we took
four newtons, that was just the height of this rectangle
and we multiplied by five meters and that was just
the width of this rectangle and if you multiply a height times width, you know what you get, you
get the area of a rectangle, so what we found was when
the force is constant one way to find the work done
is by using the work formula but another way to find the
work done is just to find the area enclosed by the graph. So from this line that
determines the force down to this x-axis, if you find the area, that's gonna equal the work done. Now you might be like, alright, well, fat lot of good that does us, we can already find it with this formula. Why would I ever want
to know that the work is equal to the area under a force graph? I'll show you why. Because in this case, yes, it was easy, we had a formula, we could
just plug the force in here, we could just plug the
displacement in there, we get our value, but think about this. What if our force was not constant? What if we had a varying force? In that case, what force
would we plug in here? It'd be changing. One way to handle that scenario is using calculus, but if you don't know calculus that doesn't do you any good. Fortunately, there's
another way to determine the work done by a varying force and that's to take this idea seriously. That's why I showed you this. It turns out the area
underneath any force versus position graph is gonna equal the work, not just ones where the force is constant, even where the force is varying, if you can find the area
underneath that graph, that's a quick and easy way to get the work done by that force. So what I mean is this. Instead of just pushing
with a constant four newtons for the entire duration of this trip, let's say you started
pushing with four newtons but you were getting tired
and your force was diminishing and you were pushing with
a weaker and weaker force until your force became zero newtons. This would be zero newtons of
force when it hit this axis. Let's say the hamburger
still made it five meters. You might be confused. How could it make it five meters if we're pushing with less force? Well it may have taken
more time to get there, but let's say it still
made it five meters. How do we find the work
done by our force now? Well I'm still gonna make the claim that the work done is gonna be
equal to the area underneath this force versus position graph, but you might be skeptical, you
might be like wait a minute, all we really showed in that previous example was that the work
done for a constant force was equal to the area underneath. How do I know for this
varying force that the area underneath this graph is still
gonna give me the work done? Well physicists and
mathmeticians are clever. What they did is they said this, alright, they were like,
the only thing we know is that for a constant
force the area underneath is equal to the work,
so let's just say this, instead of just considering
this case where I diminish my force continuously, let's
say I pushed with a whole bunch of constant forces but
for a small displacement, so I started with four newtons
but I pushed for like only 10 centimeters with four
newtons, and then I dropped that down to maybe 3.9 newtons
and I pushed for another 10 centimeters, and I keep
doing this over and over, reducing my force but keeping it constant, and then dropping it again
and here's why that's useful. Because the work done
for this rectangular case is just gonna equal the area
of all of these rectangles added up, right, because
they're each constant forces so I know the work done
is just equal to the area underneath, so if I add
up the area of all these rectangles I get the work
done during that rectangular process and here's the really clever idea. If we made these rectangles
infinitesimally small they would add up to the
total area just underneath this triangle, it'd be
the exact same area. Now the area under this
triangle would be exactly the same as the area under
all of these rectangles and it'd be the exact same
process we had before, because think about it, you'd
be pushing with four newtons for like a millimeter but if
it's infinitesimal I mean it's even smaller than that,
but for the sake of just conceptually thinking about it, let's say you push with four
newtons for a millimeter and then for the next
millimeter you pushed with 3.99 newtons and then for the next millimeter you pushed with 3.98 newtons. That's basically just you
diminishing your force continuously and going through this exact process that we described earlier. But now we know the area
under all these rectangles is gonna just equal the work done, but that's the same area and process that's just the area
underneath this triangle. Now, you might not buy that. You might be like, wait a minute, look it, these edges are throwing everything off, look at this little edge, this shoots out a little too far, that's a
little overestimating the area, and then you gotta little hole in here, that's underestimating the area, you got these all over the place. Is that really gonna be equal to the area? And it will be if you let
these widths of the rectangles become infinitesimally small,
the error you're incurring from all these little
overshoots and undershoots are also gonna get infinitesimally small, and if the error is getting
infinitesimally small that means that the area under all these infinitesimal rectangles
is gonna exactly equal the area under this triangle
which is great news for us, because that means that the area under any force versus position graph, it didn't have to be a triangle, we could use infinitesimally
small rectangles to represent the area under any graph. That means the area under
any force versus position graph is gonna equal the
work done by that force and now we have a powerful
tool to determine the work done for cases where the
force is not constant. This formula here where
you have work equals Fd cosine theta, this only gives you true answers if you use a constant force. If the force is constant
you can use this equation, but if the force is varying
you couldn't just use this, but now we derived this. This is great. This showed me that if I
can find the area under my force graph, my force
versus position graph, I can find the work done. So, in other words, for
this case right here, if I push with four
newtons on my hamburger and then I reduce that force to zero and it went five meters,
how much work was done? Well now I know how to find it. I just need to get the area
under this force versus position graph and it's a triangle and I know how to find
the area of a triangle. So the area under this triangle is 1/2 base times height,
well so I'll have 1/2, the base here is gonna be five
because this is five meters, so I have five meters and
the height of this triangle is four newtons and so I've
got four newtons times five meters times a 1/2 is gonna
give me positive ten joules of work done during this process
where my force diminished continuously, and that's why
this idea is so powerful, because if your force versus
position graph happens to take a shape that you
know how to find the area of, like a rectangle or a
triangle or some combination of rectangles and triangles,
then you can quickly get the work done without
having to know anything about calculus because you
just get the area underneath. Let me just clear up any confusion. You don't actually have
to draw any rectangles. That was just to prove this relationship. Once you know this relationship
you don't need to draw or think about those little
infinitesimal rectangles. That was just how we
were proving to ourselves that even for cases where
the force was varying the area underneath, it's
still gonna equal the work. So there's one more thing
you gotta be careful about. When we say the area underneath the graph is equal to the work
done, we mean the area from the line that represents
the force to that x-axis, and you might be like, well of course that's what you mean,
what else could you mean? Well sometimes it gets a
little unclear when you have a case like this. So let's say your force started down here. So we'll say we first started
pushing on our hamburger to the left and then we're like oh there's no one there, awkward, we'll start pushing to the right, so our force starts off negative and then it becomes positive. Now if we wanted to find
the total work done during this entire process, how would we do it? We can still use this
idea that the work done is gonna equal the area
underneath the curve, but for this first portion what do I do? How do I find the area
underneath this portion? Some people are like
underneath, alright, I'm gonna go underneath, we'll keep going this way, how much area's down here,
but that's just crazy. That goes on infinitely. I did not do infinite
work on this cheeseburger. No. The area I'm talking about is
gonna be this area right here, so I gotta take all this area right here. That area is finite. It's from this line that represents the force to the x-axis. That's the area we're talking about, just like over here the
area we'd be talking about is from this line that represents
the force to the x-axis. It's all of this area. So how would I find these areas? They're both triangles. I can do this, so this one's
gonna be 1/2 base times height, so it's gonna be 1/2 the
base, this base right here is one meter, so it'd be one
meter times the height. I've got negative height. Look it, it's down here. It's negative two newtons. That means I'm gonna have a negative area. Is that okay? Yep, that just means I did negative work on this cheeseburger during that time so I'm negative one joule. During the first meter
of me exerting force on that hamburger, I did
negative one joule of work, and then for this portion of the trip I'll find the area underneath. It's also a triangle, so
1/2 base times height. I'll have 1/2, the base
this time is not three, the base of this triangle is
only two, because it goes from one to three, so that's two
meters times the height, well that is four
newtons, so I'll multiply by four newtons and then I get for that portion of the trip, I
did four joules of work. So, positive four joules
of work for this portion, negative one joule of
work for that portion. The total work done for
the entire three meters would be four joules minus one joule and that would be positive three joules. So, recapping, if your
force is constant you can just use this formula
to find the work done. It's just Fd cosine theta,
but you can also find the work done by determining
the area underneath a force versus position
graph and this is useful because this works even
when the force is varying which would render this
equation somewhat unusable but if the shape of the
graph is one that you know how to find the area of,
you can still find the work done by determining the area underneath the force versus position graph.