So in this video, I'm going to show you how to calculate the work that's done on an object by using a force versus displacement graph. So some problems are just going to give you a diagram or graph like this that plots the force in the y-axis with the position in the x-axis. And they'll ask you to calculate how much work is done on an object by some force. We've actually seen something very similar to this in a previous chapter. So let's just get to it. The whole idea here is that the work that is done by any force, whether it's constant or variable, is really just the area that is under that f x graph. What that means, that phrase under the graph, is it just means the area that's between the graph and the x-axis. So really instead of using f d cosine theta a bunch of times to calculate work, all we do is we just figure out what's the area that is underneath the graph between where you are on the function and the x-axis. So there's an area right here. And then once it goes below the x-axis, there's also another area that's here.

So the whole idea here is that, again, instead of using f d cosine thetas, we're really just going to be breaking up this graph into a bunch of rectangles and triangles and using some pretty basic geometry equations to solve it. Let's check this out. So to calculate the work that's done by the force, we're just going to take the area that's underneath the curve. And what we can do here to make this a little bit easier for us is break this up into a bunch of simpler shapes. So if I break this up like this, I've got a rectangle like here, and I'm going to call this area one. So then I've got this triangle like this that's area 2. And then I've got another triangle that's going to be down here. I'm going to call this area 3. So the whole idea is that to calculate the whole work that's done on this object, I'm just going to be adding up all of those areas. Area 1, area 2, and area 3. So the area 1 is the rectangle. Area 2 is the triangle. And then area 3 is also a smaller triangle that's below the x-axis. Alright?

So let's get to this. Area 1. So to use area to figure out the area of a rectangle, we're just going to use base times height. So this is base times height. So I'm just going to look at this rectangle here and figure out the base and height by using the numbers on the axis. So I've got a base of 4 and then I've got a height of 30. So this is going to be 4 times 30, and I've got area 1 is equal to 120 joules. Let's move on to the second one. For area 2, we've got a triangle. So we're going to have to use the area for a triangle which is 12bh. So I've got 12bh. And if I go ahead and take a look here, the base of this triangle goes from 4 to 16, so that's 12. And then the height of this triangle is the same 30 that I was using before. So it's just going to be 12×12×30. And I get area 2 is equal to 180 joules. Alright? Now finally for the last area, area 3, that's the one that's sort of like underneath the x-axis over here. I'm going to use an area for a triangle, 12bh. This is going to be 12. Now I've got the base of this triangle is 4, and the height of this triangle now is not going to be 10 because we're actually going down into the negative axis here. So our height that we plug in is actually negative 10. So this is going to be 12×4×(-10), and what you end up getting is you end up getting negative 20 joules. So we can see here that when you have areas above the x-axis, those are going to be positive works because your force is positive. When you have areas below the x-axis, those forces are sorry, are going to be negative, and so you're going to have negative work. So, basically, we've got our 3 areas. Now we're just going to add them all up together and just, calculate. Right? So this is work done. This is going to be 120 plus 180 plus negative 20. So when you add all altogether, you're going to get the work that's done is equal to 280 joules, and that's your answer. So this is we can see here that this graph stuff is actually super straightforward. If you ever have a variable force, instead of using f d cosine thetas and trying to calculate like, you know, f averages or something like that, just try and see if you can create this sort of like f versus x graph. And so therefore, you can just calculate Anyways, that's it for this one, guys. Let me know if you have any questions.