# Work, Force and Distance

by Jared Rovny

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00:01 Now, we've introduced not only force but how work as a physical concept can help us connect force and energy, which we introduced as well.

00:10 Now we're going to talk a little bit about force and distance and work, and how to find work from force and distance when you might have a slightly more complicated situation.

00:19 So, for example, we mentioned already that if you have a force that's acting over a distance, you get work, but what if the force is changing at each location of distance.

00:30 What if you have one force for some distance and then a different force for another distance or maybe even worse, as you're moving your force is changing as might be the case in a spring, for example.

00:40 As you stretch the spring, the amount that it pulls back on you changes depending on how stretched the spring is.

00:46 So, we need to think of a way to find the amount of work done for some force, if that force is changing over a distance, d.

00:54 In general, if you wanted to do calculus, you could always take an integral to find such a thing for more complicated expression.

01:02 But without resorting to calculus, we can make some general statements about how to find the work from a force and a distance.

01:08 If I plot, as I have here, a distance versus force graph, where my distance, where I am, is represented on the x-axis and the force, how much force I'm applying, is represented on the y-axis as I move my object.

01:22 We already know that if the force is constant over the entire distance as I have here in this graph, a nice horizontal blue line because the force is not changing as I move a distance, d, then we can find the work just by multiplying the force by the distance.

01:37 But again, we can think about the geometry of this.

01:40 Just look at this box that I have here.

01:42 The area under this blue line that I've drawn is d, the width of this rectangle times the height, which is the force.

01:50 And this means that the area under that line is equal to the work done.

01:57 Fortunately, even if I have a changing force that's changing throughout my graph, sometimes I have one higher force then the force gets smaller and then I have a lower force across some long distance, d, it is still true that the area under my force graph is equal to the amount of work done by my force over the entire distance.

02:17 And so, as the force is changing, one way to find the amount of work done is again to break up the area under your force graph into simple geometric shapes that you can find the areas of and then use whatever you're given in a given problem of whatever forces over whatever distances and all the different dynamics together to find each area here, the area of each of the three rectangles as well as the area of the triangle.

02:42 Once you've found that area, you'll have the work done.

02:46 And so, if we represented these areas as A1 and A2, etc., the work would just be the sum of all the areas of the different shapes.

02:54 Remember that this is exactly like what we talked about with the velocity versus time graphs.

02:59 If you have a velocity but that velocity is changing as time goes on and you wanna find the change in position, the displacement caused by that velocity, you do the same trick in some of the earlier lectures.

03:10 So if you practice and got comfortable with that method, this is just the exact same thing repeated one more time.

### About the Lecture

The lecture Work, Force and Distance by Jared Rovny is from the course Work.

### Included Quiz Questions

1. 25 J, because the area of a triangle is 1/2 (base)(height)
2. 50 J, because the area of a triangle is 1/2 (base)(height)
3. 25 J, because the area of a square is 1/2 (base)(height)
4. 50 J, because the area of a square is 1/2 (base)(height)
5. 50 J, because the area of a triangle is (base)(height)
1. The work done is represented by the area of a rectangle.
2. The work done is represented by the area of a triangle.
3. The work done is represented by the area of two squares.
4. The work done is represented by the line from start to finish.
5. The work done is represented by the area of an upward curve.
1. The work is the area of the quarter of a circle.
2. The work is the circumference of the quarter of a circle.
3. The work is the area of half of a circle.
4. The work is the volume of a quarter of a circle.
5. The work is the volume of an eighth of a sphere.

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