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Work Done by a Variable Force

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  • 0:02 Work Is Moving an Object
  • 3:15 Calculating Work Using…
  • 4:43 Calculating Work Using…
  • 5:47 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Doing work on an object is a simple concept: we apply a certain force over a certain distance. But in real life, that force is rarely constant. Therefore, we need to understand variable forces and be able to calculate them accurately.

Work Is Moving an Object

We do work on objects when we move them. Pick your cat up off the couch, and you've done work on it. Lift your friend's TV into a moving truck, and you've done work on it. When we first learned about work, we were told that the amount of work done on an object is equal to the force on that object over the distance it is displaced. Simplified, we can write this equation as work = force x distance.

This sounds pretty good, right? But there's a problem here. Think about lifting that TV or picking up your cat. Is the amount of lifting you do constant as you move the object? Probably not. Even if you try really hard to apply a steady force on that object, it's still somewhat variable over the time you apply it.

This complicates things a bit, because calculating the amount of work done is not as straightforward. For example, let's say you move a box across the floor, but the amount of force you apply decreases as you go along. For this variable force, we're going to need more than just a simple equation. But before we dive into that, let's look at work on the graph to help us understand what we're actually calculating.

Calculating work done
graph of work done with constant force

It might help to think of work done as an area on a graph, where force is on the y-axis and distance is on the x-axis. When the force is constant, we have a straight line coming out from the y-axis and stopping at the distance the object moved. To find the work done, we simply find the area under the 'curve.' In this case, since the 'curve' is a horizontal line, the area is in the shape of a rectangle.

Therefore, to calculate the area, we simply multiply one side by the other, which is equal to the force times the distance. The result is the work done on an object.

That was pretty easy, right? But remember, that's the work done by a constant force. What would the graph look like if the force changed as the object was moved? There are actually many different ways that the force could change, but we'll start out with the simplest scenario.

Let's say you pushed a box across the floor with a constantly decreasing force. You pushed and pushed until your force reached zero and the box stopped. If you plotted the force over the distance you pushed the box, you would have a straight line with a negative slope.

Work done with a decreasing linear force
graph of work done with decreasing force

Looking at the graph, it's easy to see that the area under the curve is a triangle. Using basic geometry, we can find the amount of work done by simply calculating the area of the triangle. That's pretty simple, so let's add a twist.

This time, you push the box with the same decreasing force, but before it reaches zero, you get a phone call and have to stop. If we plotted this force, we'd get the same straight line with a negative slope, but now it's floating up in the air.

Work done with a decreasing linear force and an interruption
graph with decreasing force and stop

Not to worry. We can still solve this problem with basic geometry. The area under the curve is actually the combination of a triangle and a rectangle. All we need to do is find the area of both shapes and add them together.

This method will allow you to find the work done by any force that changes linearly, but what can we do if the force changes in a more complex way?

Calculating Work Using Rectangular Approximation

If we reduced our box-pushing force by a changing rate as we moved across the floor, we might end up with a curved line on the graph, like this. Calculating the area underneath the curve is going to require more than basic geometry. Or is it?

Work done with a decreasing variable force
graph of work done with decreasing variable force

One method we can use to approximate the area under a curved line is to break it up into rectangles. We draw the rectangles so that the curve passes through the midpoint of the top edge.

Calculating work done on a decreasing variable force
graph of variable decreasing force with rectangles drawn in

Drawing the rectangles this way helps to reduce the error of our approximation. This is because the portion of the rectangle above the curve is balanced out with a piece that's 'missing' under the curve.

The width we choose for our rectangles depends on how accurate we need our approximation to be. A few wide rectangles won't fit the curve very well, so they're not as accurate as using many narrow rectangles. However, using lots of narrow rectangles means we'll have a lot of calculations to do when it comes time to find the area. Either way, the decision requires us to make a trade-off that will be based on the needs of the particular problem we're trying to solve.

Once we've decided on the size of the rectangles, we can find the area of each one with basic geometry. By adding up the areas of all the rectangles, we get an approximation of the total area, and therefore, the work.

But what if we need an exact solution and not just an approximation?

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