Back To CourseAP Physics 1: Exam Prep
12 chapters | 136 lessons
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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.
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.
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.
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.
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?
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?
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.
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?
If we take rectangular approximation to the extreme, we end up with a calculus concept, called integration. Imagine drawing an infinite number of rectangles under the curve by making them more and more narrow. Eventually, we wouldn't see any difference between the shape made by the rectangles and the shape made by the curve. The problem is that adding up all the areas of an infinite number of rectangles would take an infinite amount of time. That's where integration comes in.
Integration gives us a way to find an exact solution if we know the function that describes the force. Sometimes the problem we're trying to solve will tell us what the function is, and sometimes we'll have to figure it out. Either way, the actual process of integrating a function requires some knowledge of calculus that goes outside the scope of this lesson. However, if you have access to a computer, there are many tools available that will do the integration for you. With the help of technology, integration can still be a useful tool for finding the work done by a variable force.
Any time we do work on an object, we are applying a force on the object over a given distance. When that force is constant, we can find the amount of work done by simply multiplying the force times the distance the object is moved.
However, in reality, most forces are not constant, which requires us to use other methods. If we think of work done as the area under a curve, we can see why it's often not possible to simply multiply the force times the distance to find our answer.
If the force is linear, then you can use basic geometry to find the area under the curve by dividing it into simple, geometric shapes. If your varying force is non-linear, and a close approximation of the work will suffice, then we can divide the area under the curve into small rectangles. While this doesn't give us the exact amount of work done, if we make our rectangles small enough, we can get it pretty darn close.
However, if an exact answer is needed for a non-linear force, then we need to use calculus. But before we can integrate, the force must be defined by a function. We may also need to use technology, such as a computer, to help us along. Whichever method you choose, it's important to understand these concepts because, more often than not, work is done by a variable force.
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Back To CourseAP Physics 1: Exam Prep
12 chapters | 136 lessons