Video: Free-Body Diagrams

In this lesson, we will learn how to analyze free-body diagrams and use them to determine the net force and unknown forces acting on objects.

11:47

Video Transcript

In this video, we’re talking about free body diagrams. As we’ll see, these are diagrams that show us the forces that act on a particular isolated object. Free body diagrams help us understand how an object will or will not move under the influence of forces. And they’re especially useful in scenarios with multiple interconnected masses. One of the best ways to start learning about free-body diagrams is to see them in action. Let’s say we have this classic physics scenario of a block sliding down an incline plane. To understand the motion of this block, it will be helpful to learn about its acceleration.

Now, in and of itself, we may not know anything about how this object accelerates. But if we refer to Newton’s second law of motion, this law tells us that the acceleration an object undergoes is related to the forces applied to it. So if we can learn about the forces on this block, we’ll know something about its acceleration. And this is where free-body diagrams get involved. We’ve already been specific about which object we want to understand the motion of. It’s this block. That is, it’s not the incline the block is sliding on or the block in inclined system, but just the block itself. So the first step in creating our free-body diagram of the forces acting on this block will be to isolate it, to draw it all by itself.

So here’s our block drawn by itself. Now, what we want to do is start counting up the forces that are acting on this block. Starting off that list, we know for one thing that gravity is a force on the block. And then partly counteracting that force is the normal force, sometimes also called the contact force, between the incline in the block. And then, let’s assume our incline plane is not smooth, but instead is rough. That is, it applies a frictional force to the block. At this point, we’ve named all the forces that are active on the block, gravity, the normal force, and the frictional force. Our next step is to draw these forces in on our diagram of our block using arrows.

Now, the reason we use arrows that will have a certain length in a certain direction is because these three forces we’ve named, just like all forces, are vectors. That is, they point a certain way. And they have a particular magnitude or value in that direction. Now, when we draw a free-body diagram, one way to do it, is to pick a point at the centre of the object of interest and draw in all the forces as though they come from that point. Taking this approach, the force of gravity would be drawn in something like this, an arrow starting at that point and going downward. The normal force would be normal or perpendicular to the angle of the incline. That is, it will point in this direction. And the frictional force, like always points in the direction opposing the object’s motion. So that would be up the incline like this.

Notice that, just like we expected, not only do these arrows point in different directions representing the direction of the force, but they also have different lengths representing different force magnitudes. We can see, for example, that the arrow representing the force of gravity is longer than the arrow representing the frictional force. This means that, relative to the frictional force, gravity is stronger. It has a greater magnitude. Now, before we go further, we should label these arrows. Otherwise, we might get the forces they represent confused. The arrow pointing downward is the gravitational force; we’ll call that 𝐹 sub 𝑔. The arrow pointing up into the left is the normal force; we’ll call it 𝐹 sub 𝑁. And then up into the right is the frictional force; we’ll call that 𝐹 sub 𝑓.

As we look at these three forces, we can see that they’re not all at right angles or perpendicular to one another. In particular, the angle between the force of gravity and the normal force and the force of gravity and the frictional force is not 90 degrees. Now, this is a bit of an issue because, remember, we want to be able to figure out what is the net force acting on this object and use that to figure out how it may be accelerating. But it will be difficult to solve for the net force on our block if the individual forces acting on it are hard to combine. But there’s a step we can take to make this easier. What we can do is define a coordinate axes, 𝑥𝑦-directions, on our free-body diagram.

For example, we could say that the positive 𝑦-direction of these axes that we’re defining points the same way as the normal force. And then we could say the positive 𝑥-direction points in the direction of the frictional force. And even though it doesn’t completely look like it, these axes are at right angles to one another. With those drawn in, we can see that the normal force and the frictional force are entirely either along the 𝑦- or the 𝑥-axis of our graph. But the gravitational force is an exception. It’s split between these two. Part of it is in the 𝑥-direction, and part of it is in the 𝑦. But what we would like to do is to break up or divide this gravitational force into the components in the 𝑥-direction and the 𝑦-direction.

Here’s how we can do that. We can say that this force vector, the gravitational force arrow, is the hypotenuse of a right triangle where the two other sides of this triangle are the 𝑥 and 𝑦 components of this force. So if the overall gravitational force is represented by this arrow as we’ve drawn it, then the 𝑦 component of that force is represented by this one and the 𝑥 component by this one. By dividing the gravitational force into the directions that we’ve called 𝑦 and 𝑥, respectively, we’ve made it easier to compare this force with the other ones acting on our free body. By doing that, we make it much simpler to solve for the net force acting on this block, which then tells us a bit about its acceleration.

Now, what we’ve drawn here is a very specific free-body diagram for this particular block sliding down this particular plane. But the steps we followed in drawing up this diagram are ones that can apply to any object. Off to the side then, let’s write in just what were those steps that we followed. The first thing we did was we isolated our object of interest, in our case, the block sliding down the plane. Part of doing this involved in drawing a simple sketch of that block. Now, this block was already pretty simple, but in some cases we may see a free-body diagram that’s even more simplified. It’s possible to represent an object, a mass, by a dot, for example. We didn’t do it exactly that way. But there is some variation in the free-body diagram generation process.

Once we had our simplified sketch, though, we moved on to step two. And that was to list the forces that we’re acting on our object. It’s important to specify that they’re on the object and not forces the object exerts on something else and recall that, in our case, those forces were gravity, the normal force or contact force, and the frictional force. Once all the forces were accounted for, our next step was to draw them in as vectors, that is, arrows on our diagram. Those arrows pointed at certain direction, depending on the direction of the force. And their length indicated the relative strength of that force. Next, we saw that it was important to put labels on the heads of these arrows. So we could tell the forces apart.

Since all the arrows represent forces, the only way these labels deferred was in their subscript, 𝐹 sub 𝑔 compared to 𝐹 sub 𝑁 compared to 𝐹 sub 𝑓. With these steps complete, we moved on to our final one. And that is to draw in coordinate axes by our free-body diagram. That way, we’ll know which direction is positive 𝑦 and which is positive 𝑥. With this four-step process, we understand how to draw a free-body diagram. And as we saw in the case of our block, this is all motivated by a desire to understand the motion of the object of interest. When we understand the forces on that object, and in particular the net force, we can better understand how it may or may not be in motion. Let’s get a bit of practice now with these ideas through an example.

A box is pulled along a surface by an applied force of 32 newtons as shown in the diagram, which is not to scale. The net horizontal force on the box, acting to the right is 24 newtons. What is the magnitude of the force 𝐹?

Okay, so looking at our diagram, we see our box right here on top of this surface in blue. Our statement tells us that the box is pulled along the surface by an applied force of 32 newtons we see there. And it’s the force acting in the other direction 𝐹 whose magnitude we want to solve for. Now, even though we’re not told it explicitly, what we have in this diagram is a free-body diagram of our box. That is, we have a depiction of all the forces acting on this object drawn in as vectors. So we see, for example, that there’s this 16-newton force acting down. We can guess that that’s the weight force or the gravitational force on the box. And then there’s this equal and opposite 60-newton force acting up. That’s likely the normal or contact force. And then, as we saw before, the 32-newton force is an applied force on the box. And the force 𝐹, of course, is our unknown that we want to solve for.

In order to solve for 𝐹, we’ll need to account for all the forces we see in this free-body diagram. But there’s something very helpful about these forces that makes our task simpler. And that is that two of these forces, the 16-newton force acting up and the one acting down, are perpendicular at right angles to the force 𝐹. That means they have no influence on forces in that direction, in the horizontal plane left or right. Even if they did, by the way, the fact that these two forces are equal in magnitude but opposite in direction means that they effectively cancel one another out. 16 newtons of force pushing up and 16 newtons of force pushing down add up to a net force of zero. So the only forces we need to keep in mind are the 32-newton force and the one we want to solve for, 𝐹.

We can start solving for 𝐹 by using some information given in the statement. We’re told that the net horizontal force on the box acts to the right and has a magnitude of 24 newtons. That means that if we combine the force 𝐹 acting to the left and the 32-newton force acting to the right, then together they yield 24 newtons acting to the right. Now, before we go further, let’s decide on a horizontal direction that we’ll call positive. Let’s say that motion to the right is in the positive direction. And that means that motion to the left is in the negative. Using this convention, if we add together all the forces in the horizontal direction, here’s what we find.

First, there’s our 32-newton force, and that’s a positive value because it’s acting to the right. Then we subtract from that force the force 𝐹, the one we want to solve for. The reason for this minus sign is 𝐹 points to the left, which we’ve decided is in the negative direction. We’re told that when we add these two forces together, the sum or the net is 24 newtons acting to the right and, therefore, a positive number. This is the equation we’ll use to solve for our unknown force 𝐹.

To do that, let’s begin by adding 𝐹 to both sides of the equation. When we take that step, we can see we have a plus 𝐹 and a minus 𝐹 on the left-hand side. So they cancel one another out. And then following that, let’s subtract 24 newtons from both sides, which, we can see, results in 24 newtons minus 24 newtons adding up to zero on the right-hand side. And that leaves us with this equation. 32 newtons minus 24 newtons is equal to the force 𝐹. And that, we can tell, is eight newtons. That’s our final answer. Eight newtons is the magnitude of the force 𝐹.

Let’s now summarise what we’ve learned about free-body diagrams. Starting off, we saw that a free-body diagram, sometimes abbreviated FBD, shows all the forces that act on a particular object. Free-body diagrams we saw are useful for understanding the net force on an object and, therefore, its motion. And finally, we learned a four-step process for drawing free-body diagrams. Step one is to isolate the object of interest. Step two is to list all the forces acting on that object. These may include forces like gravity, tension, normal or contact force, and so on. Step three is to draw those forces in on our isolated object. We use arrows where the arrow direction indicates force direction and the length of the arrow indicates the relative force strength. Once the arrows are drawn in, we label them with the name of the force they represent. And lastly, we pick and then draw in a pair of coordinate axes. So we know which way is positive 𝑥 and which way is positive 𝑦. This is the process for drawing a free-body diagram.

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