Video: Finding the Net Force on an Object from a Free-Body Diagram

A free-body diagram representing the forces acting on an object is shown. What is the direction of the net force that acts on the object? Which of the lines shown correctly represents the magnitude of the net force that acts on the object compared to the magnitude of the force that acts toward the west?

05:12

Video Transcript

A free-body diagram representing the forces acting on an object is shown. What is the direction of the net force that acts on the object?

Okay, so in this question, we’ve been given a free-body diagram here, and we need to find the direction of the net or resultant force acting on the object. Now, to answer this question, we first need to realize that in a free-body diagram, the object that all of the forces are acting on can be thought of as a point, in this case the point at which all of the lines are intersecting. And that’s because we’ve been showing four different forces acting on this object.

So, if this is our object here, then there’s a force acting to the east on it. There’s a force acting to the north. There’s a force acting to the south. And there’s a force acting to the west. And that’s why it looks like there are lines intersecting at the point where the object is. In reality, they’re four separate arrows showing four different forces. Now, if we want to find the net force acting on the object, then the first thing that we can do is to think about the components of the net force.

Specifically, we can find the net force in the north-south direction and the east-west direction, separately. And then, we can combine the net forces in these two directions afterwards, if necessary. So, starting with the north-south component, we can see that the arrow representing the northward force has the same length as the arrow representing the southward force. In other words then, the magnitude or size of the force acting towards the north is the same as the magnitude or size of the force acting towards the south.

And hence, these two forces cancel each other out. They’re acting in opposite directions and they have the same magnitude as each other. And so, the resultant force just in the north-south direction is zero. This means there is no north-south component and our net force overall on the object is only going to have an east-west component. And so, all we need to do is to compare the magnitudes of the westward force and the eastward force.

We can see that the arrow representing the westward force is much longer than the arrow representing the eastward force. And this tells us that the magnitude of the westward force is much larger than the magnitude of the eastward force. The force on the object acting towards the west is much larger than the force acting towards the east. And because, as we saw earlier, there is no north-south component to the resultant force, we can therefore say that the direction in which the net force is acting on the object is towards the west.

Moving on to the next part of the question then, we are now being asked which of the lines shown correctly represents the magnitude of the net force that acts on the object compared to the magnitude of the force that acts towards the west.

Okay, so in the previous part of the question, we already saw that the net force on the object only has an east-west component. Specifically, it acts towards the west. And so, for now we’re going to ignore the forces acting towards the north and the south. The only thing we care about is the force acting towards the west which has this magnitude, and the force acting towards the east which has this magnitude. We can even draw them separately to make things a little bit clearer. So, here’s the westward-acting force and here’s the eastward-acting force, drawn separately to scale.

In other words, the length of the eastward-acting force is the same as the length given in the diagram, and the same is true for the length of the westward-acting force. This is the same length of this one here. Now, what we’ve been asked do is to find the magnitude of the net force compared with the force acting toward the west. And once again to scale, we’ve been given the westward-acting force. We need to find whether line number one, two, three, or four correctly represents the magnitude of the net force compared with the magnitude of the force acting towards the west.

Now, to answer this question, we need to realize that the westward- and eastward-acting forces are acting in opposite directions. Therefore, even though the net force on the object is towards the west because the larger force was the westward force compared with the eastward force. The magnitude of the net force is going to be smaller than the magnitude of the westward force because some of that whole westward force, some of this force with this magnitude here, is counteracted by the eastward force with this magnitude. And it’s because they act in opposite directions.

Therefore automatically, we can rule out option number four because that length is actually longer than the westward force. And hence, this is suggesting that the net force has a larger magnitude than the westward force. In reality, the magnitude of the net force is going to be equal to the magnitude of the westward force minus the magnitude of the eastward force because we can think about it as adding forces together. Firstly, we add the westward force, and then we add the eastward force acting in the opposite direction.

And so, the vector that should represent the final or resultant force is a vector starting here and finishing at the end point once we’ve added both of the required forces. And therefore, the resultant force should have this magnitude here. So essentially, what we’ve done here is to add forces vectorially. In other words, add them together, accounting for their directions. And so, we find our resultant force with a size or magnitude that’s less than the magnitude of the westward force, which makes sense because some of that is negated by the eastward-acting force, the force acting in the opposite direction.

And if we now translate all of this to the diagram given to us below, we can see things a bit more clearly if we add the eastward-acting force drawn to scale to the diagram given to us. Because this tells us that the magnitude of the resultant force is from here to here. And we can clearly see that it’s only line number one that has the correct magnitude. Every other line is far too long. And therefore, at this point, we found the answer to our question. Line number one correctly represents the magnitude of the net force that acts on the object compared to the magnitude of the force that acts towards the west.

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