Video: Resultant Force

A free-body diagram representing the forces acting on an object is shown. What is the net vertical force acting on the object, taking the upward force as positive? What is the net horizontal force acting on the object, taking the force toward the right as positive? What is the magnitude of the net horizontal force acting on the object?

06:07

Video Transcript

A free-body diagram representing the forces acting on an object is shown. What is the net vertical force acting on the object, taking the upward force as positive? What is the net horizontal force acting on the object, taking the force toward the right as positive? What is the magnitude of the net horizontal force acting on the object?

Okay, so in this question, we see that we’ve been given a diagram. Now, this diagram is a free-body diagram, which essentially represents the forces acting on an object. Where we consider the object to be a point-like object. In other words, we shrink the object down just to a point. And we say that all of the forces in the diagram are acting at that point.

Now, we see that there are some vertical forces, whether they’re upward or downward. And we also see some horizontal forces, rightward or leftward. And interestingly, because all the forces acting in the vertical direction, the up-down direction, are perpendicular to or at right angles to the forces acting in the left-right direction. We can therefore consider the up-down forces separately to the left-right forces. Because any force that is perpendicular to another force can be considered to be working independently of the other force. And the reason for this is that any vertical force will not affect the horizontal motion of the object. And any horizontal force will not affect the vertical motion of the object. This is why we can work out the net vertical force acting on the object and the net horizontal force acting on the object separately.

So let’s start by finding the net vertical force on the object. In order to do this, we simply need to consider all of the forces acting upward and all of the forces acting downward on the object. So we see that there’s a 40-newton force acting upward on the object and a 20-newton force acting downward on the object. These are the two forces we need to consider when finding the net vertical force. Because net force means the overall or resultant force once we combine all of the forces acting in the vertical direction. And the way to combine this, as we’ve already been told in the question, is to firstly take the upward force as positive. So we say that the 40-newton force is a positive 40-newton force. And this must mean that if the upward force is positive, then any forces acting in the downward direction must be in the opposite direction and therefore negative. And hence, this 20-newton force, when we account for it in our calculations, must be written as a negative 20-newton force.

And hence we can say that the net force, which we’ll call 𝐹 subscript net, in the vertical direction, we’ll add a comma 𝑣 to the subscript, is equal to the positive 40-newton force, which is the force that acts upward plus. Because remember, we add forces in order to find the net force. But then, to this force we’re adding a negative 20-newton force, which might seem confusing. But remember, like we said already, in order to find a net force, we add up all of the forces that we’re considering. And we account for each of those forces’ directions by calling forces in one direction positive and forces in the opposite direction negative.

And so when we add a negative force, mathematically, that’s the same as subtracting 20 newtons from 40 newtons. And so the resultant force that we find is 20 newtons. But remember, this force is positive. And therefore, we’ve got a 20-newton force acting upward as the net force on our object. In other words then, if this here is our object. Then having one 40-newton force upward and one 20-newton force downward is equivalent to if the object just had one 20-newton force upward. Which is basically the net force. And so at this point, we found the answer to our first question. The net vertical force acting on the object, taking the upward force as positive, is 20 newtons.

Now, the second question asks us, what is the net horizontal force acting on the object, taking the force toward the right as positive? So here we’re doing the exact same thing. Except we’re finding the net force in the right-left direction, the horizontal direction. And so we need to consider all of the forces acting towards the right, the 15-newton force and the five-newton force. And all of the forces acting toward the left, the 30-newton force and the 15-newton force. Additionally, we’ve been told to take all of the forces acting toward the right as positive. And that means that any force acting in this direction is positive. And any force acting in this direction must be labeled as negative.

With that in mind, we can say that the net force in the horizontal direction is equal to — well, firstly, add all the positive forces. So that’s 15 newtons plus five newtons, which are both forces acting toward the right. And then to this we will add the forces acting toward the left. So the first leftward acting force has a magnitude of 30 newtons. And so we add negative 30 newtons. And we also add negative 15 newtons. And when we combine all of these forces, we will find the net force in the horizontal direction. When we evaluate the right-hand side, we find that it’s equal to 15 newtons plus five newtons minus 30 newtons minus 15 newtons. And that ends up being negative 25 newtons.

Now because it’s a negative force, this means that the net force in the horizontal direction is toward the left. Because, remember, we said any forces acting toward the right must be positive. And so what we can say is that if this is our object and there are four horizontal forces acting on it. The 15-newton force, the five-newton force, the 30-newton force to the left, and the 15-newton force toward the left as well. Then all of those combined is equivalent to just having a 25-newton force acting toward the left, or a negative 25-newton force. And hence, our answer to the second question is negative 25 newtons. At which point, we can move on to the last question.

The last question asks us, what is the magnitude of the net horizontal force acting on the object? Now, we’ve already found the net horizontal force acting on the object. We’ve just done this. It’s negative 25 newtons. However, this question is asking us to find the magnitude of that force. Now, the magnitude of the force is simply given by this part, the size of the force. And the size of the force is 25 newtons. The negative sign only accounts for the direction, which tells us it’s acting toward the left. Since we agreed at the beginning that rightward forces are positive. But if we’re being asked to state just the size or magnitude of the net horizontal force acting on the object, then this magnitude ends up being 25 newtons.

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