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A free-body diagram representing the forces acting on an object is shown.
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What is the net vertical force acting on the object, taking the upward force as positive?
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What is the net horizontal force acting on the object, taking the force toward the right as positive?
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What is the magnitude of the net horizontal force acting on the object?
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Okay, so in this question, we see that we’ve been given a diagram.
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Now, this diagram is a free-body diagram, which essentially represents the forces acting on an object.
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Where we consider the object to be a point-like object.
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In other words, we shrink the object down just to a point.
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And we say that all of the forces in the diagram are acting at that point.
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Now, we see that there are some vertical forces, whether they’re upward or downward.
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And we also see some horizontal forces, rightward or leftward.
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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.
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We can therefore consider the up-down forces separately to the left-right forces.
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Because any force that is perpendicular to another force can be considered to be working independently of the other force.
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And the reason for this is that any vertical force will not affect the horizontal motion of the object.
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And any horizontal force will not affect the vertical motion of the object.
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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.
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So let’s start by finding the net vertical force on the object.
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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.
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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.
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These are the two forces we need to consider when finding the net vertical force.
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Because net force means the overall or resultant force once we combine all of the forces acting in the vertical direction.
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And the way to combine this, as we’ve already been told in the question, is to firstly take the upward force as positive.
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So we say that the 40-newton force is a positive 40-newton force.
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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.
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And hence, this 20-newton force, when we account for it in our calculations, must be written as a negative 20-newton force.
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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.
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Because remember, we add forces in order to find the net force.
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But then, to this force we’re adding a negative 20-newton force, which might seem confusing.
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But remember, like we said already, in order to find a net force, we add up all of the forces that we’re considering.
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And we account for each of those forces’ directions by calling forces in one direction positive and forces in the opposite direction negative.
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And so when we add a negative force, mathematically, that’s the same as subtracting 20 newtons from 40 newtons.
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And so the resultant force that we find is 20 newtons.
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But remember, this force is positive.
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And therefore, we’ve got a 20-newton force acting upward as the net force on our object.
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In other words then, if this here is our object.
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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.
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Which is basically the net force.
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And so at this point, we found the answer to our first question.
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The net vertical force acting on the object, taking the upward force as positive, is 20 newtons.
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Now, the second question asks us, what is the net horizontal force acting on the object, taking the force toward the right as positive?
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So here we’re doing the exact same thing.
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Except we’re finding the net force in the right-left direction, the horizontal direction.
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And so we need to consider all of the forces acting towards the right, the 15-newton force and the five-newton force.
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And all of the forces acting toward the left, the 30-newton force and the 15-newton force.
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Additionally, we’ve been told to take all of the forces acting toward the right as positive.
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And that means that any force acting in this direction is positive.
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And any force acting in this direction must be labeled as negative.
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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.
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So that’s 15 newtons plus five newtons, which are both forces acting toward the right.
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And then to this we will add the forces acting toward the left.
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So the first leftward acting force has a magnitude of 30 newtons.
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And so we add negative 30 newtons.
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And we also add negative 15 newtons.
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And when we combine all of these forces, we will find the net force in the horizontal direction.
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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.
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And that ends up being negative 25 newtons.
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Now because it’s a negative force, this means that the net force in the horizontal direction is toward the left.
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Because, remember, we said any forces acting toward the right must be positive.
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And so what we can say is that if this is our object and there are four horizontal forces acting on it.
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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.
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Then all of those combined is equivalent to just having a 25-newton force acting toward the left, or a negative 25-newton force.
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And hence, our answer to the second question is negative 25 newtons.
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At which point, we can move on to the last question.
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The last question asks us, what is the magnitude of the net horizontal force acting on the object?
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Now, we’ve already found the net horizontal force acting on the object.
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We’ve just done this.
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It’s negative 25 newtons.
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However, this question is asking us to find the magnitude of that force.
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Now, the magnitude of the force is simply given by this part, the size of the force.
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And the size of the force is 25 newtons.
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The negative sign only accounts for the direction, which tells us it’s acting toward the left.
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Since we agreed at the beginning that rightward forces are positive.
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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.