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, in this question, we can see that we’ve got a diagram showing a box and the forces acting on this box. We can see that on this box, there’s firstly a 16-newton downward force, most likely the box’s weight, that is exactly balanced by a 16 newton upward force. However, apart from this, there’s also a 32-newton force to the right and a force 𝐹 which is unknown to the left. We’ve also been told that the box is being pulled along the surface. And this is that surface. And it’s being pulled along by an applied force of 32 newtons, which is this force here.
The fact that the diagram has not been drawn to scale simply tells us that the length of these arrows here representing all of the forces are not to scale. But anyway, so, just because this box is pulled towards the right by this 32-newton force does not mean that the net force on the box is 32 newtons. Because if the box is moving in this direction, we can look very carefully and see another force, the force 𝐹, acting in the direction opposite to the box’s motion. And in fact, in the question, we’ve been told that the net horizontal force on the box, which is acting to the right, is 24 newtons.
In other words, the overall result of all of the forces combined is a net horizontal force of 24 newtons — we’ll draw this here — acting towards the right. However, because the two 16-newton forces are acting perpendicular to the direction that is the right. In other words, the 16-newtons forces are acting up and down at 90 degree angles to the direction that is the right. They are not going to have any contribution towards the net force that is towards the right. And furthermore, the two 16-newton forces exactly cancel each other out, which means that we can ignore them. And the only two important forces are the 32-newton force acting to the right and the force 𝐹 acting towards the left.
In other words, if we choose to say arbitrarily that the direction towards the right is positive, and therefore anything acting towards the left is negative. Then, we can say that the positive 32-newton force, which is pulling the box to the right. Added to the negative force 𝐹, because 𝐹 is acting towards the left, it’s a negative force, will give us the resultant force which we know to be 24 newtons towards the right, and therefore a positive force. And hence, we’ve set up the equation containing the force towards the right and the force that’s resisting this motion towards the left, as well as the resultant force that occurs due to these two forces interacting.
Which means that when we’ve got this equation, all we need to do is to rearrange to solve for the force 𝐹. We can do this by firstly adding the force 𝐹 to both sides of the equation because when we do this then on the left-hand side, we’ve got a positive 𝐹 and a negative 𝐹. Those add together to give zero. And so, all we’re left with on the left-hand side is a 32-newton force. And then, on the right-hand side, we’re left with 24 newtons plus the force 𝐹 that we’re trying to find.
And then, finally, all that’s left for us to do is to subtract 24 newtons from both sides. Because when we do this, this time on the right-hand side, we’ve got a positive 24 newtons and a negative 24 newtons. Those add to give us zero. And we’re just left with the force 𝐹 on the right. And on the left, we’ve got 32 newtons minus 24 newtons, which, when we carry out the subtraction, ends up being eight newtons. In other words, the force 𝐹 has a magnitude of eight newtons.
But hang on. Didn’t we earlier label any forces acting to the right as positive and any forces acting to the left as negative? And if so, shouldn’t this be a negative value? Well, no, the reason for this is that we accounted for the fact that the force 𝐹 acted towards the left when we set up our initial equation. If you recall, we said that 32 newton, the force acting to the right, minus the force 𝐹 is equal to the resultant force, which is 24 newtons towards the right. And in that situation, this negative sign in front of the force 𝐹 accounted for the fact that 𝐹 was acting towards the left.
In other words, the value of 𝐹 that we found is only the magnitude of the force 𝐹, or the size of the force 𝐹, which is exactly what we need to find based on the question. And hence, we can say that this box, which is being pulled along by a 32-newton force to the right, and has a net force on it of 24 newtons to the right, must therefore also have a force 𝐹 which has a magnitude of eight newtons acting towards the left.