Video Transcript
Two people have a tug-of-war match. Each person tries to pull the other just by leaning backward, as shown in the diagram. Both people have the same mass as each other. Both people are pulled toward each other by the rope. Which of the following statements best describes the result of the people pulling on the rope? (A) Both people accelerate toward each other and meet each other at the midpoint between them. (B) Both people move at a constant speed toward each other and meet at the midpoint between them. (C) Both people move alternately toward and away from each other. (D) Neither person moves. (E) Both people fall over backward.
So we’ve got this diagram that shows two people pulling on a rope in opposite directions in a tug-of-war match. Since both these people are pulling on the rope, then we know that each of them is exerting a force on the rope. This person over here on the left will be exerting a force that acts toward the left. Meanwhile, the force exerted on the rope by this person over here on the right will act toward the right. Now we’re told in the question that each person tries to pull the other just by leaning backward. This means that the force each person exerts on the rope is purely as a result of that person’s weight.
We’re also told that both of these people have the same mass as each other. The weight of an object, which we’ve labeled as 𝑊, is equal to that object’s mass 𝑚 multiplied by the gravitational field strength 𝑔. This quantity 𝑔 has a constant value on Earth, which means that two objects on Earth with the same mass will have the same weight as each other. So then we know that each of these two people has the same weight, which means that they each exert the same magnitude force on the rope. We’ve labeled this magnitude as 𝐹 subscript pull as it’s the magnitude of the force with which each person is pulling on the rope.
Note that the forces exerted by each person are equal in magnitude because their weights are equal. But this doesn’t necessarily mean that the force exerted by each person is equal to that person’s weight. The actual force exerted depends on other factors, such as attraction with the ground, for example.
So this person on the left is exerting a force on the rope that acts to the left with a magnitude of 𝐹 subscript pull, meanwhile this person over here on the right exerts a force on the rope with the same magnitude 𝐹 subscript pull, but this force acts to the right.
The question then tells us that both people are pulled toward each other by the rope. This implies that not only are each of the people exerting a force on the rope, but the rope is also exerting a force on each of the people. This force exerted by the rope on the people acts to pull the people toward each other. This means that the force exerted by the rope on this person on the left must act toward the right. Similarly, the force exerted by the rope on the person on the right must act toward the left.
The reason that we’ve got these two forces exerted by the rope on the people is a consequence of Newton’s third law of motion. This law is often summarized as saying that every action has an equal and opposite reaction. To understand what this means in practice, let’s consider two objects that we’ve labeled as 𝐴 and 𝐵. We’ll suppose that object 𝐴 exerts a force on object 𝐵 that acts to the right and has a magnitude of 𝐹 subscript 𝐴. Now this force is an example of an action, which Newton’s third law tells us must have an equal and opposite reaction. This reaction means that object 𝐵 also exerts a force on object 𝐴. And this force acts in the opposite direction to the force that 𝐴 exerts on 𝐵. So in this case, that’s to the left. Let’s label the magnitude of this force that 𝐵 exerts on 𝐴 as 𝐹 subscript 𝐵.
So we’ve seen that these two forces act in opposite directions to each other, and that’s this opposite part of Newton’s third law. The other part of the law is this word “equal.” What this means is that the magnitude of the force 𝐹 subscript 𝐵 is equal to the magnitude of the force 𝐹 subscript 𝐴. So in summary then, Newton’s third law of motion tells us that if an object 𝐴 exerts a force on object 𝐵, then object 𝐵 also exerts a force on object 𝐴 that’s equal in magnitude but opposite in direction.
We can apply Newton’s third law of motion to these two people having a tug-of-war match. This person on the left exerts a force on the rope that has a magnitude of 𝐹 subscript pull and is directed to the left. Newton’s third law tells us that this action has an equal and opposite reaction, which means that the rope exerts a force on the person that acts to the right and has the same magnitude 𝐹 subscript pull. If we then consider the person on the right, we see that the exact same logic applies. This person exerts a force on the rope again with the magnitude 𝐹 subscript pull but this time acting to the right. So then, Newton’s third law tells us that the rope exerts a force on the person which acts to the left and has the same magnitude again, 𝐹 subscript pull.
We can see then that in this diagram on each side of things, the forces are balanced. On the left-hand side, the rightward force exerted by the rope on the person balances the leftward force exerted by the person on the rope. Likewise, on the right, the leftward force exerted by the rope on the person balances the rightward force exerted by the person on the rope. With all this in mind, now let’s take a look at the answer options available to us.
Let’s begin with option (A), which says both people accelerate toward each other and meet each other at the midpoint between them. Let’s recall that Newton’s second law of motion says that the net force 𝐹 on an object is equal to that object’s mass 𝑚 multiplied by its acceleration 𝑎. Now we’ve just seen that in this case the forces all balance, which means that there is no net force. Then from Newton’s second law, we know that a net force of zero means that there must be zero acceleration. Since answer option (A) is claiming that both people accelerate toward each other, then we know that this answer cannot be correct.
Let’s now consider the statement in answer option (B), which says both people move at a constant speed toward each other and meet at the midpoint between them. Now at a first glance, this perhaps seems more plausible than the statement in option (A) because it’s not saying that the people accelerate but rather that they move at a constant speed toward each other. That means that we can’t automatically discount this statement just on the basis of Newton’s second law.
However, there’s a couple of ways we can see that this statement doesn’t make sense either. We’re told that these two people each try to pull the other just by leaning backward, so initially they stood stationary in place and leaning backward. But if each person is initially stationary and just leaning back, then in order to end up moving at a constant speed, there must be some initial acceleration to get them moving in the first place. However, because of Newton’s third law, we know that there’s no net force acting on either person. And also because of Newton’s second law, we know that no net force means there can be no acceleration.
Now there’s also another problem we might spot with the statement in option (B). Since each person is trying to pull the other simply by leaning backward, then this means that these two people aren’t pulling the rope through their hands, but rather they keep their hands fixed in place and just lean back. This means that if each of these people were somehow to move toward each other, then since their hands aren’t moving on the rope, then as they get closer and closer, the rope between them would have to get more and more slack, which means it wouldn’t be able to keep exerting this force 𝐹 subscript pull on the people.
We can see that this doesn’t make any sense whatsoever because if the rope stops exerting this inward force on each of the people, then there’s nothing here that could conceivably make them move toward each other at all. All of this means that we know these people won’t move toward each other at a constant speed. And so we can discount the statement that’s given in answer option (B).
Now let’s consider the statement in option (C), which says both people move alternately toward and away from each other. Now if these people are moving toward and away from each other in turn, this means that they must be accelerating and decelerating each time they change direction. However, we’ve already seen that there’s no net force acting and that from Newton’s second law this means that there can’t be any kind of acceleration. So the statement in option (C) can’t be correct.
Now let’s move on to the statement in option (D), which says that neither person moves. Now this statement seems to make some sense because we know that initially each person is stationary and simply leaning backward. And we also know that there is no net force acting on either person. No net force means no acceleration, and so an object that’s initially stationary will continue to be stationary. So the statement in option (D), that neither person moves, looks like it could be our answer.
Just to be sure though, we should also check out answer option (E), which says both people fall over backward. Now this outcome is what we might expect to happen if it wasn’t for this inward force provided on each person by the rope. This act of falling over backward would essentially amount to an acceleration as a result of gravity. However, we know from Newton’s third law that there is this inward force provided by the rope, and this balances the outward force exerted by each person’s weight. These balanced forces mean that each person experiences no acceleration. And this includes the act of falling over backward. We know then that the statement in option (E) is not correct.
This leaves us with the statement in answer option (D). If two people of equal mass have a tug-of-war match, and each tries to pull the other just by leaning backward, then the overall result is that neither person moves.
