Two planets of equal density are near each other as shown in the diagram. Which of the following diagrams correctly represents the gravitational forces in action, ignoring any objects other than the two planets?
Okay, so we’re given this diagram showing two planets that are near each other. We’re told that these planets have equal density. And we can see from the diagram that this blue planet is larger than the orange planet. Since density 𝜌 is defined as mass 𝑚 divided by volume 𝑣, then by multiplying both sides by 𝑣 so that the 𝑣’s on the right-hand side cancel out, we can see that mass is equal to volume multiplied by density. Since both planets have the same value of density but the blue planet has a larger value of volume 𝑣 than the orange one does, then we know that the mass of the blue planet must be greater than the mass of the orange planet.
Now in this question, we’re being asked about the gravitational forces that these two planets exert on each other. We can recall that the reason that each planet does exert a gravitational force on the other is because these planets have mass. We should notice by the way that we’re told that we can ignore any objects besides these two planets. So all we need to worry about is these two objects exerting a gravitational force on each other. In order to work out which of these five diagrams that we’re given in the question correctly represents these gravitational forces, we can appeal to Newton’s third law of motion. Let’s clear a little space on the screen so that we can recall what this law tells us. Newton’s third law is often summarized as saying that every action has an equal and opposite reaction.
Now, in this statement of Newton’s third law, the word equal refers to a magnitude, while the word opposite refers to a direction. The law is talking about two objects that exert a force on each other, such as the blue planet and the orange planet from this question. Now we know that the blue planet has a mass and therefore exerts a gravitational force on the orange planet. This force will be directed from the center of mass of the orange planet toward the center of mass of the blue planet. Now, this force constitutes an action, which Newton’s third law tells us has an equal and opposite reaction. What this means is that since the blue planet exerts a force on the orange planet, then the orange planet must also exert a force on the blue planet.
We already knew that this would be the case. Since the orange planet also has a mass, then we know that it must also exert a gravitational force on the blue planet. The key thing that Newton’s third law of motion is telling us is that this force must be equal in magnitude and opposite in direction to the force exerted by the blue planet on the orange one. So the gravitational force exerted by the orange planet on the blue planet must be directed from the center of mass of the blue planet toward the center of mass of the orange one.
Not only this, but these two forces must also be equal in magnitude. Since the magnitude of a vector quantities such as force is represented by the length of the arrow that we draw for it, then that means that these two arrows we’ve drawn here for the gravitational forces exerted by each of the two planets must have the same length as each other. With this in mind, let’s now compare this scenario against the five different diagrams that we’re given as potential answers.
We’ll begin by looking at diagram (A). We can see that in this diagram we do indeed have each planet exerting a force on the other and these two forces are opposite in direction. In terms of magnitude though, we’re told that 𝐴 is equal to 𝐵, where 𝐴 and 𝐵 are the distances that each of these force vectors extends beyond the radius of their respective planet. That means that the magnitude of this force exerted on the blue planet by the orange planet is larger than the magnitude of this force exerted on the orange planet by the blue planet, since the blue planet is larger in radius than the orange one. Since the two forces shown in this diagram don’t have equal magnitudes, then this contradicts Newton’s third law of motion. And so the diagram in option (A) cannot be correct.
Let’s now move on to diagram (B). This diagram shows only one force directed from the orange planet to the blue one. Newton’s third law tells us that this action must have an equal and opposite reaction. But since there is no second arrow directed from the blue planet to the orange one, then this action has no reaction at all. This means that diagram (B) cannot be our answer.
Moving on to diagram (C), we can notice that this does show two arrows with opposite directions, and they’re labeled with lengths 𝐴 and 𝐵. We’re also told that 𝐴 equals 𝐵, which means that these two arrows have equal lengths, and so they represent forces with equal magnitudes. So then diagram (C) shows two forces with equal magnitudes and opposite directions, which is exactly how it should be according to Newton’s third law. It’s looking like diagram (C) may well be our answer then. But to be sure, we should also check out the remaining two diagrams.
Diagram (D) again shows two forces that are oppositely directed, and they’re labeled with magnitudes 𝐴 and 𝐵. We’re told that in this diagram the ratio of these magnitudes 𝐴 over 𝐵 is proportional to the ratio of the radii capital 𝑅 over little 𝑟. That is, the magnitude of the force exerted on the blue planet divided by the magnitude of the force exerted on the orange planet is proportional to the radius of the blue planet divided by the radius of the orange planet. Now we know that this can’t be the case because Newton’s third law of motion tells us that these magnitudes must be equal, not in some ratio that is proportional to the ratio of the planet’s radii. This means that we can eliminate diagram (D).
Lastly in diagram (E), we can see that we’ve again got two oppositely directed forces and we’re told that the ratio of magnitudes 𝐴 over 𝐵 is proportional to the ratio of radii little 𝑟 over capital 𝑅. Just as with diagram (D), we know that this can’t be correct because according to Newton’s third law, these forces must have equal magnitudes. So we can also eliminate diagram (E). This leaves us with our answer that the diagram which correctly represents the gravitational forces in action is the one given in answer option (C), which shows two forces that are equal in magnitude and opposite in direction.