Video Transcript
Which of the lines on the graph shows how the magnitude of the gravitational force between two objects varies with the distance between their centers of mass?
In this example, we’re considering two objects. Say that they have masses 𝑚 one and 𝑚 two. And we want to understand how the gravitational force of attraction between these masses varies with the distance between their centers of mass, what we’ve called 𝑟. Our graph shows us several possibilities. There’s a blue line, a green line, a black line, and a red line and a purple line. These curves are all plotted on axes that show us force in newtons against distance in meters. This distance on the horizontal axis is the distance 𝑟 that we’ve drawn here in our sketch.
A key to helping us answer this question is to recall that the gravitational force between two objects is an example of an inverse square law. What that means is the gravitational force, we’ll call it 𝐹 sub g, between two masses is proportional to the inverse of the distance between their centers of mass squared. We want to be able to identify which color curve on our graph shows this relationship between distance and force.
To begin doing this, let’s think about the smallest and the largest possible value for this distance 𝑟 we could have. As the distance 𝑟 between the centers of mass of two objects gets smaller and smaller, we realize the smallest it could get would be to approach zero. If we let 𝑟 in this proportionality relationship go to zero, then that means we’ll be dividing one by a number that is very close to zero. And so this fraction will get closer and closer to infinitely large. In other words, as 𝑟 approaches zero, the gravitational force of attraction between two masses becomes infinitely big.
This tells us something important about the lines on our graph. If we go to where the distance, measured in meters, is zero, any curve that shows us a finite value for the force in newtons can’t be correct. For example, both the blue and the green curves pass through the origin. This means that when distance is zero, the force is zero as well. We’ve seen that this is not the case, so we won’t choose the blue or the green curve as our answer.
The black curve, here, and the purple curve, here, have the same problem. Both of these indicate a finite gravitational force of attraction between the masses when the distance between their centers of mass is zero. Therefore, we can also eliminate these curves from consideration.
As we see, we’re left now with just the red curve. To confirm that the red line is the one we really want to choose, instead of thinking specifically about the smallest possible value for 𝑟, let’s think about the largest this distance could be. There’s really no upper limit for this distance; it could become infinitely big. As the number 𝑟 gets larger and larger in our denominator, we can see that the gravitational force of attraction between two masses would get smaller and smaller.
On our graph then, we would expect that as the distance 𝑟 gets very, very large, the force in newtons will become very, very small. It will approach zero. Note that we indeed see the red curve doing this. The red curve approaches zero at large distances. This confirms us in our answer choice. On this graph, it’s the red line that shows how the magnitude of the gravitational force between two objects varies with the distance between their centers of mass.
