Video: Investigating the Relation between the Orbital Speed and the Orbital Radius for Objects in Circular Orbits

Which line on the graph shows the relation between orbital speed and orbital radius for objects moving along circular orbits due to gravity?

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Video Transcript

Which line on the graph shows the relation between orbital speed and orbital radius for objects moving along circular orbits due to gravity?

The question asks us to identify which of the curves on this graph, which has orbital radius on the horizontal axis and orbital speed on the vertical axis, corresponds to the correct functional relationship between those two quantities. Recall that by equating the centripetal force and the gravitational force for an object in a circular orbit. We find that the orbital speed is equal to the square root of the universal gravitation constant times the mass of the body being orbited divided by the radius of the orbit.

Since our question is only about the relationship between orbital speed and orbital radius, we can treat the mass of the body being orbited, that is capital 𝑀, as constant. Then, since capital 𝐺 is also constant, we can rewrite our formula as orbital speed is equal to the square root of a constant divided by the orbital radius, where we’ve used the fact that a constant times a constant is just another constant. Now, using the fact that the square root of a constant is just another constant, we can rewrite this form as orbital speed is equal to a constant divided by the square root of the orbital radius.

We don’t actually care about the identity of this constant. So, we can rewrite this equation as a qualitative proportionality relationship. And that relationship is that 𝑣 is proportional to one over the square root of 𝑟. We’ve written the relationship this way to focus on the connection between orbital speed and orbital radius and avoid getting confused by constants that are not relevant to this particular question. Let’s now use this functional form to make predictions for the orbital speed corresponding to large orbital radii, that is the right edge of the graph, and to small orbital radii, that is the left edge of the graph.

We can then match those predictions to the appropriate line. Let’s see what happens as 𝑟 increases. As the orbital radius gets larger, the square root of the orbital radius also gets larger. So one over the square root of the orbital radius gets smaller because as the denominator of a fraction grows, the size of the fraction shrinks. So when orbital radius increases, we expect orbital speed to decrease. Although because orbital radius can never be infinite, the orbital speed can never be zero.

Looking at the graph, clearly, the green line doesn’t work because it never changes, so it isn’t decreasing with increasing radius. The blue, orange, and red lines all decrease with increasing radius. But it also can’t be the red line because the red line reaches zero. And we know that the orbital speed can never be zero. Let’s now see what happens as the orbital radius shrinks.

As 𝑟 decreases, so does the square root of 𝑟. So, the denominator of our fraction is shrinking. And so, the fraction itself is growing. And since the fraction is proportional to orbital speed, the orbital speed is also growing. We also know that as the denominator of a fraction gets closer and closer to zero, the value of that fraction increases without limit. So here, too, we expect the orbital speed to increase without limit as the orbital radius gets closer to zero. The orange curve has a maximum value as the radius approaches zero. So, the orange curve is not the right curve because the orbital speed doesn’t increase without limit.

This leaves the blue line as the correct answer. Indeed, the blue line shows an orbital speed that gets smaller and smaller as the radius increases and gets larger and larger without limit as the radius decreases. So, the answer is that the blue line shows the correct relationship between orbital speed and orbital radius for objects moving in circular orbits due to gravity.

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