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
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.