Video Transcript
The graph shows how the
acceleration due to gravity varies with radial distance inside and outside of three
planets. Each planet is a perfect sphere and
has a constant density. Which of the following is true
about all of the planets? (A) All of the planets have the
same surface gravity. (B) All of the planets have the
same radius. (C) All of the planets have the
same volume. (D) All of the planets have the
same mass. (E) All of the planets have the
same density. Which planet has the largest
radius? (A) They all have the same radius,
(B) planet B, (C) planet A, (D) planet C.
On our graph, our planets are
represented by three different colors. Planet A is red, planet B is blue,
and planet C is green. All of the planets overlap on the
diagonal line that runs from the origin to the radial distance of 8000
kilometers. We can see that there are two
distinct shapes on the graph. The diagonal line is the
relationship between the acceleration due to gravity and radial distance inside of
the planets. On the graph, we’ve labeled it to
start and end for planet C. The planets end at three different radial distances:
6000 kilometers for planet A, 7000 kilometers for planet B, and 8000 kilometers for
planet C.
The curved portion of the graph
represents outside of the planet and has once again been labeled on the graph for
planet C. We should know that the shape of the curve for each planet outside is
similar. The shapes of the graph come from
the equations for the acceleration due to gravity inside of a planet and the
acceleration due to gravity outside of a planet. The acceleration inside the planet
is equal to four-thirds 𝐺, the universal gravitational constant, 𝜌, the density of
the planet, 𝜋𝑟, where 𝑟 is the distance from the center of the planet to a point
within the surface. The acceleration outside of a
planet is equal to 𝐺, the universal gravitational constant, times 𝑚, the mass of
the planet, divided by 𝑟 squared, where 𝑟 is the distance from the center of a
planet to a point that’s outside the surface.
In our first question, we’re asked
to determine what is true for all of the planets. If all the planets have the same
surface gravity, then we would expect them to have the same acceleration due to
gravity at the surface. The surface on our graph is the
point at which our planet switches from inside to outside. That would be in the three
different locations represented by the vertical dotted lines. For planet A, the surface is at
6000 kilometers, planet B is at 7000 kilometers, and planet C 8000 kilometers. The acceleration due to gravity at
the surface for planet A is just above nine meters per second squared. At planet B, it’s just below 11
meters per second squared, and planet C is just above 12 meters per second
squared.
Since all three planets have
different accelerations due to gravity at their surface, we can say that answer
choice (A) is not true. If all the planets have the same
radius, then on the graph, they would switch from inside to outside at the same
radial distance. However, as we discussed earlier, A
is at 6000, B is at 7000, and C is at 8000 kilometers. Therefore, answer choice (B) is
also not true.
To determine whether or not they
have the same volume, let’s look at the equation for volume. The volume of a sphere, as we’ve
assumed each planet is a sphere, is equal to four-thirds 𝜋𝑟 cubed. Four-thirds and 𝜋 are both
constants. That means that, depending on the
radius, our volume may be the same or different for each planet. As we just stated, the three
planets have different radii. Therefore, they have different
volumes. So answer choice (C) is also not
true.
To determine if the planets have
the same mass, let’s look at the field outside of the planet. For any radial distance, let’s
choose 12000 kilometers from the center of each planet, we can see that each of them
will have a different acceleration due to gravity. In our formula outside of the
planet, we can see that the acceleration due to gravity is based on the mass, along
with the distance away. Since each planet has a different
acceleration due to gravity at the same radial distance, they must each have
different masses, with planet C being the biggest mass and planet A being the
smallest mass. We can therefore eliminate answer
choice (D), leaving us with answer choice (E) that they all have the same
density. But let’s check to make sure that’s
correct.
The slope of the line inside of our
planet, 𝑎 divided by 𝑟, is equal to four-thirds 𝐺𝜌𝜋, where four-thirds, 𝐺, and
𝜋 are all constants. Therefore, if our planets have the
same densities, then they all have the same slope. Looking at our graph, we can see
that all three planets do indeed have the same slope. Answer choice (E), all of the
planets have the same density, is true.
For the second question, which
planet has the largest radius, let’s go back to our graph. Remember that we said that the
radius is going to be the radial distance at which we go from inside our planet to
outside our planet. And we said that this happened at
6000 kilometers for planet A, 7000 kilometers for planet B, and 8000 kilometers for
planet C. 8000 kilometers is the largest number, which belongs to planet C. The
planet with the largest radius is answer choice (D) planet C.
