Question Video: Examining the Acceleration due to Gravity of Three Planets Having Varying Properties Physics

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

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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.