Question

Two planets, planet A and planet B, have the same surface gravity. However, planet B has twice the radius of planet A. How does the mass of planet B compare to the mass of planet A?

A)The mass of planet B is one-fourth the mass of planet A.
B)The mass of planet B is twice the mass of planet A.
C)The mass of planet B is equal to the mass of planet A.
D)The mass of planet B is four times the mass of planet A.
E)The mass of planet B is one-half the mass of planet A.

Answer 1

To maintain the same surface gravity with twice the radius, planet B's mass must be four times that of planet A, to counteract the reduced gravitational force that would result from the larger radius.

Step-by-step explanation:

When two planets have the same surface gravity, but planet B has twice the radius of planet A, the mass of planet B must be larger to compensate for the larger radius. Since gravitational force Fgravity is inversely proportional to the square of the radius (r), if planet B's radius is doubled (r x 2), the gravitational force would be (1/2)² or 1/4 of what it would be if the radius were unchanged. Therefore, to have the same surface gravity with twice the radius, the mass of planet B must also be increased by a factor that cancels out this decrease. If planet B has twice the radius of planet A, the mass would need to be increased by a factor of four (since gravity is inversely proportional to radius squared) to maintain the same gravitational pull. Therefore, the correct answer is D) The mass of planet B is four times the mass of planet A.

Answer 2

Answer: D) The mass of planet B is four times the mass of planet A.

Step-by-step explanation:

If the surface gravity is the same, this means that the attractive force exerted by both planets upon any body on the surface ,is the same.

This means that FgA and FgB are equal each other, so, applying the Universal Law of Gravitation to both planets, we can write:

FgA = G mMa / ra² = FgB = GmMb/rb²

Equating both sides, and simplyfing common terms, we have:

Ma/ra² = Mb/(2ra)²

Solving for Mb:

Mb = Ma . (4ra)² / ra² = 4 Ma