Question

An earth's satellite near the surface of the earth takes about 90 min per revolution. A satellite orbiting the moon also takes about 90 min per revolution. Then which of the following is true? ($\rho$m is the density of the moon and ρe is the density of the earth).

• A. $\rho_m<\rho_e$
• B. $\rho_m>\rho_e$
• C. $\rho_m=\rho_e$
• D. No conclusion can be made about the densities

Answer 1

Answer: (C)

$\rho_m=\rho_e$

Answer 2

This is because the time period of a satellite in orbit depends on the mass of the celestial body it's orbiting and the radius of its orbit. In this case, both satellites have the same time period of 90 minutes per revolution, indicating that the ratio of the cube of the semi-major axis of their orbits (a 3) to the sum of their masses (M) is the same.

For Earth's satellite: ​ ae 3​​=constant

For Moon's satellite: ​ am 3​​=constant

Since both satellites have the same time period and the same constants of proportionality, we can equate the two equations: Me ​ ae 3​​=Mm ​ am 3​​

Given that the mass of the moon (Mm ​) is much smaller than the mass of the Earth (Me ​), the only way for this equation to hold true is if the densities of the moon (ρm ​) and Earth (ρe ​) are equal, i.e., ρm ​=ρe ​.

The correct option is(C): $\rho_m=\rho_e$