Two satellites orbit Earth with the same mass, but satellite A orbits at a distance of 2 Earth radii whereas satellite B orbits at a distance of 3 Earth radii. By what factor is satellite B's gravitational force smaller than A's? (Hint: You are finding the missing number to the statement: Satellite B's force is ___times the size of Satellite A's force.) Express your answer to two significant figures.
the gravitational force between the Earth and a satellite
... is inversely proportional to the distance between the enters of mass
A is 3 Earth radii from the center of the Earth
B is 4 Earth radii from the center of the Earth
B / A = 3^2 / 4^2
To determine the factor by which satellite B's gravitational force is smaller than satellite A's, we can use the concept of inverse square law for gravity. According to the law, the gravitational force between two objects decreases as the square of the distance between their centers increases.
Let's calculate the ratio of the forces:
The gravitational force of satellite A, FA, is directly proportional to the mass of satellite A, mA, and the mass of Earth, ME, and inversely proportional to the square of the distance, rA, between satellite A and Earth.
So, FA ∝ (mA * ME) / rA^2
Similarly, the gravitational force of satellite B, FB, is also directly proportional to the mass of satellite B, mB, and the mass of Earth, ME, and inversely proportional to the square of the distance, rB, between satellite B and Earth.
So, FB ∝ (mB * ME) / rB^2
To find the factor by which satellite B's force is smaller than A's, we need to compare the two forces:
FB / FA = [(mB * ME) / rB^2] / [(mA * ME) / rA^2]
Canceling the common factors (mass of Earth and mass of satellite),
FB / FA = (mB / mA) * (rA / rB)^2
Given that both satellites have the same mass,
FB / FA = (rA / rB)^2
Substituting the given values:
rA = 2 Earth radii
rB = 3 Earth radii
FB / FA = (2 / 3)^2
FB / FA ≈ 0.44
Therefore, satellite B's gravitational force is approximately 0.44 times the size of satellite A's force.