Planets A B and C are all in one line where planets A and C are the same distance apart from planet B. Planet A has a mass of m. a planet B has a mass of m. Planet C has a mass of 4 m. Explain why Planet A experiences a gravitational force by Planet C that is equal to that by Planet B using the equation for gravitational force.
The equation for gravitational force is F = G * (m1 * m2) / r^2, where F is the gravitational force between two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
In this case, the masses of planets A and B are both m. Let's assume that the distance between planets A and B is r1, and the distance between planets B and C is r2.
Since planets A and C are the same distance apart from planet B, r1 = r2.
Now let's calculate the gravitational force between planet A and planet B, and between planet B and planet C:
Gravitational force between A and B (FAB) = G * (m * m) / r1^2 [Equation 1]
Gravitational force between B and C (FBC) = G * (m * 4m) / r2^2 [Equation 2]
Since r1 = r2, we can rewrite Equation 2 as follows:
FBC = G * (m * 4m) / r1^2
= 4 * (G * (m * m) / r1^2)
= 4 * FAB
This means that the gravitational force between planet A and planet B is four times smaller than the gravitational force between planet B and planet C. Therefore, planet C has four times the mass of planet A, but they both exert the same gravitational force on planet B.