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 the gravitational force Planet A experiences by Planet B using the equation for gravitational force.
The equation for gravitational force is given by Newton's law of universal gravitation:
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 them.
In this scenario, Planet A is located between Planets B and C. In order for Planet A to experience a gravitational force that is equal in magnitude from both B and C, the distances between A and B (r1) and A and C (r2) should be the same.
Given that Planet A is equidistant from B and C, we can set r1 = r2.
Now, let's compare the gravitational forces between A and B (F_AB) and between A and C (F_AC):
F_AB = G * (m * m) / r1^2
F_AC = G * (m * 4m) / r2^2
Since r1 = r2, we can equate these two forces:
G * (m * m) / r1^2 = G * (m * 4m) / r2^2
Since the masses (m) and gravitational constant (G) cancel out, we are left with:
1 / r1^2 = 4 / r2^2
Cross multiplying, we have:
r2^2 / r1^2 = 4
Taking the square root of both sides, we get:
r2 / r1 = 2
This equation tells us that the ratio of the distance between A and C (r2) to the distance between A and B (r1) is 2:1. So, if the distance between A and B is x, then the distance between A and C is 2x.
Since gravitational force depends on the distance squared, the gravitational force between A and B will be spread over a smaller area (r1^2), while the gravitational force between A and C will be spread over a larger area (r2^2).
Therefore, to ensure that the gravitational forces from B and C on A are equal, the mass of C must be four times greater than the mass of A.