The star has a mass n_1 times that of the planet, which orbits this star. For a person on the planet, the average distance to the center of the star is n_2 times the distance to the center of the planet. In magnitude, what is the ratio of the star's gravitational force on you to the planet's gravitational force on you?
The star has a mass n_1 times that of the planet, which orbits this star. For a person on the planet, the average distance to the center of the star is n_2 times the distance to the center of the planet. In magnitude, what is the ratio of the star's gravitational force on you to the planet's gravitational force on you?
To find the ratio of the star's gravitational force on you to the planet's gravitational force on you, we can use Newton's law of universal gravitation.
Newton's law of universal gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, it can be expressed as:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects,
- G is the gravitational constant,
- m1 and m2 are the masses of the two objects, and
- r is the distance between their centers.
Let's designate the mass of the star as M and the mass of the planet as m. The average distance from your location to the center of the star is n_2 times the distance to the center of the planet. Let's denote the distance to the center of the planet as r.
Therefore, the distance to the center of the star is n_2r.
Now, we can calculate the ratio of the star's gravitational force on you (F_star) to the planet's gravitational force on you (F_planet):
F_star / F_planet = (G * (M * m) / (n_2r)^2) / (G * (m * m) / r^2)
Canceling out the G and m terms, we have:
F_star / F_planet = (M / (n_2^2 * r^2)) / (1 / r^2)
= M / (n_2^2 * r^2) * r^2
= M / (n_2^2 * r^2)
Therefore, the ratio of the star's gravitational force on you to the planet's gravitational force on you is M / (n_2^2 * r^2).