Planets A and B orbit the same star. Planet B is four times further from the star than planet A and has twice the mass of planet A. The gravitational force of the star on planet B is ____ the force on planet A.
The force is proportional to M/R^2
F(B)/F(A) = M(B)/M(A) * [R(A)/R(B)]^2
= 2*(1/4)^2 = 1/8
To determine the gravitational force of the star on planet B in relation to the force on planet A, we need to consider 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 them.
The formula for gravitational force is:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force between the two objects,
G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this scenario, planet B has twice the mass of planet A and is four times further from the star. Let's assume the gravitational force on planet A is represented as F_A, and the gravitational force on planet B is represented as F_B.
Given that planet B is four times further from the star than planet A, the distance ratio is (distance_B / distance_A) = 4.
So, if the distance between planet A and the star is represented as r_A, then the distance between planet B and the star would be (4 * r_A) since planet B is four times further.
Now, let's analyze the mass ratio. Planet B has twice the mass of planet A, and the mass ratio is (mass_B / mass_A) = 2.
Using these ratios, we can calculate the gravitational force on planet B relative to the force on planet A.
F_B / F_A = (G * (m_B * m_star) / (4 * r_A)^2) / (G * (m_A * m_star) / r_A^2)
Note that the gravitational constant (G) and the mass of the star (m_star) are common in both equations and cancel out.
F_B / F_A = (m_B / 4 * r_A)^2 / (m_A / r_A)^2
F_B / F_A = (m_B^2 / (4^2 * r_A^2)) / (m_A^2 / r_A^2)
F_B / F_A = (m_B^2 / (16 * r_A^2)) / (m_A^2 / r_A^2)
F_B / F_A = (m_B^2 / m_A^2) * (r_A^2 / 16 * r_A^2)
F_B / F_A = (m_B^2 / m_A^2) * (1/16)
Thus, the gravitational force of the star on planet B is (m_B^2 / m_A^2) * (1/16) times the force on planet A.
Note: Squaring the mass ratio ensures that we account for the effect of mass on the gravitational force, and dividing by 16 accounts for the effect of distance since planet B is four times further from the star compared to planet A.