Express algebraically the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun. Why, since the ratio is so small, doesn't the Sun pull the Moon away from the Earth?

Newton's Law of Gravitation states that the gravitational force (F) between two objects is given by the formula: F = G * (m1 * m2) / r^2 where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.

We will denote the force between the Earth and the Moon as F_EM. Using the formula in step 1: F_EM = G * (M_E * M_M) / r_EM^2 where M_E is the mass of the Earth, M_M is the mass of the Moon, and r_EM is the distance between the Earth's center and the Moon's center.

We will denote the force between the Sun and the Moon as F_SM. Using the formula in step 1: F_SM = G * (M_S * M_M) / r_SM^2 where M_S is the mass of the Sun, and r_SM is the distance between the Sun's center and the Moon's center.

To find the ratio of F_EM to F_SM, we will divide the equation for F_EM by the equation for F_SM: (F_EM) / (F_SM) = [(G * (M_E * M_M) / r_EM^2)] / [(G * (M_S * M_M) / r_SM^2)] Notice that the gravitational constant G and the mass of the Moon M_M will cancel out in both the numerator and denominator: (F_EM) / (F_SM) = [(M_E) / r_EM^2] / [(M_S) / r_SM^2] Now, we can find the simplified expression for the ratio: (F_EM) / (F_SM) = (M_E * r_SM^2) / (M_S * r_EM^2)

From the expression in step 4, we can see that while the gravitational force of the Sun (M_S) is significantly larger than that of the Earth (M_E), the distance between the Sun and the Moon (r_SM) is also much greater than the distance between the Earth and the Moon (r_EM). This results in the ratio of gravitational forces being small, which means that the gravitational force between the Earth and the Moon is much stronger relative to the force between the Sun and the Moon. The reason the Sun doesn't pull the Moon away from the Earth is due to the balance of forces acting on the Moon. Even though the Sun has a stronger gravitational force on the Moon, the Earth's gravitational force is acting closer to the Moon, making it more dominant. This balance of forces keeps the Moon in orbit around the Earth rather than being pulled away by the Sun.