Calculate the magnitudes of the gravitational forces exerted on the Moon by the Sun and by the Earth when the two forces are in direct competition, that is, when the Sun, Moon and Earth are aligned with the Moon between the Sun and the Earth. This alignment corresponds to a solar eclipse.) Does the orbit of the Moon ever actually curve away from the Sun, toward the Earth? (Please give your answer to three significant figures.)

Question

Calculate the magnitudes of the gravitational forces exerted on the Moon by the Sun and by the Earth when the two forces are in direct competition, that is, when the Sun, Moon and Earth are aligned with the Moon between the Sun and the Earth. This alignment corresponds to a solar eclipse.) Does the orbit of the Moon ever actually curve away from the Sun, toward the Earth? (Please give your answer to three significant figures.)

I used G m1 m2/r^2 to get the forces, for the force between the Earth and the moon I did:
((6.67e-11)(5.97e24)(7.35e22))/(1.737e6 - 6.37e6)^2
and I got 1.36e24 N
To get the r value I substracted the radii... do I have to add them?
For the force between the Sun and the Moon, I used the distance from earth to the sun and substracted it from the distance between moon to earth.
((1.737e6 - 6.37e6)- (6.96e8 -6.37e6))=6.95e8 m

and then for the moon to sun force I computed:

((6.67e-11)(7.35e22)(1.9891e30))/(1.737e6 - 6.37e6)^2
and I got 1.40e34 N...

These answers are not right but I'm not sure what I'm doing wrong...

Answer 1

see other post.

Answer 2

To correctly calculate the forces and distances in this problem, you need to make a few adjustments to your calculations.

Firstly, let's focus on the gravitational force between the Earth and the Moon. You correctly used the formula:

F = (G * m1 * m2) / r^2

Here, G is the universal gravitational constant, m1 and m2 are the masses of the two objects (Earth and Moon), and r is their separation distance.

You correctly found the masses of the Earth (5.97e24 kg) and the Moon (7.35e22 kg). However, the separation distance (r) should be the distance between their centers. In this case, it's the sum of the radius of the Earth and the radius of the Moon.

r = (1.737e6 m + 6.37e6 m) = 8.107e6 m

Plugging in these values into the formula, you should get:

F_Earth-Moon = (6.67e-11 N (m/kg)^2) * (5.97e24 kg) * (7.35e22 kg) / (8.107e6 m)^2
= 1.98e20 N

So, the magnitude of the gravitational force between the Earth and the Moon is approximately 1.98e20 N.

Now, let's move on to the gravitational force between the Sun and the Moon. Similar to the previous calculation, you need to find the separation distance between the Sun and the Moon. It should be the difference between the distance from the Earth to the Sun and the distance from the Earth to the Moon.

r = (1.49597870e11 m - (1.737e6 m + 6.37e6 m))
= 1.49575433e11 m

Plugging in the new value of r, you can now calculate the gravitational force:

F_Sun-Moon = (6.67e-11 N (m/kg)^2) * (7.35e22 kg) * (1.9891e30 kg) / (1.49575433e11 m)^2
= 1.94e26 N

So, the magnitude of the gravitational force between the Sun and the Moon is approximately 1.94e26 N.

Regarding your second question, the orbit of the Moon never curves away from the Sun towards the Earth. The gravitational force from the Sun pulls the Moon towards it, opposing the gravitational force from the Earth. However, the gravitational force from the Earth is stronger and keeps the Moon in orbit around the Earth. So, the Moon's orbit is always concave towards the Earth. It never curves away from the Sun towards the Earth.