Moon orbits Earth. How large is the force required to pull it away from the Earth?

Question

Please including physics formula and explanation, thanks!

Answer 1

Well, the force between the earth and moon is given by

F = GMm/(r^2) where

G= constant of gravitation = 6.67428 x 10^-11
M = earth mass = 5.79 x 10^24 kg
m = lunar mass = 7.35 x 10^22 kg
r = mean earth-lunar separation = 384350 meters

So, that means

F = 1.923^26 newtons.

This is an enormous force. A force greater than this will cause the moon to drift away from the earth.

HTH

Charles

Answer 2

Actually, it is more a question of "impulse" (Force * time). You can apply a small force for a long time or a very strong force for a short time.

The Moon is pulling away from Earth and the force that does it comes from Earth! The tidal bulges that are raised by the Moon on Earth's ocean lag behind the sub-lunar point because of Earth rotation. Therefore, the bulge pulls (a tiny bit) on the Moon, with the (even smaller) resultant force being in the direction of Moon's motion in its orbit.

This pulls the Moon a bit forward which would tend to make the Moon go faster than its orbit requires. Therefore, the Moon climbs to a higher energy orbit. The Moon is receding at 4 cm per year.

Earth's escape speed at a distance equal to the the mean orbital distance of the Moon is a bit more than 1.445 km/s.
The Moon's orbital speed is 1.022 km/s.

So we have to accelerate the Moon by 423 m/s (I've switched from km to m). How fast do you want to do that? Let us say (to make calculations easier) that you want to do that in one million seconds (a little more than 11 and a half Earth-days).

Then the acceleration would have to be 423 m/s divided by one million seconds = 0.000423 m/s^2.

What force is needed for that much acceleration?

F = m a
We have a = 0.000423 m/s^2
The Moon's mass (m) is 7.35x10^22 kg

F = m a = 0.000423 * 7.35x10^22 = 3.11x10^19 Newtons

Impulse = 3.11x10^19 N * 1,000,000 s = 3.11x10^25 Ns

If you wanted to do it a thousand times faster (1000 seconds instead of a million), you'd need a force a thousand times stronger. (same impulse)

If you want to do it over 330 years (10^10 seconds), then you'd need a force of 3.11x10^15 N (still same impulse).

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The Saturn V first stage rockets had a thrust of 34,000,000 Newtons and they could keep it up for 150 seconds.

Answer 3

Charles's answer is close enough, except I'd point out that the Moon is 384,350 km from the Earth, not 384,350 meters.

Technically, the force necessary would be slightly less since Charles failed to take into account the motion of the Moon. It's motion generates some centrifugal force. I'd tend to ignore that as well, since the actual amount of force is 1.922x10^26 Newtons, scarcely any difference from Charles's answer.

For a closed orbit, the overall energy level is negative. To 'pull away' from the Earth, the minimum energy level has to be 0.

E=1/2mv^2 - Gm1m2/r

The Moon's speed is about 1.03 km/sec.

Force is the derivative of work or energy. In other words, to find the force with respect to the radius:
F = Gm1m2/r^2

If you're talking about force provided by thrusters, you could reduce the amount of force necessary by increasing the amount of time the thrusters are on. Whatever the force, there will be a change in velocity and a change in the radius. As each is slowly increased, the amount of force necessary to pull the Moon from its orbit decreases. Eventually the velocity and radius will equal some value where the force provided by the thrusters will be sufficient.