Moon-Earth System's Final Distance Calculation

In summary, the conversation discusses the future rotation of the moon and its effects on the Earth's day length. It is suggested that the Earth and moon can be treated as uniformly dense spheres and that the final distance between them can be determined based on conservation laws. However, the assumption of both kinetic and rotational energy being conserved leads to too many constraints, so the equation must be modified to account for tidal effects. The conversation concludes with a suggestion to start by writing an equation for the combined angular momentum of the Earth moon system.
  • #1
oswaler
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[SOLVED] Moon's future rotation

Homework Statement



COnsider the Earth and moon as if they formed an isolated system. The length of the day increases every year, a feature attributed to tidal effects. At some distant time in the future the Earth moon system will be orbiting such that the same face of the Earth will be pointed at the same face of the moon and the distance between the Earth and moon will reach a final value. Assuming that the Earth and moon can be treated as uniformly dense spheres and that the masses will be what they are at present, what will be the final distance between the Earth and moon?

Homework Equations





The Attempt at a Solution



According to conservation laws, energy and angular momentum should be conserved since according to the problem there is no external force to provide a torque. From the question, at the final distance, the orbital period of the Earth will = the orbital period of the moon. I'm not quite sure where to go from there though. A push in the right direction would be appreciated.
 
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  • #2
If you assume both kinetic plus rotational kinetic energy and angular momentum are conserved, then you have two equations in three unknowns, the unknowns being the rotation rates of the Earth and moon and the radius of the earth/moon orbit. If you introduce the constraint that they are tidally locked - this means that the rotation rate of the Earth is the same as the orbital period of the moon AND the rotation period of the moon is the same, then you have introduced too many constraints. You won't be able to solve the equations. You need to relax the conservation of kinetic energy. Tidal effects produce heating. You can't assume kinetic energy is conserved.
 
  • #3
Thank you, that makes sense. I think at this point I'm mainly having trouble getting started with the equations.
 
  • #4
Write down an equation for the combined angular momentum of the Earth moon system first. Include both rotational parts and orbital parts. Put the numbers in and get the total. That's where to start.
 

1. What is the current rotation of the Moon and how will it change in the future?

The Moon currently rotates on its axis at the same rate that it revolves around the Earth, which is why we always see the same side of the Moon. However, due to tidal forces from the Earth and other celestial bodies, the Moon's rotation is gradually slowing down. This means that in the future, the Moon's rotation will become even slower, and eventually it will become tidally locked to the Earth, meaning we will only ever see one side of the Moon.

2. Will the Moon ever stop rotating?

No, the Moon will never completely stop rotating. The forces that are slowing down its rotation will eventually reach an equilibrium, causing the Moon to always show the same side to the Earth. However, it will still technically be rotating, just at the same rate as it revolves around the Earth.

3. How long will it take for the Moon to become tidally locked to the Earth?

The Moon's rotation is currently slowing down at a rate of about 2.2 microseconds per century. Based on this rate, it is estimated that it will take about 50 billion years for the Moon to become tidally locked to the Earth.

4. What will happen to the Moon's orbit as its rotation slows down?

As the Moon's rotation slows down, its orbit will also change. Its orbit will become more elliptical, meaning it will be closer to the Earth at some points and farther away at others. This is because the Earth's gravity will have a stronger effect on the Moon's orbit as its rotation slows down.

5. How will the Moon's rotation affect its appearance to us on Earth?

As the Moon becomes tidally locked to the Earth, we will only ever see one side of it. This means that the Moon's appearance will never change from our perspective. The side facing away from us will always be hidden, and the side facing us will always be visible, with the exception of minor changes due to libration (a wobble in the Moon's axis of rotation). This will also affect the phases of the Moon, as we will only ever see the same side lit up by the Sun during its orbit around the Earth.

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