Energy conservation for a point at the Earth's surface

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Discussion Overview

The discussion centers on the conservation of energy for a mass located at the Earth's surface, specifically at the equator, and how potential energy changes due to the Sun's position affect the system. Participants explore the implications of these energy changes and the role of Earth's rotation and gravitational forces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • The original poster (OP) calculates potential energy differences at noon and midnight due to the Sun's position and questions what balances this difference in potential energy.
  • One participant suggests that over a complete rotation, the net energy change is zero, implying that energy is conserved within the system.
  • Another participant highlights that energy transfer occurs between masses within the same system, correcting the OP's initial consideration of the system.
  • A different participant notes the influence of the Earth's orbit around the Sun, introducing the concept of centrifugal force and its effects on gravitational forces.
  • One participant challenges the OP's assumption of an unbalanced Earth, suggesting that this assumption is incorrect.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions made regarding the system and the effects of gravitational forces, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

Some participants point out limitations in the OP's reasoning, including the consideration of the wrong system and the neglect of energy transfer between masses. The discussion also touches on the effects of the Moon and tidal forces, which may complicate the analysis.

Who May Find This Useful

This discussion may be of interest to those exploring concepts of energy conservation, gravitational forces, and the dynamics of systems involving celestial bodies.

Kamikaze_951
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Hi everyone,

This is my first post here and I am really sorry for that question, but I have found the answer nowhere.

Consider a mass at the Earth's equator that is static in the Earth's referential during an entire day. Put the Earth at one of its equinoxes to simplify the problem. Then, at noon, the potential energy due to the sun of a mass m at that point is :

E_p noon = -GMm/(distance Sun-Earth - radius of Earth)

At midnight,

E_p midnight = -GMm/(distance Sun-Earth + radius of Earth)

Clearly, there is a difference in potential energy. By energy conservation, it should be balanced (by another kind of periodic energy variation). However, if the mass stand on solid ground, its rotation speed is the Earth rotation speed and it should be constant during the day (not going up and down) and therefore, the difference in kinetic energy is null.

My questions : What balances this difference in potential energy? Is there a mistake in my reasoning?
 
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You have only got half the rotation of course ... when you do a whole rotation the net energy change is zero.

Not happy? Well that really just means that, wherever the energy went, we got it back.

As a mass "falls" from the dark side of the Earth towards the light side (towards the sun) is gains energy which goes into "lifting" the equivalent mass on the light side into the dark side. Since the masses are balanced, and they are connected by the rest of the Earth, there is no gain or loss.

You get the same for any rotation in gravity ... or any conservative force field.
 
Your post just made me realize that I considered the wrong system (Sun+mass rather than Sun+Earth) and forgot to take into account the energy transfer between two masses of the same system.

Your answer was very helpful and I thank you a lot for it.
 
No worries - it's actually a very common mistake.
 
If you look at force differences of this size, keep in mind that the Earth orbits the sun, which leads (in the reference frame of earth) to a centrifugal force outwards - "down" at day and "up" at night.

Note that the effect of variable gravitational forces is bigger if you use the moon. Tides exist due to these differences, and there are power plants which use those tides.
 
The OP's question assumes an unbalanced earth, which is wrong.
 

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