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In the story I'm writing, magic mostly works by mages moving energy from one place to another, or converting it from one type to another. One particular mage is naturally telekinetic—she doesn't have to do all the calculations (mass of the object, how much energy do we need to move it, etc.) in her head to move stuff around like other mages do. She also doesn't need to think about where she gets the energy for her telekinesis; she instinctively uses the pressure from the ground beneath her.

If this mage draws kinetic energy from a semisphere of ground beneath her that's three meters in radius, about how much energy does she have at her command at any given time? Since I don't actually know any physics, I don't know how to do this math, but I would guess it would involve taking the average mass of a piece of ground that size, and plugging it into Newton's law of gravity, correct?

Also, what happens to the ground after this mage takes energy from it? Does the temporary removal of its downward acceleration make it rise up, or is there no perceptible change?

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    $\begingroup$ I would stay away from kinetic energy. Instead, imagine drawing thermal energy from an area (it gets colder and the change in thermal energy is the energy your mage has to work with). If you really want to, you can imagine using kinetic energy, but stay away from potential energy like you are describing. You can imagine leeching kinetic energy from a moving object by applying a force opposite to its direction of motion, and applying the same amount of energy expended to the work you want done, but this may be too powerful - you could stop a projectile AND use that energy for magic! $\endgroup$ – akirilov Oct 4 '16 at 16:53
  • $\begingroup$ Some numbers: 3m radius semisphere = 56m^3, packed dirt = 1700kg/m^3, so 3m radius semisphere of packed dirt = 95200kg. $\endgroup$ – Rob Watts Oct 4 '16 at 17:02
  • $\begingroup$ To expand on the problem with using potential energy - imagine lifting a book in the air, then sucking out all the potential energy. The book would be stuck...floating? It could never fall back down (otherwise you create energy out of nothing!) The only reasonable way to handle it would be to harness the energy as the book is falling back down (taking the kinetic energy and converting it to work elsewhere), much like a water wheel uses the kinetic energy of moving water to do useful work. $\endgroup$ – akirilov Oct 4 '16 at 17:04
  • $\begingroup$ @akirilov You wouldn't be sucking away potential energy. The dirt below you is being pulled down by gravity at 9.8m/s^2, but it doesn't go down because the dirt below it resists that force. So this mage would absorb that force instead so that the dirt below the semisphere doesn't have to resist the force. $\endgroup$ – Rob Watts Oct 4 '16 at 17:12
  • $\begingroup$ This is precisely the definition of potential energy. If you're resisting force, that's one thing (spoiler: it would explode), but if you're taking out the energy as OP describes, you would be breaking physics. Assuming the force option, the dirt you're affecting is applying downward pressure on the dirt below it, and everything is in equilibrium and motionless. Removing the force of gravity from the dirt above would remove the pressure on the dirt below it, breaking equilibrium (there is now internal pressure and force from below, but no balancing force above), causing it to blast upwards. $\endgroup$ – akirilov Oct 4 '16 at 17:14
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It depends on exactly how you harness this kinetic energy, as individual air particles move at around 509m/s, but knock into each other very quickly, if you could harness this energy then you have a LOT of energy at hand.

There are about 2.5 x 10^25 molecules of air per meter cubed then in a 3 meter radius sphere around you, you have about 4/3*pi*27 = 113.1 meters cubed of air,

And using various equations you can show that the kinetic energy per air molecule at room temperature and normal atmospheric pressure is 6.1 × 10-21

Now, 6.1 × 10-21 x 2.5 x 10^25 x 113.1 = 17,247,750 joules of energy, and would cool the air around your mage to absolute zero, so use with caution, but that is a LOT of energy, so you could use that number as your 100% efficient grand master levels of energy, then have less powerful mages with lower efficiencies of that 17.2 million joules of energy

(I know you asked for energy from the ground, but this is so much more energy and would definitely be preferrable)

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  • $\begingroup$ As a side note, cooling a sphere of air to that degree would have catastrophic consequences. At the very least, it would probably kill your mage and anyone else within at least a city block instantly. $\endgroup$ – akirilov Oct 4 '16 at 21:38
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    $\begingroup$ @akirilov oh I know, for one, massive low pressure area, wind rushes around you, you die, then the temperature would be so low it would burn your skin, but this is an "ideal" situation, that is the energy available, you don't have to use it all, and as I said in my answer "use with caution" $\endgroup$ – Alex Robinson Oct 4 '16 at 21:48
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Alright, here's a less hand-wavey explanation of the physics involved and why it will cause problems:

There are several important forces acting on any given clump of dirt:

Down:

  • Gravity
  • Downward force (pressure) from dirt above

Up:

  • Normal force from dirt below

Other:

  • Internal pressure (this is a combination of many forces, but breaking it down here is irrelevant to the answer)

The (Up) and (Down) forces are normally balanced, and constitute an external pressure balancing the internal pressure. Removing (harnessing) the force from gravity has the following effects:

  • The (Up) force is greater than the down force.
  • The external pressure is reduced, meaning internal pressure is greater than external pressure

The net result of this is the clump of dirt you are harnessing gravity from "explodes" outwards and upwards.

You can hand-wave this by saying the mage harnesses all forces acting on it for the duration of the spell. This is kind of odd, but should maintain equilibrium.

Alternatively, I would suggest other ways of harnessing energy, such as leeching heat from an area.

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To move an object from point A to B, you need...

Energy(joules) = Weight(kg) X Distance between point A and B(meters) (all energy to move outside this line cancel out. So if you move it up, you can move it down that much for free basically.)

You can harness heat at a rate of...

0.0012 joules per cm^3 per degree Celsius (for normal air). (roughly 34 joules per cubic foot per degree) or

4.186 joules per gram per degree Celsius (for water) (roughly 118,534 joules per cubic foot per degree. Water retains heat very well)

If you draw kinetic energy from the ground, you get 0 joules (no movement = no energy). Of course if you want to slowly kill your planet, you can harness it's angular kinetic energy (from rotation). Near infinite for one person, but will slowly kill your planet over time.

A more balanced alternative is to say that using telekinetic magic drains energy from the caster's muscles equal to what they would have exerted if they moved it themselves (as if there was a path they could walk along, so ignoring climbing like stuff to go up).

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You could convert the 95tons of dirt the mage is standing on into pure energy based on E=mc^2, which is a lot, but then the mage falls into the hole

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