Moving planets.

[Scion of the Mists]

7 minutes ago, king of nowhere said:

In that case, make yourself heavier than the planet, push, the planet will move one meter away, then become light and stop pushing, you will fall back on the planet and the planet will have effectively moved.

Yeah, this is what I was thinking.  Except than instead of moving the planet a fixed distance, you're really imparting a small velocity onto it, which will persist indefinitely (as there's no drag in space).  

Regarding energy conservation, you're already "cheating" that every time you push on anything because you're converting investiture into kinetic energy.  

[CaptainRyan]

7 hours ago, Scion of the Mists said:

I believe this would mean that an Allomancer could accidentally flatten themselves by pushing against a coin as well, right?  

I don't know if an Allomancer can produce enough force to flatten themselves that way? If the coin was against a wall or something and the Allomancer used Duralumin but did not have access to Pewter than I could see them harming themselves (though probably not flattening themselves). In the example with the anvil I was assuming (though I did not state it and for that I apologize) that the Allomancer could generate enough force to stop the anvil (in reality I do not think even Elend, using Duralumin, had that kind of power).

8 hours ago, Scion of the Mists said:

Additionally, how do you define what a single object is in terms of it acting as an anchor.

It is whatever is bracing the metal object/Allomancer and preventing it/them from being moved when a force is applied. This could be friction (an Allomancer has to overcome the force of friction when pushing on a coin, no?), it could be the screws that hold a lantern to the wall, it could be a person who is holding on tightly to their gun and, therefore, the Allomancer has to push harder to knock the gun away as opposed to if the gun was simply lying on the ground. Whatever increases the force needed to move the object counts as an "anchor".

8 hours ago, Scion of the Mists said:

Why would the the entire planet be an anchor and not just the dirt around your feet.

You are correct in that a person standing on the ground does not actually use the entire planet as their anchor. Their anchor is the durability (for lack of a better word) of the ground underneath them or even their own body if the ground is more durable than they are. An anchor only provides as much resistance as it can withstand force. The person holding a gun while an Allomancer pushes on it? That "anchor" is only as strong as their grip and/or their arm socket haha. The ground provides as much "anchoring" as the object can withstand.

Perhaps I am missing some key concept here or am maybe underthinking this but I have never really had an issue with understanding how the mechanics of pushes/pulls work. I think they are fairly well suited to physics once you accept that investiture is allowed to be converted into energy. 

[Scion of the Mists]

20 hours ago, CaptainRyan said:

Perhaps I am missing some key concept here or am maybe underthinking this but I have never really had an issue with understanding how the mechanics of pushes/pulls work. I think they are fairly well suited to physics once you accept that investiture is allowed to be converted into energy. 

Assuming equal forces are applied to both the Allomancer and the object, my key question is: how is the magnitude of that force determined?  Is it dependent only on the masses of the Allomancer/object (my initial thought), or does it take into account the "anchor"?  

[CaptainRyan]

9 hours ago, Scion of the Mists said:

how is the magnitude of that force determined?

My head canon is that the F in the equation is determined by the Allomancer's "strength". The "stronger" the Allomancer is at Allomancy, the greater the F they can produce*. Vin remarks that Elend is able to push/pull with much greater strength than herself and the reason for this is that Elend had "stronger" Allomancy due to his abilities not being diluted by genetic drift. 
*It is also important to take into account the distance the Allomancer is from the object (the farther away the Allomancer is the less force they can exert).

Example (all numbers are arbitrary and are merely used for demonstration purposes): Elend, a 50 kilogram Allomancer, can generate 20 Newtons of force at a distance of 1 meter using a steel push. Elend pushes on a 0.1 kg coin that is lying on a frictionless surface 1 meter away. Assuming Elend is also standing on a frictionless surface, at what acceleration do Elend and the coin move?

Example #1
50kg * A(elend) = 20 N = 0.1kg * A(coin)
A(elend) = 20 N / 50 kg = 0.4 m/s^2
A(coin) = 20 N / 0.1 kg = 200 m/s^2

Now, assuming the same parameters above except that the coin is now glued to a wooden board with the glue and board weighing an additional 0.9 kg, what is the acceleration of Elend and the coin?

Example #2
A(elend) = 20 N / 50 kg = 0.4 m/s^2
A(coin + glue + board) = 20 N / (0.1 kg + 0.9 kg) = 20 N / 1 kg = 20 m/s^2

So, in these examples, I lay out, according to my head canon, that the "strength" of the Allomancer combined with the distance they are from the metallic object determines the force involved. Then, the mass of the metallic object, plus the mass of anything else that is resisting the push of the Allomancer, are used to calculate the acceleration of the object.

Let's do one more thought experiment! Supposed the parameters are the same as our last example except there is no glue holding the coin to the board (we will also assume the glue's mass is negligible so we can keep the same mass as example #2). In this scenario, Elend pushes on the coin and, initially thanks to either the angle or the friction between the coin and the board, the coin and board remain together. Then, after some amount of time, either the board rotates or the force of friction is overcome and the board is no longer resisting the push. That would mean, ignoring the decrease in Elend's "push strength" as the coin gets farther away, we would initially see an acceleration of 20 m/s^2 but once the board moves out of the way (the "anchor" is removed), the acceleration would increase to 200 m/s^2!

[Scion of the Mists]

On 4/7/2018 at 0:07 AM, CaptainRyan said:

Example #1
50kg * A(elend) = 20 N = 0.1kg * A(coin)
A(elend) = 20 N / 50 kg = 0.4 m/s^2
A(coin) = 20 N / 0.1 kg = 200 m/s^2

Example #2
A(elend) = 20 N / 50 kg = 0.4 m/s^2
A(coin + glue + board) = 20 N / (0.1 kg + 0.9 kg) = 20 N / 1 kg = 20 m/s^2

This is how I originally thought of it, with the force generated being independent of any anchoring.  However, it the books, it shows that anchoring the coin changes the magnitude of the force on the Allomancer:

Quote

Vin reach reached toward the proper thread, and pushed slightly.  The coin flipped out of Kelsier's fingers traveling directly away from Vin.  She continued to focus on it, pushing against the coin through the air until it snapped against the wall of a nearby house.  Vin was thrown violently backward in a sudden jerking motion.  

 

[CaptainRyan]

1 hour ago, Scion of the Mists said:

This is how I originally thought of it, with the force generated being independent of any anchoring.  However, it the books, it shows that anchoring the coin changes the magnitude of the force on the Allomancer:

Quote

Vin reach reached toward the proper thread, and pushed slightly.  The coin flipped out of Kelsier's fingers traveling directly away from Vin.  She continued to focus on it, pushing against the coin through the air until it snapped against the wall of a nearby house.  Vin was thrown violently backward in a sudden jerking motion.  

 

*mind explodes*

There has to be a reasonable explanation for this! ... Right?! My equations are utterly useless now! I am undone!!

*deep breaths*
*reassembles exploded mind*

Ok, let's think about this then. A good example of what happened to Vin might be someone who is pushing their hands out in front of them. If they do that with nothing but air in front of them then their hands move forward with little resistance. If, however, they push their hands out in front of them and suddenly meet a brick wall then they are pushed back. The coin is similar in this case. If the Allomancer is pushing a coin through air then there is little to no push back against the Allomancer. If their coin strikes a wall while they are still pushing on it then they are thrown back just as if they had pushed against the wall with their own hands.

These two scenarios are similar in that the same amount of force (arm strength or Allomantic strength) is being brought to bear. The difference is that the arms/Allomancy are pushing either very little mass (the air/the coin) or a comparatively tremendous amount of mass (the wall/the coin + anchor). In the case of the former the person/Allomancer is not affected very much at all whereas in the case of the latter the person/Allomancer is thrown backwards. There was no change in force applied in either scenario, there was only a change in the amount of mass the force was being applied to.

So, mathematically, what does that look like? I have not the foggiest at the moment haha. I will think on this and write back later. Thank you @Scion of the Mists for the wonderful discussion this has been. It has given me much to think about!