Law of the lever - Explained Physically Thanks for reading.
I've seen the "Archimedes Proof" of the law of the lever, and many similar ones, but...at the end of the day, all they're doing is getting observations about the way rotating bodies behave, and using those observations as axioms.
By defining torque to be the cross product of force and distance, we're just creating new terminology, but we aren't really explaining physically why it is that a force applied at a greater distance would be better at spinning something than a force applied at a lesser distance.

For example, say the green man and the pink man weigh exactly the same, but the green man is half as far from the pivot point as the pink man is. The rod will rotate towards the pink man.
Why?
I think that the reason that the green man doesn't have as much "spinning" power as the pink man does must have to do with the molecules of the bar. In a sense, the pink man will cause more bending in the bar, and it'll want to straighten out towards his direction (forgive me if that doesn't make any sense, it's just a thought).
Can someone please explain, from the approach of the atomic interactions in the rigid body connecting both the pink man and the green man to the fulcrum, and without using the concept of torque, why it is that the bar rotates towards the pink man?
And, if the torque equation could then be derived from that physical explanation, that would be awesome!
In summary, I'd like to know how to approach the concept of torque with atomic theory. I'd like to understand the interatomic electromagnetic forces in a rigid rod, how momentum travels along a rigid rod when we apply forces perpendicular to its length, and how that leads to the force being "amplified" by larger distances from the pivot point.
 A: 
For example, say the green man and the pink man weigh exactly the same, but the green man is half as far from the pivot point as the pink man is. The rod will rotate towards the pink man.


Why?

The answer is because the pivot force acts away from the center of mass, causing a rotation.

So now your question is

*

*Why is there only translation (no rotation) when the line of action of a non-zero force goes through the center of mass?

The answer to that has to do with the definition of the center of mass. by definition, if you apply a small force on every particle in a body, it is equivalent to the total force through the center of mass. Hence why it is called center of gravity sometimes, since every particle of the body experiences gravity in the same way.
And you can intuitively understand that if all the particles experience the same small force, the body is not going to rotate. It is pushed equally everywhere in a parallel fashion.
In this scenario, the center of mass is the unique point in space where a single force through it is equivalent to a uniform distributed force over the entire body. In both cases, no rotation exists.
A: Forget about atomic structure, that is an entirely different thing to show. Start with a rigid bar sitting on a pivot. Suppose that the contents of the situation bring about whatever happens in it. If the rigid bar is on the pivot with the pivot directly in the middle of the bar, since we are supposing the contents of the situation bring about what happens, but there is no difference between the left hand and the right hand sides of the bar, in relation to the pivot, there should be no more reason for one side to go down than the other, and there should be no more reason for one side to go up than the other. (This, come to think of it, assumes that the rigid bar is uniform along its length, it also assumes that left and right don't make a difference, and that there is no other difference in the situation, such as a greater gravitational field on one side of the pivot than the other. It also assumes that the bar and the pivot continue and are not replaced every logically distinguishable moment by new existences, which may, for all we know, have arbitrarily different properties from the original bar and pivot. These sorts of assumptions are, at any rate, easy to make.) So, from the contents of the situation, there seems no reason why one side of the bar should go down or up rather than the other side of the bar.
But supposing that in spite of everything on one side of the pivot being exactly similar to everything on the other side---except for one side being on the left and the other being on the right, but then why couldn't you just go round the other side, which would reverse this difference? so why should one side being on your left and the other on your right make any difference?---the bar doesn't balance but one side keeps going up, and the other side keeps going down.Or sometimes one goes up and sometimes the other. And after all, from a purely logically possible point of view this seems something we could find happening--In that case, I think, we would be at a loss how to understand this situation, from its contents. That is we have two apparently indistinguishable situations (except one being on the left and one being on the right, but if you have more than one situation there has to be SOME difference between them, otherwise you couldn't tell them apart, and there must be some difference between all situations for them to be more than one situation.) But if they don't produce the same result, how could their same contents produce a different result? So we suppose that this can't happen, and it is found in experience that it actually doe not happen. If two situations are exactly the same, with the same contents, you get the same result--or otherwise you should re-examine both situation to discover some difference you have missed (This is not supposed true of quantum mechanics, but it works normally, in everyday experiences).
So, back to our bar and pivot;
everything on one side of the pivot is exactly similar to everything on the other side of the pivot; we suppose that the contents of situations should themselves bring about what happens in the situations. There are no differences between the contents, so the tendency of the bar to fall or rise on either side of the pivot is exactly matched by that part of the bar on the other side of the pivot, so they remain horizontal and balanced. Now let us suppose we move the pivot so its not exactly in the middle of the bar. The bar on one side of the pivot is longer than that on the other side, so it will have more weight than the part of the bar on the other side---This is again assuming the bar has physical properties evenly distributed along its length, and that these sorts of properties continue through the situation, and no other difference is introduce. Weight is a tendency for something to fall and this can be compared between instances. One way of comparing it is through the physical effort required to lift various weights, another way is to compare the effect of different objects on a spring. Another is to compare them on a balance (!)(this is because, if we suppose an equal balance, with everything the same on both sides we can compare the effect of one object at one end with an object at the other end as to which end, with the objects added, goes up or down--and why should one object consistently go down more than another object at the other side of the balance unless it had the property of going down more strongly than the other object?) But these different ways of comparing objects tendencies to fall turn out to have the same result; i.e. an object that feels heavier will, roughly and by and large compress a spring or any other matter more than a lighter feeling object, and will also out balance the same lighter object. Logically deductively this does not have to be the case, but it makes sense if you suppose the contents of situation themselves bring about what happens in them, and that objects and their properties continue, and a stronger or weaker tendency for objects to fall is a property an object can have and continue to have in these various logically distinguishable situations. But let's return to our bar and pivot. We are supposing that the bar is rigid, and uniform along its length, and each part of the bar has a weight or tendency to fall that is equal to all the other similar parts of it, that these things continue, and that more of the bar is now on one side of the pivot than there is on the other side. So there will be more tendency to fall on that side of the pivot than on the other side, so that side should fall.
Now the new question is; why, if we suppose two equal weights on the bar, equally distanced from the pivot should one side of the bar fall when it is moved further from the pivot? But, as before, there will be a greater part of the bar, and so tendency to fall on the side of the pivot as the pivot is moved along the bar towards the other weight and away from the now falling side.
But there is also another aspect involved  because the weights will move different distances the further or nearer they are to the pivot than the weight on the other side. So, supposing they are the same weights, more will be done in respect of moving a weight a distance, on one side of the bar than on the other. But if the contents of the situation themselves bring about what happens and one weight moves twice the distance of the other, then unless it is twice as difficult to move the other than the one that moves twice as far, then the one that moves twice as far would be able to move a grater than the other weight half that distance.--- On the view that the contents of the situation are what bring about what happens, the whole situation, in this extended sense, has to 'balance'. But logically deductively there is no reason why this should happen. We might find energy just being created, and perpetual motion machines being no problem. But neither do I agree that in that case, our principle (that the contents of situations should themselves bring about (be sufficient to bring about what occurs in them) must be known merely through individual experiences--if something is itself sufficient it doesn't need another justification, it wouldn't be itself sufficient if it did. This means it isn't an a-priori principle either, and it isn't known independently of experience to be true, but is based on trying to see how the contents of situations could be themselves sufficient to bring about what occurs.
