The more paradoxical scales balancing with the heavier weight further from the pivot than the lighter weight
The device described here is probably getting a bit subtle for a hands-on exhibit, but I include it here because it relates to an existing hands-on exhibit.
The paradoxical scales was an exhibit I saw at the old Bristol Exploratory. It consisted of a two-armed balance, with weights that could be positioned anywhere along the arms. Now everyone who's played on a see-saw knows that a light person can balance a heavier one if they sit further away from the pivot than the heavier one. But in this exhibit, it didn't matter where on the arms you put the weights. Two identical weights would always balance the scales, no matter where you put them. And two different weights would never balance the scales, no matter where you put them.
This behaviour makes us think harder about ordinary scales where the position of the weights does matter. It's easy to say 'principle of moments', but harder to understand where the principle comes from.
The important thing about ordinary scales (or a see-saw) is that, as the scales tip, weights far from the pivot move through greater vertical distances than weights near the pivot. This happens because the arms tilt as they move up and down. When the scales are perfectly balanced and set in motion by a push, any work done by one weight in descending must be exactly balanced by the work done on the other weight as it rises. Applying the principle of conservation of energy, we find that a small weight far from the pivot can balance a larger weight near the pivot.
The paradoxical scales don't have an ordinary pivot, but instead have a special arrangement of parallel links which means that although the arms can still move up and down, they don't tilt as they do so - they are always horizontal.
This parallel motion of the paradoxical scales means that as the scales tip, weights on either side of the scales move through identical distances, regardless of their positions on the arms. Energy conservation now tells us that the position of the weight on the arm is irrelevant as far as balancing the scales is concerned.
The more paradoxical scales
Following this train of thought, if we can build some scales where the motion of the arm is greatest near the pivot and decreases away from the pivot, then we have 'more paradoxical' scales where the heavier weight has to be further from the pivot than the lighter weight to balance the scales. I built such a pair of scales out of the construction toy K'nex.
We can do this by altering the parallel linkage in the paradoxical scales to a non-parallel linkage, as in the picture below.
The picture shows the pivot linkage and the inner ends (FE and DC) of the two lattice-girder arms. The pivot linkage is pivoted on the supporting framework at the top and bottom. The two upper links AF and AD are separate, but the lower link EBC is one rigid piece - in the model it is a lattice girder to make it rigid. EBC pivots on the support structure at B. Each arm is attached to one end of the lower link, and to one of the upper links.
Because EF and CD are shorter than AB, their orientation changes as they move up and down. Consider CD: as it moves up, it rotates clockwise; as it moves down, it rotates anticlockwise. This is illustrated in the picture below.
Between them, these two motions mean that the vertical motion of an arm attached to CD is greatest near the central linkage and decreases away from the linkage (up to a point, as we'll see). As a result of this pattern of movement, a light weight near the middle of the scales can balance a heavier weight further away from the middle of the scales on the other arm. This is the 'more paradoxical' behaviour.
The neutral point
As one moves along an arm away from the central linkage, the movement of the arm (caused by movement in the central linkage) gets progressively smaller, and then starts to increase again. Thus there must be a neutral point where the arm does not move at all as the central linkage changes shape. Applying our energy arguments again, we predict that a weight placed at the neutral point cannot do any work on the scale arm, and therefore needs no weight on the other side of the scales to balance it. The picture below shows the scales happily balanced with a weight on the neutral point of one arm and no weight on the other.
To understand the neutral point, consider that a weight on an arm has 2 effects: it applies a downward force, and applies a moment at the root of the arm. For an arm attached to CD, the force tends to move CD down, and the moment tends to rotate CD clockwise and thus (because of the way that the linkage works) move it up. The force is the same wherever the weight is, but the moment grows as the weight is placed further out. When the weight is near the root of the arm, the force wins and CD tends to move downwards. When the weight is far out along the arm, the moment wins and CD tends to move upwards. At the neutral point, the two effects balance and application of the weight doesn't move CD at all.
(The geometry is rather complicated and I would bet that the neutral point moves along the arm as the scales rock. However, for any position of the scales there will be a neutral point for which an infinitesimal angular displacement of the central linkage gives a zero vertical linear displacement of the neutral point.)
Beyond the neutral point
Beyond the neutral point the movement of the arm once again increases with distance from the central linkage. Weights placed in this region behave non-paradoxically: the lighter weights need to be placed further from the pivot than do the heavier ones.
The real pivot
One way to think about all of this is to say that the real pivot is not the central linkage any more: it is the neutral point on the arm. One can place two weights on the same arm on either side of the neutral point and get them to balance, following the non-paradoxical rules of ordinary scales. It's no longer surprising that a weight at the neutral point doesn't need another weight to balance it: a weight placed on the pivot of a normal scales doesn't need another weight to balance it either.
The one-armed scales
Following this way of thinking, we can remove one arm of the scales altogether, and do all our balancing on the remaining arm, putting weights either side of the neutral point. Thus one has scales which behave approximately like normal scales but which don't appear to have a central pivot. In practice, instability sets in for anything but rather small displacements, so the demonstration is rather subtle.
The parametrically variable scales
If the distance AB in the central linkage is made adjustable, it would be possible in principle to build a single set of scales that showed normal, paradoxical, or 'more paradoxical' behaviour. The picture of the pivot is repeated below. With AB = 0, we have normal scales. With AB = EF = CD, we have the paradoxical scales, and with AB > EF and AB > CD, we have the 'more paradoxical' scales. The practical problem is that without also making AF and AD adjustable in length, for some settings the scales would balance with the arms very non-horizontal.