I'm talking about a Weighing Balance shown in the figure:
Press & Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.
Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude & opposite in direction. Then why does it come back to it position? What's making it to come back?
If it would only be the weights exerting torque, the balance would be in equilibrium at all angles. What makes the balance go back to the horizontal position is the fact, that the center of mass is below the beam. consider this picture
The needle exerts a torque too, so you have more torque on the side, where the plate is higher. You can have more subtle configurations (like in your picture, where the beam is rounded below) but the mechanism is the same.
The horizantal beam on such scales is intentionally placed below the rotational axis. As long as the weights are in equilibrium the torque is equal on both sides.
But as soon as the position changes e.g. tipping the left scale down, the torques differ because only the tangential part of the gravitational force vector in relation to the rotational axis contributes to the torque around it. When tipping down the left scale, torque on the left side gets smaller and torque on the right side gets bigger, therefore the right side moves down again until equilibrium is reached (besides some swings to accommodate for the temporary impulse energy).
This effect gets the more pronounced as the distance of the horizontal bar approaches the half length of the bar.
This effect would not be if the horizontal bar went exactly through the axis.
It behaves this way because that's how it was built. By adjusting the mass distribution, we could make a scale that flops to one side, is roughly balanced at all angles, etc. However, those scales would not be useful, so the scale isn't built that way.
It might be assumed from the left/right symmetry of the picture that the system cannot decide which way to go, and so is at an equilibrium point. This equilibrium will be stable if a small perturbation (rotating the beam a small angle) raises the center of mass. It will be unstable if a small perturbation lowers the center of mass.
Beyond that, it is difficult to say how the center of mass moves simply by looking at your picture because we do not completely understand the mass distribution and the location of the pivot point.
When finding the center of mass, we can ignore any stationary pieces because we are only interested in the change of the height of the center of mass. Additionally, if the pans hang freely down, it appears as if one will rise by the same amount the other falls, and thus they will not change the height of their center of mass when considered jointly. They can also be ignored.
Let's assume the rest of the scale rotates rigidly. In that case, the center of mass of the rigid portion we're considering will be constrained to a circle with its center at the pivot point. If the center of mass is exactly at the bottom of the circle, we have a stable equilibrium. Otherwise, it is unstable.
Nice question! if the following analogy applies : imagine a seesaw on a half-sphere fulcrum (top of the picture). if it inclines e.g. to left side (bottom) - the length from the right edge to fulcrum ($L_2$) increases, the lever rule kicks in ($F_2>F_1$) and the weight of the right side brings the seesaw back to equilibrium (top) (which is then broken again by inertia)
balance beam diagram
Torque about a given axis equals force times distance of line of action of the force from the given axis.
(This is a copy of my answer to Why does a beam balance restore? I added it here because I think the other question is essentially asking for clarification of this one so may eventually be deleted. If a friendly editor would like to shrink the diagram a bit so it fits more neatly on screen, please go ahead.)
it seems to me that when I do the calculations for the torques considering the pivot as a single point in space they are equal. However I must assume that real scales are built such that any rotation through an angle $\theta$ will cause the pivot point to shift slightly from center causing unequal torques. The effect of this net torque is to restore the beam to the horizontal no matter which direction in theta you rotate (e.g. whether you push down on the left or right piece, both case result in a restoring torque). I would also add that the oscillation die down due to frictional damping, the whole system is approximated by simple harmonic motion with damping.
Actually with the lever sticking up above the balance scale shown; balance axis point; it adds gravitational weight to whichever side is initially positioned downward, and also since this side is down, it should weigh more being closer to the ground... and should tend to keep this side down... However is doesn't...
If you put a weight, on some point of a wheel... to make it balance; you put an identical weight at an identical point equal and opposite... Spinning the wheel at high rpm will tell you quickly, if you did right. Then if idle, it should sit still, in any upright position...
If two identical boats were tied together with a rope and were sailing parallel, in a parallel steady wind, in friction-less water and etc... With one boat ahead, and off to the side; they... would maintain their positions and the friction-less rope is meaningless.
I don't really know the answer... but you might align the scale north & south; then again east and west; to see if there is a difference due to Earth's centrifugal effect? It seems to keep the oceans fairly balanced, (disregarding the moon.)
If it is true that matter has "matter waves"; they may be equaling out repelling against the ground... Whereas with the large heavy wheel above mentioned, this effect would probably be negated...?
If this experiment were laid upon a flat horizontal plane... and different forces were applied equally like the sail boats... I have an idea the results would be: the mechanism would hold still, wherever positioned; with similar circumstances.
