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More specifically, why does having two objects of different weights make the balance lean to one or the other then stop partway? Why does the balance not just keep tilting until it falls to the side? How can something be heavy enough to tip the balance but not heavy enough to make it tilt the rest of the way?basic balance

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Balances work because they only need to tell when things are in balance, not how far out of balance they are. As such, the only behaviors needed are those which are very close to equilibrium.

When one is that close to equilibrium, small things matter. For example, as you tip, you roll the fulcrum ever so slightly to one side (the more knife edged the fulcrum is, the more sensitive it is). That roll does shorten the lever arm of the heavier mass and length the length of the lever arm on the lighter mass. Its a small amount, but when you get very close to equal weights (or, more precisely, equal torques), it's a stabilizing effect that results in the balance settling on a position (rather than moving freely as one might expect if one assumed the fulcrum was an unmoving point)

Naturally, if one has two weights which are within 1/100th of each other's mass, a shift of only 1/100th in the length of the lever arms is enough to bring torques into equilibrium. The balance will settle on the angle which yields that 1/100th of the length shift.

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