enter image description here

I have made a simple sketch of how I think the system looks like. My problem is:

I always thought that the angle the balance makes is a function of the difference between the two masses (or the difference in torques): a small difference in mass will give a small angle, and in the limit the angle will be equal to zero when both masses are equal.

Now I cannot derive that from the picture. As I see it now, the torques will always be different, no matter at which angle the balance is tilted, since both torques scale with the same sine. This leads to a equilibrium only when the balance is completly vertical (or in my figure pressing against the triangle), which is contra-intuitive. Moreover, when the masses are the same, equilibrium would be obtained at every angle as can be seen from my figure (torques are equal independant of the angle of the balance), which would contradict with the working of a balance.

I see that my question kind of overlaps with this one, but the accepted answer doesn't solve my problem.

(One way to solve my problem, would be to draw a half circle instead of the triangle in my figure, so the lever of the torque becomes a function of the angle. But somehow in my head the schematic drawing of a balance is with a triangle in the middle. Is this perhaps the simple solution I'm looking for?)

  • 3
    $\begingroup$ Based on the picture that you drew, your conclusion is correct: if the torques are different, then they will remain different even if the angle changes. However, the thing you have to realize is that the picture you drew is not quite how a balance works. Read the "teeter-totter/mass balance" comment in the link you provided for an explanation what additional factors need to be considered. $\endgroup$ Apr 1 '14 at 0:01
  • $\begingroup$ possible duplicate of Why does the weighing balance restore when tilted and released $\endgroup$ Apr 1 '14 at 14:33
  • 1
    $\begingroup$ In real life balances are always designed so the centre of mass is below the pivot. This is discussed in a couple of the answers to the previous question about balances. $\endgroup$ Apr 1 '14 at 14:34
  • $\begingroup$ So if I just add an extra mass under the center of rotation, it can be done with the triangle? And are there balances that work with a "half-circle" (gears or so) instead of the triangle to play with the lever arm as driving force instead of playing with the extra mass under the center of rotation? $\endgroup$
    Apr 2 '14 at 17:26