Angle of Balance with Unequal Weights: Equation?

[lanzara]

Does anyone know what the equation would be for the angle (from horizontal) of a simple two pan balance with unequal weights? This would be the simplest type of balance - no damping mechanism, etc. I have looked in many places, but I can't seem to find a specific equation for this angle for unequal weights on a balance.

Thank you.

 

[Bystander]

Sum the torques about the pivot as functions of the pivot angle; this sum equals zero when the balance is at rest.

 

[lanzara]

angle terms

It seems that the angle terms cancel unless I'm making a mistake, which is certainly possible. Has this been done anywhere that I can see a resulting equation?

 

[Bystander]

You've made the assumption most people make when analyzing a balance, that the pivot point is located at the center of mass of the balance mechanism (arms plus suspension plus pans). Such a balance is at equilibrium (with equal masses on the pans) at ALL rotational positions about the pivot. To actually get "balance" action for comparing masses, the pivot is moved to a point either higher than, or lower than the COM. The higher position is the normal balance configuration, and the lower is what is known as a trip balance. The distance between COM and pivot determines the sensitivity of the balance.

Nose around in the dusty shelves of the physics reference texts and handbooks in the library, and you will find assorted ancient tomes on laboratory practice with all sorts of helpful analyses of vacuum systems, balance calibrations, and other forms of good, old-fashioned, "smash-mouth" physics.

See "A Dictionary of Applied Physics," Glazebrook.

 

[HallsofIvy]

If I read your question correctly, you have a balance beam with pivot at the middle (center of mass) but with un-equal masses in the pans, no friction or damping, and are asking what angle the bar will make with the horizontal?

The answer is a right angle. If there is no damping or friction, the beam will continue dropping on the heavier side until it is vertical.

 

[lanzara]

Reply to answers

With all due respect, your answers show that we all have initial assumptions that may be wrong unless tested.

If you actually have a very simple, two pan balance (the old fashion swing kind) and test it, you will see that it does make an angle, which isn't necessarily a right angle for every difference in weight.

Now, I know that your first reaction is that it is due to friction, but let me describe a simple experiment:

Take for example the following experiment with a simple two-pan balance in an initial horizontal equilibrium. Add enough weight to one pan of the balance so that the pan just touches the surface of the table. Now add two additional weights that are equal but one hundred times larger to each pan. What happens to the balance when these two large weights are added to both sides? This is a simple problem, but it also shows our biases when we rely on intuition and forego scientific measurement and inquiry. When the large weights are added, the pan that was touching the surface of the table will rise off of the table.

I have looked at hundreds of texts to find one that gives the angle of a balance - to no avail! I know that it seems like it should be out there, but apparently it isn't!?

 

[Bystander]

Glazebrook, vol. III, p. 107; Weissberger, "Physical Methods of Organic Chemistry."

 

[lanzara]

I wonder if its not too much trouble if you could write the equation and I also wonder if it shows that the angle of the balance will decrease when extra weight is added to each side as in the example I gave above.

 

[Bystander]

Originally posted by lanzara
I wonder if its not too much trouble if you could write the equation

It wouldn't be that much trouble, nor would it be that useful to you without the context in which it appears --- transcribing that context IS too much trouble.

and I also wonder if it shows that the angle of the balance will

Strike "will" and insert "can," meaning, that it is possible, depending upon the balance geometry.

decrease when extra weight is added to each side as in the example I gave above.

Couple points: 1) you are trying to address questions about the topic you present in Physics Resource; 2) real balances are NOT comprised of meter sticks, strings, and rocks as in treatments in freshman physics courses; 3) your description of "Langmuir Binding" does not include any description of the balance apparatus, and I ain't John Edward, Madame Cleo, nor Kenny Kingston --- I cannot intuit what is or ain't useful information for you; 4) the analysis you require WILL involve someone applying the discussion in Weissberger to a detailed description of your balance; 5) a better analysis would include reference to Glazebrook's, and to Partington's (An Advanced Treatise On Physical Chemistry) discussions of balances and balance techniques.

Weissberger is available in any library (okay, most libraries), or at least through inter-library loan; chapter and verse are going to depend upon which edition, volume I probably does the trick for all editions.

If you have NO access to Weissberger, Partington, or Glazebrook, PM me and we can discuss possible arrangements for the assistance you seek.

 

[lanzara]

I think that you are confusing my posting of an article that I wrote about the "Desensitization of a Balance With Langmuir Binding of Weights" which is basically an experimental article with my current request to find an equation to give the angle of a balance with unequal weights.

 

[lanzara]

As to the detailed description of my balance: it is a simple two-pan balance with swing pans attached to a triangular shaped fulcrum inside a metal circle. The horizontal length of the arm from one side to the other is 13 inches and the vertical length from the top to the bottom of each pan, which is attached by 3 chains, is 13 inches.