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### Author Topic: On what principle does a weighing balance operate?  (Read 38600 times)

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #25 on: 30/11/2011 21:03:20 »
I resisted  the temptation add diagrams thinking it was a challenge to clarify the matter with text alone.

#### Bored chemist

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##### On what principle does a weighing balance operate?
« Reply #26 on: 30/11/2011 21:20:44 »
In the next development of the thought experiment I explain that my pointer is of a non-traditional shape.
Specifically it is exactly the same shape as the beam. It just has a little arrow drawn on it  labelled "Down".

There's nothing magical about what you call the mass that hangs downwards below the suspension point. A pointer will clearly do the job of ensuring that the beam comes to rest horizontal.
But the beam itself will also do that job.

#### Geezer

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##### On what principle does a weighing balance operate?
« Reply #27 on: 30/11/2011 21:23:36 »
Look mum! No pointer.

The forces (F) on the pan pivots are equal but the beam is not level. L1 is greater than L2, so the counter-clockwise torque is greater than the clockwise torque. The net torque is zero when L1 and L2 are equal, at which point the beam will be horizontal.

EDIT: BTW, this works when the pointer has zero mass, but it works just as well when the beam has no mass either.

« Last Edit: 30/11/2011 22:39:32 by Geezer »

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #28 on: 01/12/2011 08:18:47 »
For the purpose of calculation we will take an idealised scale of the following dimensions and calculate the rotational forces operating on the beam

The beam will have a length of two meters and height of four centimetre's and will have a suspension point in the dead centre for the fulcrum and similar suspension points precisely in line at each end for the pans .

The beam mass will be zero and the mass of the pans will be 1/9.82 Kg.

The beam will be suspended at the fulcrum point so that the pans are 10cm above the working surface.

Let us next push the left hand pan down to the surface, the right hand pan will rise and exert a torque of

(1-(.1^2))^.5 = 0.99498 newton meters tending to rotate he beam clockwise while the left hand pan will exert a similar torque tending to rotate the beam anti clockwise hence the system will be stable and the left hand pan will remain down.

Now let us calculate the effect of gravity, taking the radius of the Earth as 6,366,197.8 meters the left hand pan will be this distance from the centre of the Earth while the right pan will be 6,366,197.6 meters away.

Applying Newton's inverse  square law the gravitational attraction on the right pan will be reduced by one part in (6,366,197.8/6,366,197.6)^2 =1.0000000631 hence there will be a net force of 0.99498*0.0000000631=0.0000000628 Newton's tending to hold the left hand pan down.

Now we come to the effect of raising the fulcrum point as it would normally be on any practical set of scales, If the fulcrum point is raised by one cm relative to the line of the pan suspension points when the left hand pan is pushed down the effective length of the left hand side the beam is reduced by one part in a ten thousand  while that of the right hand beam is increased by the same amount hence a restoring force tending to move the beam to a horizontal position  of 0.0002 Newton meters is generated vastly more than any gravitational effects.

But the left hand pan wont stay down.  The beam will go back to horizontal.

If all three fulcrum points are in-line the balance will still work despite the fact that both arms remain the same length regardless of the beams inclination.

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #29 on: 01/12/2011 08:24:17 »
MikeS
"If it's not a differential gravity effect what is it?"

Let me answer this specific point as I think it is something you have not considered.
When the beam is pivoted at a point above the line of the pan suspension points if for instance the left hand pan tends to drop the beam moves to the right reducing the effective length of the beam on the left hand side and increasing it on the right hand side.
This produces a negative feedback effect causing the beam to stabilise in the horizontal position if the masses in the two pans are equal.

It's true that I hadn't originally thought of the effective lengths of the arms changing due to the geometry of the fulcrum points.
But
If all three fulcrum points are in-line the balance will still work despite the fact that both arms remain the same length regardless of the beams inclination.
I think I am right in believing in this design torque is always equally balanced and therefore plays no part.
« Last Edit: 01/12/2011 08:58:37 by MikeS »

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #30 on: 01/12/2011 08:44:57 »
This is the question that still hasn't been answered.

Let's consider a simple balance beam with all three pivot points in line.  One pan is slightly heavier than the other.  The beam will come to stable equilibrium with one pan lower than the other.  Why?

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #31 on: 01/12/2011 09:06:41 »
But it will not with all three pivot points in line and one pan heavier than the other the heavier pan will descend until it meets a stop.
This of course assumes their is no friction in the system and an equal mass of beam above or below the fulcrum points

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #32 on: 01/12/2011 09:21:34 »
Mmmmm.  Do you have any evidence for that?

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #33 on: 01/12/2011 10:00:02 »
No I thought this was a theoretical discussion and I have not had access to any laboratory type balance since I left school seventy years ago.
Perhaps our Bored Chemist could confirm it.

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #34 on: 01/12/2011 10:03:25 »
syphrum

"Practical balances are constructed with the central knife-edge lying a little
below the plane of the terminal knife-edge."

This is the opposite of the way that most of you have described the pivot points of a balance and the opposite of
Geezers diagram.  L2 is greater than L1.

If a balance can work with the central pivot point either above or below the terminal pivot points then I see no reason why it should not work with all three pivot points in line.

In which case this is wrong
But it will not with all three pivot points in line and one pan heavier than the other the heavier pan will descend until it meets a stop.

« Last Edit: 01/12/2011 10:06:17 by MikeS »

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #35 on: 01/12/2011 10:40:39 »
The balance will certainly work with all pivot points in line but with equal weights in each pan it will not automatically restore its self to a horizontal beam state if either pan is pushed down it will stay down.
As far as I recall our school balances had a lever to lift the central post after the pans were loaded, if they were equally loaded the beam would come up horizontal but of course they had some self restoring force built in.
Even in the absence of this feature they would still would still have come up in this manner but this state would have been unstable and if either were pushed down the would have stayed down.
Reading the article you quoted I notice that the central pivot is placed below the line of the pan pivots which in effect gives a positive feedback making the balance point unstable but stability is restored by the use of a pointer which gives a negative feedback effect to counteract this.
This combination is done to increase the sensitivity at small deflections.
We have of course been discussing the most basic form of balance but commercial designs are a little different

« Last Edit: 01/12/2011 10:58:04 by syhprum »

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #36 on: 01/12/2011 11:26:30 »
syphrum

Ok so let's consider a simple balance beam with all three pivot points in line and the balance beam incorporates some kind of restoring force.  One pan is slightly heavier than the other.  The beam will come to stable equilibrium with one pan lower than the other.

Do you agree?

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #37 on: 01/12/2011 11:44:16 »
With some sort of restoring force ? a pointer perhaps as your postioning of the pivots gives none.
In this case yes

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #38 on: 01/12/2011 11:54:27 »
The question is why does the balance reach stable equilibrium with one pan lower than the other?  Why does the heavier pan not continue until it reaches the stop?

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #39 on: 01/12/2011 12:17:31 »
It depends how stability has been achieved, in the original case where stability had been achieved by the positioning of the pivots the non linear change in apparent length of the arms with the deflection angle would limit how far the heavier pan drops, the restoring force exerted by a pointer is also not in a linear relationship to the deflection angle .
The mathematics of the situation would involve trigonometric functions into which I don't really wish to go.

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #40 on: 01/12/2011 12:37:09 »
syphrum

Ok so let's consider a simple balance beam with all three pivot points in line.

Would you consider the pans to contribute a restoring force without the requirement of the mass of a needle?

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #41 on: 01/12/2011 13:24:53 »
No in this case there would be no restoring force so the heavier pan would go down to limit or equally weighted pans would just sit where they were placed, but let me continue
As stated by Bored Chemist there is no difference between a scale stabilised by the weight of a pointer to one stabilised by the positioning of the pivots so I will use the later to explain how trigmetric functions determine the deflection angle hence how far the weighted pan drops .
As the weighted pan drops the torque it develops varies as the cosine of the angle of the beam from the horizontal i.e starts high and falls away whereas the torque generated by the pointer being set at 90° to the beam varies as the sine of the angle of the beam from the horizontal i.e starts low and climbs.
when these two sources cancel out that is the angle at which the beam rests.
« Last Edit: 01/12/2011 13:27:16 by syhprum »

#### Bored chemist

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##### On what principle does a weighing balance operate?
« Reply #42 on: 01/12/2011 19:05:37 »
"Ok so let's consider a simple balance beam with all three pivot points in line. "
Why should we?
It's not as if anyone would actually make a balance like that (unless they were relying on a pointer to keep it in check).
If they did that, it wouldn't work.

#### Geezer

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##### On what principle does a weighing balance operate?
« Reply #43 on: 01/12/2011 19:13:25 »
Ok so let's consider a simple balance beam with all three pivot points in line.

Would you consider the pans to contribute a restoring force without the requirement of the mass of a needle?

Absolutely not. If the weights are equal there is no torque in any position.

Your question is the same as "what would happen if I picked up a perfectly balanced bicycle wheel by it's axle?" Obviously, it won't rotate because there is no reason why it should rotate.

If the fulcrums on a scale are in line, and the center of mass of the entire beam assembly (including the pointer) is conincident with point of rotation of the beam, there will be no restoring force. If the weights are equal, the balance will remain in any position it is put in. If the weights are unequal, the heavier one will tip the balance as far as it can go.

A scale must have a restoring force because a balance is actually comparing the the restoring force with the difference in the weights on the pans. The displacement of the pointer from center actually calibrates that amount on a graduated scale.
« Last Edit: 01/12/2011 19:16:12 by Geezer »

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #44 on: 01/12/2011 20:03:54 »
Bored Chemist

The balance in the article you directed us to actually had the fulcrum below the suspension points of the pans and relied on the weight of the pointer to provide stabilisation.
This combination of positive and negative feedback was done to increase the sensitivity to small deflections

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #45 on: 01/12/2011 20:16:20 »
Geezer

The wheel of an inverted bicycle could well be used as a model balance, a small weight could be placed say where the valve comes out to emulate the pointer and pseudo pans attached either in line with the axle or symmetrically above or below it for experiment.
A simple and readally available model
« Last Edit: 01/12/2011 20:18:28 by syhprum »

#### Geezer

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##### On what principle does a weighing balance operate?
« Reply #46 on: 01/12/2011 20:59:38 »
Geezer

The wheel of an inverted bicycle could well be used as a model balance, a small weight could be placed say where the valve comes out to emulate the pointer and pseudo pans attached either in line with the axle or symmetrically above or below it for experiment.
A simple and readally available model

Oh, you mean something like this?

(The pseudo pans are calibrated galvanised buckets attached to a piece of string.)

(Astute correspondents will observe that the distance between the string and the axis of the wheel is constant.)

« Last Edit: 01/12/2011 21:33:34 by Geezer »

#### syhprum

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##### On what principle does a weighing balance operate?
« Reply #47 on: 01/12/2011 21:36:30 »
No I think the buckets should be attached to the spokes so that you can adjust the suspension points relative to the axle
« Last Edit: 02/12/2011 08:14:08 by syhprum »

#### Geezer

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##### On what principle does a weighing balance operate?
« Reply #48 on: 01/12/2011 21:46:13 »
No I think the buckets should be attached to the spokes so that you adjust the suspension points relative to the axle

Yes. That would make it more like a beam balance, but I like my version better!

The only thing that produces a restoring force is the weight of the valve which tends to prove that there are many ways to produce a restoring force, but you gotta have one.

We could even eliminate (or counterbalance) the unbalanced mass of the valve and replace it with a torsion spring and get the same result.
« Last Edit: 01/12/2011 21:55:01 by Geezer »

#### MikeS

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##### On what principle does a weighing balance operate?
« Reply #49 on: 02/12/2011 08:42:26 »
Thank you all for taking part you have convinced me.  A balance needs a restoring force.

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##### On what principle does a weighing balance operate?
« Reply #49 on: 02/12/2011 08:42:26 »