Simple Quiz... Missing Weight

1,2,2,5 and then units of 10, 20, 20 and 50. Decades ago I authored an article in ME on making a balance and suggested using silver steel since the composition and thus density is better controlled than mild steel, as is the diameter so that only the length needs to be accurately cut to give a specific weight. I measured several batches of silver steel to give a density of 4.50 oz/cu"

Rod

Just so.

I've not checked, but noting 1, 3, 9 and 27 are rising powers of 3, I predict the 5th weight needed to answer Neil's question is 81 ounces.

Actually this set of 5 weight goes well beyond 4lbs - up to 7lbs 9oz if I'm right. (I hope no-one has forgotten my maths is terrible.)

Not too difficult getting a 1oz weight from 2lb with a balance, but building the rest of the set in powers of 3 will be fun! Another disadvantage of the 5 weight system is the arithmetic needed to work in pounds and ounces from a base 3 set of weights. No problem though - imperial weights and measures were riddled with strange conversions, and no-one complained about the sums, allegedly.

Dave

The simplest solution would be to go back to the historically commonplace offset beam balance. It was the weighing system used for thousands of years that used the minimum of materials and parts, was robust and was easy to carry if you were travelling.

Martin C

It would seem likely that, the balance being old is Imperial, so the weights would increment a a geometric progression with a factor of 2.

So one of each of: 1/2 oz, 1, 2, 4, 8, 16 would allow items weighing up to 31 1/2 ounces to be measured in 1/2 oz increments. . Duplicating some or all of the weights would extend the range.

Methods of determining the missing weights have already been described, so as long as one weight is available, the others ought to be to be derived by using temporary ones to halve or double them datum weight.

For greater precision the way is to drill and tap a hole and fit a grubscrew. The weight can then be adjusted to the desired level more accurately by adding lead shot under the grubscrew.

This was the method advocated by the local Weights and Measures Inspector when we needed a set of standard weights for calibrating dynamometers. He was apologetic that the 25 and 50 lb weights would not be closer than a few grains!

Howard.

I wouldn't use scrap, which tends to come in the wrong sizes. Instead use dry sand, lead shot, ball bearings, small nuts and bolts, aniseed balls, hundreds and thousands.... even Hopper's coins, if he will lend them to you.

Andrew is right in saying the question is incompletely specified; you didn't tell us the tolerance, Neil!

George

My answer uses Base 3 arithmetic to get the smallest number of different weights needed, but is seriously inconvenient because deriving the weights 1, 3, 9, 27 and 81oz is tricky. Not quickest.

Deriving weights in Base 2 with a balance is easy. So first step is to decide what they are. Without trying to minimise the number of different weights required by adding known weights to the unknown side.

The maximum weight to be measured is 4lbs, which is 64oz

Binary digits rise in powers of two, thus:

64 is binary 1000000
32 is binary 0100000
16 is binary 0010000
8 is binary 0001000
4 is binary 0000100
2 is binary 0000010
1 is binary 0000001

Therefore any weight up to 63oz in 1 oz steps can be represented by adding one of any of the set 1,2,4,8,16,32

For example:

24 is binary 00011000, or weight 16 + weight 8
55 is binary 00110111, or weight 32 + weight 16 + weight 4 + weight 2 + weight 1
63 is binary 01111111, or all the weights

The given 2lb is weight 32, no need to make another

Going down, balancing weight 32 with scrap gives a heap weighing the same as weight 32. Putting weight 32 to one side, the scrap is redistributed into two balanced piles, each of weight 16. Use one pile to make the actual weight 16, put it to one side, and rebalance the remainder to give two piles each of weight 8. Repeat, saving one pile to make the actual weight 8, and rebalance the other to get two piles of weight 4. And so on until the balance isn't sensitive enough to register differences. In this case, stop at weight 1. The weights so far only add up to 63oz, so make a second weight 1. A total of 8 weights are needed.

Then 4lbs (64oz) is binary 01111111 + 00000001

The whole system works in ounces, but as that's not complicated enough for Imperial Measure purists the system is partitioned into slugs, tons, quarters, hundredweights, stones, pounds, quarters, ounces, drachms, and grains. Common sense rules a quarter of acid drops isn't the same weight as a quarter of coal. It's simple as long as you don't look closely...

Dave

Edited By SillyOldDuffer on 03/07/2022 20:00:22

Neils scales are equally at home with the metric system as pounds and ounces, only the weights are different.

I believe that if you have a known weight and a ruler you can derive any weight you want using m=fxd

Take for example I have a 1kg steel weight but I want a 0.5kg weight from some scrap brass ( or what ever material available). I cannot measure the steel weight and make it half the size as the density is different. But if I hang the 1kg steel weight on a balance at 100mm and a piece of brass at 200mm, I could machine the brass weight until it is in equilibrium with the 1kg steel weight. At that point, the brass weight should weigh 0.5kg. Does it make sense?

Edited By ChrisB on 04/07/2022 08:07:48

Yes it makes sense, Chris

… Tedious to do in practice, but quite logical.

MichaelG.

+1 It's the principle of the steelyard balance, this example from the Wikipedia article is Roman:

As a way of creating accurate weights m=f x d has the disadvantage of needing an accurate linear measure plus an accurate way of positioning the balance weight on the staff. Also, machining the brass weight means it has to be taken off and replaced a few times, and moving things always upsets measurements.

Plain balances make it easier to minimise errors. The central pivot point can be found accurately with an uncalibrated divider and improved by making the fulcrum sharp edged. A long pointer mounted at the pivot amplifies any imbalance. Slight differences in the weights of the scales and their suspensions can be compensated by adding tiny uncalibrated weights to one side. Plain balances aren't vulnerable to changing temperature, humidity, or air pressure. And finely granulated 'scrap' can be used to match another weight very closely.

As usual when chasing accuracy getting it right becomes progressively more difficult. A well-made balance is sensitive enough to require protection from dust settling on the pan during measurements and there are multiple ways the operator can cock up. The principle is simple, but applying it needs training, practice and careful control of the environments. Balances don't work well on boats!

Dave

Not so simple then Chris

David

True, it may not be easy but I believe it IS simple to do. It may be tedious and might take a while true, but there's nothing like a free lunch

No special equipment a hobby machinist does not have, we all have precise measuring instruments other than a ruler. You can go crazy on accuracy if you wanted, but probably the OP does not intend to make E1 calibration weights.

This is what I suggested a few posts back. If you have a 2lb weight you can replicate a few more 2lb weights as they just have to balance the existing one. If you then work out the correct position for the 4lb combined weight balancing the 2lb weight you have the point where the ratio is 2:1. You can then progress this to produce a 1lb weight and then a 1/2lb weight, then you can start adding points for lots of other ratios along the beam and mark them off. A little bit of interpolation will let you estimate values that are not explicitly marked on the beam.

Martin C

Hi, I've dragged this counter scale form its hiding place, which I got off ebay several years ago, which looks very much like this 1964 B&H catalogue one.

It came with a rubbish pan that wasn't an original and five weights and although four of the weights have the same name on them, I doubt that the 1 oz belongs to the same set.

This one weighs to 7 LB, which according to the catalogue would've had weights from 4 LB to 1/2 oz and from 2 to 1/2 oz would've been brass. I haven't ever really used these scales, but the weights have been an addition to those that I got with this Avery scale that I guess would have maybe been used in a butchers or fishmongers shop and even then were not enough to use for full capacity but the 2 Kg lifting weight helps bring them closer, but I don't think the rusty ones were originally for the Avery anyway. I do have a 5 Kg electronic Salter scale with a stainless steel platform for the kitchen and none of the weights that I have, weigh exactly what they say on them, but I also have a fairly populated box of L. Oertling weights bought from a car boot sale 20 years plus ago and had them all checked for weight by a lady in the Lab where I worked years ago, on their highly accurate electronic scales that were checked regularly for accuracy by a company like Lloyds British, and all of them were quite a few milligrams underweight, which I checked yesterday on my Sartorius Research R 200 D electronic semi-microbalance and are pretty much the same as they were when checked before, although the 1 gram one was missing, so I made that one and had to take it back to work a few times to get it checked before I was satisfied it was close enough.

Making other size weights from one known weight is OK, but of course you need to know what your known weight actually is and what accuracy you want to achieve. The calibration weight on my Sartorius is one that came with a PS-200B pocket scale (1 to 200g) bought from Maplins a long time ago, is very accurate for a budget scale, the Sartorius has its own internal calibration and the machine can be set on four different stability conditions from very stable, stable, unstable and very unstable, I have it set on stable as I don't have a place currently where it can be considered to be very stable. The paper on the Avery scales has to be in place to bring it into the level position before anything can be weighed, which is what shops did with their bags or wrapping paper, butchers and fishmongers never put their products directly on the weighing platform in the shops that I've been in years ago, but I don't know if they do that now as I don't go in such places now.

Regards Nick.

P.S. For anyone interested https://oertling.com

Edited By Nicholas Farr on 04/07/2022 13:15:10

Neil's original post mentioned kitchen scales. I don't believe that extreme accuracy is required, and I doubt that he will ever need to weigh 63 oz as an example.

Interesting ideas.

My thought is that you have to drop down from 2lbs in steps.

Now, the way to get an accurate 1lb weight is to make three, slightly overweight ones. Keep gradually reducing so that any pairing of two exactly balances the 2lb weight. Like matching surface plates, you now know all three weigh 1lb exactly.

Now put one exact 1lb weight and use it to get three 1/2lb weights. Bear in minds that only ONE of the weights needs to be a single piece, the other two can be made up of various odd ends.

Keep going down via 8, 4,2 and 1 oz. Just make sure you keep two 1oxz weights at the end. so you can make up 4lbs as 2lbs + 1lb + 8oz + 4oz + 2oz + 1oz + 1oz.

Neil

The principle was simply that it needs to be a balance scale to use the 'any two of three' method.

The resulting series of weights will cover any number of ounces up to 64, including 63.

Neil