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Let's say I want to measure the weight of a bag of clothes (or some other object), in a situation where I don't have a weighing machine ("scales"?) available.

Objects that are available:

  • A broom.
    • The specialty of a broomstick is that its weight is evenly distributed.
    • If the broom end cannot be removed, it means there is an extra weight on one end.
  • A chair or something to balance the broom on.
  • A counterweight with a known weight of, say, 1kg.
    This could be a packaged food item, e.g. a milk package, placed in a plastic bag so it can be hanged on the stick.
  • A pencil or some other thing to make marks on the broomstick.
  • A smartphone with calculator.

Objects that might be available:

  • A (wooden) stick with even distribution of weight, e.g. a broomstick with the broom end removed.
  • A tape measure.
  • A smartphone with internet and/or an app that does the math for me (if such an app exists).
  • If you think you desperately need another object that is commonly available and does not radically change the setup of the scenario, you may include it in the answer.

None of these objects should be damaged in the process! E.g. we don't know if the owner of the broom would approve of removing the broom end.

I am looking for a method and a formula to determine the weight of the bag of clothes. So in other words, a primitive / improvised weighing machine / scales.


Note: The difference to traditional scales is that we don't have arbitrary counterweights available. It is hard enough to find one object with a known weight.


A perfect answer would have

  • Simple and easy-to-follow instructions.
  • Different versions of the instructions for the "optional" objects.
  • Some background or proof or derivation of the formula, but separate from the instructions, so this part is optional for the reader.

P.S.: Yes, I might figure this out on my own if I think really hard. But even if I do, I still think this should go into a stackexchange Q/A. The best would be macgyver.stackexchange.com, but engineering is the closest fit.


About the "on hold" / "too broad".

I seriously don't get what is wrong with the question. It is a simple problem with a simple answer.

At the time of asking, I was sure that a weighing machine can be built with a stick balanced on something, and stuff hanging on the stick. I just was not sure about the exact setup (which is embarrassing, since this is very basic stuff), and which physical properties are required from the stick object. This is why I tried to provide details without narrowing it down unnecessarily.

The goal of this Q/A is that someone with the same problem finds the information that is relevant to the solution, so they learn how to build the thing and why it works.

I could have asked "How can I build an improvised weighing machine from commonly available household materials?", but this would be too vague because "commonly available household materials" is open to interpretation, and would possibly result in a wide range of different apparatuses.

On the other hand, it was clear that it does not matter if it is a broom or another stick-like object, so there was no benefit in insisting on this. Naming it a broom-stick makes it easier to imagine, so I kept it as part of the question.

And finally I was not sure if additional items are required for the task, so I left this open.

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  • $\begingroup$ Where did you get this question? Did you come up with it on your own? Just curious... $\endgroup$ – BarbalatsDilemma Mar 9 '17 at 1:30
  • $\begingroup$ I am in a hostel and wanted to determine how much I need to pay for the laundry. I also remember a similar problem where I wanted to determine whether my bag is too heavy for the airplane baggage. $\endgroup$ – donquixote Mar 9 '17 at 2:17
  • $\begingroup$ I originally posted on physics.stackexchange.com where they said it's off topic. So I went to engineering. Now two downvotes. Great :( $\endgroup$ – donquixote Mar 9 '17 at 2:18
  • $\begingroup$ The scenario I describe sounds quite specific. But the pattern of a "broomstick scales" could be easily applied to similar, other materials. $\endgroup$ – donquixote Mar 9 '17 at 2:21
  • $\begingroup$ If you want to weigh you clothes for laundry charges, why not buy a spring balance & use the calculator feature on your smart phone to add up the weight? Amazon has some for $7 $\endgroup$ – Fred Mar 9 '17 at 3:03
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There is more than one possible configuration.

I am going to describe one that seems to be the most simple and is painfully obvious. It is really quite "duh", I don't know why I did not come up with this myself.

Here it is:

  1. Balance the broom over the chair, without any weights on either side. Mark the center of gravity (the point where it touches the chair), to make sure it does not move around.
  2. Hang on the stick: the bag of clothes on one side of the chair, the counterweight on the other side of the chair, in a way that the stick stays balanced. (The point where the broom touches the chair must remain the same as before!)
  3. Compare (*) the distance of each item to the center of gravity (the point where the broom touches the chair). The ratio of these two distances is the inverse ratio of the two weights.

This means if the distance of the 1kg counter-weight is 3x the distance of the bag of clothes, then the bag of clothes weighs 3x1kg = 3kg.

The weight and shape of the broom are really irrelevant in this scenario. So it does not matter if the broom end is removed or not.

(*) To measure the ratio of the two distances, one could of course use a tape measure. But I am sure there are other ways to do this. Remember, we don't need any absolute length, only the ratio.

One part of this can be to have a picture of a tape measure (or some kind of pixel grid with numbers on it), on your smartphone. Since absolute distances are irrelevant, it does not matter if these are real centimeters or inches.

Explanation

The two objects add torque in opposite directions, proportional to distance x weight (gravitational force). The sum of all torques (negative plus positive) must be zero, if the broom is in balance.

The sum of torque of the broom itself is already zero, because it already was in balance before.

https://en.wikipedia.org/wiki/Torque (torque = rotational force)

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