# Where on Earth does the mass of 1 kg actually produce a 1 kg reading on a digital scale?

Gravity on Earth varies by about 0.1% between poles and equator. If someone was buying/selling something mass critical e.g. gold, where is the standard place on Earth where a 1 kg mass produces a 1 kg reading as measured by a device like the following:

Based on some responses, I should give a very specific example. We take the International Prototype Kilogram mass from its repository in Paris. We then take it to one of the poles and measure it using this type of device (also shown above). We then take it to the equator and do the measurement again. The numbers are different.

Is there a place on Earth specified where the scales will read exactly 1kg?

• Nov 18 '14 at 21:20
• quora.com/… Nov 18 '14 at 23:30
• Comments are not for extended discussion; this conversation has been moved to chat. Nov 19 '14 at 21:36

EDIT: Though most of the comments have now been deleted, I state the following for completeness:

The kg is a unit of mass. Weight is a force, and like other forces it is measured (in the SI system) in Newtons. The weight of a body is given by the equation f=ma, where a is acceleration (in this case the local acceleration due to gravity.) Therefore we can rewrite this as f=mg

In the SI system, to say that something weighs 1kg is nonsense, because weight is a force, and should be quoted in units of force.

Nevertheless, there is a standard value for standard gravity g.

According to http://en.wikipedia.org/wiki/Standard_gravity:

9.80665 m/s², which is exactly 35.30394 (km/h)/s (about 32.174 ft/s², or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration.

According to the same reference, it is chosen as the gravitational acceleration at a latitude of 45 degrees at sea level.

This is of course absurdly precise given the global variation of g but there has to be an accepted standard value. It's also arguably not the most useful definition, since the majority of the earth's surface is below that latitude and of so it's likely to be slightly higher than what is needed in practice. If I had been given the opportunity to pick the value, I would have picked 9.8m/s² as a nice round number. http://en.wikipedia.org/wiki/Gravity_of_Earth shows that not just latitude but also continental landmasses influence the local gravitational acceleration.

So a 1kg mass weighs 9.80665N at standard conditions.

There is a unit called kgf. This unit is deprecated, because it complicates all kinds of calculations. Nevertheless it is an easy way for most ordinary people to visualise force, and like it or not, this unit is used in the real world. 1kgf = 9.80665N. If you buy a rope or fishing line from a local store, its breaking tension will likely be stated in kgf.

Let's look at some widely used units of pressure: Pa, Bar, atm, kgf/cm², mmHg, mmH₂O. The first two are based on the SI system. Numbers 2-4 (Bar, atm, kgf/cm²) are within a few percent of each other but all are widely used. The last three on the list all depend on the value of standard gravity in their definition. One reason for the perpetuation of units like mmH₂O is that they are easy to measure directly, for example in a water manometer. Another reason is that they simplify certain specific calculations, such as the design of water distribution systems.

When it comes to quoting quantities of gas by volume, the situation is even worse. People do measure gas in this way and it is important to know what temperature and pressure they are considering. If everyone could measure all quantities of substance in kg (mass!) and avoid talking about weight and volume, everything would be a lot less ambiguous.

• 45° is conveniently close to Paris, so not all that absurd ;-) Jul 8 '16 at 16:47

Short answer: wherever it is calibrated to be so.

The kilogram is a unit of mass.

Weight is the force of gravity on a particular mass.

It is convenient to use "kilogram" rather than "Newtons" when you are more interested in the quantity of "stuff" than the force of that stuff - and use the Newton when the force matters.

For example - an elevator manufacturer should probably quote the maximum load of the elevator car on the engineering drawings in Newtons - the critical thing is that the cables should not break(*); a chemist wants to calibrate his scales in (kilo)grams since he actually wants to know how much of a compound he is working with.

Conventionally, weights evolved in trade - "amount of stuff". Elevators came later. People therefore prefer to say "that weighs X kg".

Now the force of gravity varies by about 0.7% from the North Pole to the mountains of Peru. If you care about the measurement of a mass to better than that accuracy, there is only one remedy: calibration.

Even "cheap" digital scales (I am talking about the 20 gram scales that resolve a mg - you can get them on Amazon for about 30 dollars) include a couple of calibration weights - one for the mid range, and one for the full scale deflection.

Whenever you move your scales, change the batteries, etc, you are supposed to run a quick calibration - measuring the force of gravity on a known mass, you can now measure an unknown mass and state confidently "the force on this one was 0.9744x of the force on a 10 gram test mass; we will call it 9.744 gram".

The manufacturer may choose to send the scales out already calibrated - but frankly, for these accurate scales, just moving to the top floor of a sky scraper will cause a measurable shift in the calibration.

You might find this earlier answer or this one relevant / familiar...

And finally, following this other answer you can compute the fractional change in gravity with height (without considering rotation) as

$$\frac{\Delta g}{g} = \frac{2h}{R_e}$$

Which tells me that on the top of One World Trade Center (1776 feet tall - yes, that's symbolic; the conversion to 541 m isn't nearly as much fun) your balance will read $4\times 10^{-5}$ lower... that's just about enough to make 20.000 g read as 19.999 on my cheap Amazon scales.

(*) note they may choose to label the car in "persons", "kilos", "pounds" - whatever is most likely to prevent an accident. Accidents are bad publicity for elevator manufacturers.

1) if you used scales where one mass balances another mass like this one

Then you would not have problems with any variation in $g$.

2) I did a google search to check what people use to measure the mass of gold (and also diamond) and everything came back as digital scales like the one shown in the question.... seems like noone uses the old type shown in point 1) above any more.

These digital scales basically measure the force $F$ required to counteract the weight force when used properly. From a measurement of the force, the scale then converts this to a mass measurement using some conversion akin to $m=F/g$. On different places on Earth, you'll get different "mass" measurements since these devices use a single value for $g$.

The answer to your question as to where on Earth will we get the standard value is: It's up to the manufacturer. Take a look at slide 19 of this link. This company chooses the latitude and altitude of own laboratory to calibrate their mass scale.

Now, as a separate issue, you might also be wondering where on Earth local $g$ takes on the value of $9.80665~\text{m/s}^2$, which is the defined value according to Wikipedia. This value occurs at a latitude of about 45 degrees.

If you're interested in the value at sea level for other latitudes, which you might want for calibration purposes, Wikipedia tells me that

$$g=9.780327\left(1+0.0053024\sin^2\phi-0.0000058\sin^2(2\phi)\right)\ \text{ms}^{-2}.$$

The answer to the question is both yes and no.

Yes, the scales will display 1kg when calibrated in the same gravity zone. Otherwise, no it won´t.

The force of gravity varies simply because of the shape of the earth, and its rotation (speed). On both poles the speed is next to nothing, yet in "Middle Earth" velocity is at its highest. So, the actual shape of the globe might be round, but spinning around with all that mass makes it oval(ish). Therefore, the globe was divided in separate gravity zones, each with their different gravities. A scale will display the right weight in its own zone only!

If the scale is correctly calibrated, it will measure 1kg everywhere the variation of weight is lower than the resolution of the scale. For example, París.

Of course, this neglects atmospheric pressure and temperature variations.

In the place where it was calibrated. I'm pretty sure scales are not calibrated to fit the international standard gravity. The lab that put a standard weight on it and set it to show the desired result had the conditions ($g$, air density [related to temperature and pressure], and other less influential parameters) that are required to reproduce the measurement. An accurate scale should be calibrated to work best in the environment they are meant for. So calibrating a scale in vacuum on the south pole isn't the best idea if it'll be used at standard atmospheric pressure in a lab at the equator.

Zero setting (which you must do by yourself anyway) takes care of the initial offset, so differences in $g$ only cause relative error. This is rarely relevant for most of the cases (even in chemistry where precise measurements are important, it's usually just fine), and for specific high-accuracy measurements, the scale is usually not portable and is calibrated where it is used.

Funnily enough, the balancing scale is the one that truly measures mass (by comparison). All other scales measure force and deduce the mass by assuming some value of gravitational acceleration.