How do you measure mass? Weight is easy using a scale, but we can't measure mass that way, because then mass would be different on every planet. I know there was a Veritasium video (here) on defining what, exactly, one kilogram was, but they can only define that if they know some previous measurement (i.e., one cube of metal is 2kg)!
[Physics] How do we measure mass
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You ask, "How do we describe mass to the aliens, who don't know about our (g)?" This is an example of a class of questions referred to by Martin Gardner as "Ozma problems." The classic Ozma problem is how we describe to aliens the distinction between right and left; the answer is that we do it by describing the weak nuclear force.
Your statement of your Ozma problem seems a little ambiguous to me. Essentially you're asking how we describe to the aliens a unit of gravitational mass. (You don't say so explicitly, but it seems clear from context that you don't mean inertial mass.) Futhermore, there is a distinction bewteen active gravitational mass (the ability to create spacetime curvature) and passive gravitational mass (what we measure with a balance). Not only that, but your question could be interpreted as asking whether we can compare with the aliens and see whether the value of the gravitational constant $G$ is the same in their region of spacetime as it is in ours.
We can easily establish 1 g as a unit of inertial mass. For example, we can say that it's the inertia of a certain number of carbon-12 atoms.
The equivalence principle holds for us, so presumably it holds in experiments done by the aliens as well. This establishes that our 1 g unit of inertial mass can also be used as a unit for the passive gravitational mass of test particles.
You didn't ask about active gravitational mass, but the equivalence of active and passive gravitational mass is required by conservation of momentum, and has also been verified empirically in Kreuzer 1968. Cf. Will 1976 and Bartlett 1986.
The other issue is whether $G$ is the same for the aliens as for us. Duff 2002 has an explanation of the fact that it is impossible to test whether unitful constants vary between one region of spacetime and another. However, there are various unitless constants that involve $G$, such as the ratio of the mass of the electron to the Planck mass.
A more fundamental difficulty in the fundamental definition of mass is that general relativity doesn't seem to offer any way of defining a conserved, global, scalar measure of mass-energy. See, e.g., MTW, p. 457
Bartlett, Phys. Rev. Lett. 57 (1986) 21.
Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093
Kreuzer, Phys. Rev. 169 (1968) 1007
MTW: Misner, Thorne, and Wheeler, Gravitation, 1973.
Will, “Active mass in relativistic gravity: Theoretical interpretation of the Kreuzer experiment,” Ap. J. 204 (1976) 234, available online at adsabs. harvard.edu.
I'm not sure that it makes sense to try to measure your body weight to a precision of 100 g. For example I was just thirsty and drank a 20 ounce bottle of water, which transferred about 600 g extra mass to my stomach. Even just breathing changes your mass: if you take ten half-liter breaths per minute and your exhalations contain 5% carbon dioxide by volume, that's a mass loss of tens of grams of carbon per hour. (Moisture is probably a bigger effect there, too.) To measure a 50–100 kg mass to a precision of 0.1 kg is a fractional uncertainty of $10^{-4}$, which is about two orders of magnitude better precision than most college-course laboratory experiments. Furthermore you would expect to see changes of several hundreds of grams over the course of the day (which would be interesting, which is maybe why you're asking).
You won't in general find $10^{-4}$ precision in cheap consumer electronics. I'd expect a bathroom scale to have an absolute precision of 1%–5%, or one to seven pounds for a 150 pound person, with poorer quality loosely associated with cheaper scales.
If what you want is a well-calibrated absolute weight with three or four significant figures, the best setup for you is going to be a cantilever system with well-calibrated reference weights. That's what's at your doctor's office — sorry. If it weren't the most cost-effective way to get a reasonably accurate weight, then doctors would buy something else.
If, on the other hand, you're interested in seeing kilogram-level changes in your weight with sub-kilogram precision, you might not need the absolute weight after all. If you can convince yourself that your scale is linear for small deviations from your weight, then maybe you can take your scale's last two digits, the kilogram and decigram digits, at face value. Here's one way you could test that:
Get several similar-but-different sized weights, about the size of the mass differences you're hoping to measure. Brick pieces might work. Label them somehow: A, B, C, etc.
Put a base load on the scale so that it reads somewhere near the value that you're interested in. For instance, if you're weighing yourself, you could stand on the scale and have someone help you with the next steps.
One at a time, add your test weights to the scale. Each one will increment the reading on the scale by some amount. You'll make a data table like
reading load ------- ----- 80.0 kg just you 81.2 kg you + A 82.2 kg you + A + Band so on. From this you can find the mass of each little weight. (This is how veterinarians weigh stubborn cats, but they do it one cat at a time).
Now repeat the measurements with the same base, but with the other weights in a different order:
reading load ------- ---- 95.2 kg you plus all your weights 80.2 kg just you 81.1 kg you + B 82.3 kg you + B + A
There are a couple of things that you might learn from this procedure:
One is the random error inherent in each scale. For instance, I made the two "just you" weights different in the last digit. That's not unreasonable: essentially all digital readouts have what's called a "Schmidt trigger" that puts some hysteresis in the last digit, so that it doesn't flicker between adjacent values; however that means that the uncertainty in the last digit of a digital readout is at least $\pm1$ in the final digit.
You might also find that the same brick fragment C reliably takes the scale from 80.0 kg to 82.0 kg with one base load, but from 95.0 kg to 97.2 kg with another base load. That would mean that your scale is "nonlinear," since the same increase in signal gives a different increase in output starting from a different place. You'd have to decide how much this bothers you, if you find it.
This technique doesn't address the question of stability: presumably you're interested in measuring your weight over many days. I'd suggest essentially the same test for measuring the stability of your scale(s): find an inert weight that's close enough to your body weight that you expect the scale to be linear for nearby values, and compare your weight on the scale to that rather than simply to the reading of the scale. Depending on the precision you're interested in, you may still have some strange stuff happen. For instance some electronics will respond differently in humid weather than in dry weather; also some weights will absorb moisture from the air and have different masses in humid weather than in dry weather.
As the saying goes: quick, cheap, or correct, pick any two. You're not going to be able to get the precision that you want without some expenditure of money or time, but you can perhaps get the result that you want a little easier.
Best Answer
You measure mass by observing it's acceleration response to force (i.e by applying Newton's second law).
Now, because it is impractical to accurately measure straight-line accelerations over a wide range, we actually use periodic motions and measure frequency.
An alternative is to measure both the weight and the local value of $g$, which can be done with a small-angle pendulum ($\omega = \sqrt{g/\ell}$).