How do we measure mass?

Question

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)!

Answer 1

We almost always determine mass by measuring weight. Weight is the force on an object exterted by a gravitational field, and is proportional to the mass. On the Earth's surface W = m*g. We can convert weight to mass if our measuring scale is calibrated, usually with an object of known mass. This would work for any planet.

Even though the weight would change from planet to planet, the mass would not. But we would need to bring along a known mass to calibrate our scale on the new planet. At the moment, the kilogram, the unit of mass is defined in terms of an object stored in a vault in France. Every mass-measuring instrumenthas been at least indirectly compared to this standard kilogram.

The verisatium video talks about a new method proposed to define the kilogram, which depends on counting the number of atoms in an object. The mass of one atom is known, so simple multiplation will give you the mass of an object. This will really only give you a new way to define a standard kilogram (if for instance, the present kilogram was destroyed).

Answer 2

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.

• Mass-on-a-spring harmonic oscillator. $\omega = \sqrt{\frac{k}{m}}$ with known spring constant.
• Measure the centripetal force on a centrifuge. $F = m \frac{v^2}{r} = m \omega^2 r$, is the naive approach, but on the surface of the planet you have to be a little more clever (adding the centripetal force to the existing weight). Here you would put a scale between the test mass and the centrifuge to get $F$.

An alternative is to measure both the weight and the local value of $g$, which can be done with a small-angle pendulum ($\omega = \sqrt{g/\ell}$).

Answer 3

I would maintain that we most often explicitly measure mass by comparing the pull of the local (unknown) gravity on the mass to be measured to the pull of the same gravity on a known, reference mass or masses.

If you look at the measuring devices in stores, you sometimes see the slogan, "Honest weight; no springs!" They are proudly claiming that they are not using a direct, force measuring device, but rather a force comparing system.

A traditional chemists scale: uses an equal arm balance and a set of various masses to achieve balance between the object on one side, and the collection of calibrated masses on the other side. A suspicious individual can reverse the position of object and masses to check the geometry (Is it really equal arm?). This device will give exactly the same result anywhere on earth, on the moon,or on an accelerating elevator. Keeping the reference masses uncontaminated is a requirement...

The mass measuring device you encounter in a clinical setting: differs slightly in one detail. It relies on a small number of reference masses, but uses the geometry of the bars on which the masses slide (and the notches on the bars) to create the different forces that balance the force of gravity on the subject. Again, contamination of the masses would be a problem, as would be wear and tear on the notches on the bars...

In the absence of gravity, this method:

https://www.youtube.com/watch?v=oU3pp_4n84U

can be used to compare a known mass(the saddle alone) to an unknown mass(saddle plus astronaut) This compensates for the possible error caused by changes in the spring...