# What's the simplest way to measure mass without using gravity?

I'm putting together some rough ideas for physical experiment designs and datasets, relating to multiple hypotheses for quantum gravity.

Because I'm working on experiments to test theories of gravity, I need to be able to reliably measure the mass of physical objects, without relying on gravity for the measurement.
(for experimental control)

So, for example, I can't use scales or balances.

My first instinct is to use force or momentum/inertia, and rearrange the equations to solve for mass.
Force: F=mam=F/a
Momentum: p=mvm=p/v

But these would require me to measure either force or momentum, again, without relying on gravity to make the measurement.

Maybe it's staring me right in the face, but I'm drawing a blank.

What are some simple methods of measuring either mass, force, or momentum that don't rely on gravity?

• Why can't you measure force without relaying on gravity? Take a spring of constant $k$, pull it and measure the displacement. – eranreches Jun 4 at 23:58
• @eranreches I like the idea of a measurable spring. It has good potential. But I suppose it depends on how the constant k is derived. I need to ensure gravity is not part of the equation anywhere. – Compelling Argument Jun 5 at 0:05
• It can't depend on gravity. Springs exist even in a world without gravitation. Your spring constant is related to the Young modulus of the material with ultimately depends on microscopic properties. If the mass of the atoms appear there, it can be measured using mass spectroscopy techniques. – eranreches Jun 5 at 0:06
• @eranreches Ah, interesting. I'll do some research into that, and see what types of predictable springs I can come up with on a low budget. Thanks for the quick tip. – Compelling Argument Jun 5 at 0:08
• You can also measure the speed of sound waves in the material. It is given by $v=\sqrt{\frac{K}{\rho}}$ where $K$ is the bulk modulus and $\rho$ is the density. The bulk modulus can be meausered using powder diffraction methods (see measurement section here en.m.wikipedia.org/wiki/Bulk_modulus), and the volume is trivial to measure. – eranreches Jun 5 at 0:32

If your masses have a knowable centre, for instance a cube or a sphere, then attach them to a rigid rod. Pivot the rod perpendicular to its length - eg. horizontal rod, vertical axis of rotation. Spin it rapidly.

For example: if the point at which the rod is pivoted is exactly half way between the centres of the two masses, you will get a vibration if the masses are unequal and smooth operation if they are equal. You will be able to hear the difference.

Other ratios - if the pivoting point is $$\frac 2 3$$ of the way from one mass to the other, you will get smooth running when the ratio of masses is $$2:1$$.

One can say that the following is just a modification of the spring approach proposed in comments, but maybe the use of a gas with well-known properties can be a strong point.

Let us consider a (horizontal cylindrical) vessel filled with gas and separated by a horizontally moving piston that is in equilibrium as the gas pressure is equal at both sides of the piston. Then the period of small oscillations of the piston will depend on its mass.

You can have a fixed body (attached to a wall or something). Give this body a known amount of charge.

Give the body you want to measure the mass of, a known amount of opposite charge.

Now you do not need to measure the force, you know it from the amount of charges and distance between them. Measure the acceleration, and you can find the mass.

Now you can say that fixed wall requires gravity. But that would be true in case of a spring too as you need one end of spring fixed.