# Is it possible to measure inertial mass without gravity or rolling?

Is there a reliable (and ideally, easy to construct+tune) method of measuring inertial mass without using gravity (e.g. weighing on a scale or a freefall apparatus or a pendulum) or rolling (e.g. pushing an object and seeing how fast it moves horizontally) for sphere-shaped objects? The rolling requirement is because the spheres I'm using will have non-uniformly distributed mass (some may be hollow, some may be hollow, some may be solid, and some may have masses symmetrically (dodecahedrally or icosahedrally arranged weights) placed not at the center but not at the outside (effectively comprising a shell of high mass between two lower-mass volumes when viewed from the radius out.)

I'm attempting to compare the inertial mass of each sphere (same radius and same gravitational mass for each) while excluding gravitational effects (gravitational mass and moment of inertia) from the experiment.

• I am not 100% but it seems equivalence principle indirectly states that gravitational mass and inertial mass are both same/identical. Otherwise, if you were accelerated with same value as g, you would feel a different weight in two cases - gravity of g, and acceleration of g. Thus you would be able to differentiate between the two, which would be violation of the principle. Based upon this, if you have gravitational mass, you also have inertial mass. – kpv Feb 13 '18 at 16:51
• @kpv This is correct, but I want to observe it. – CoryG Feb 13 '18 at 16:52
• You can lift them in accelerated way. The force in excess of their weight, and acceleration imparted should give you inertial mass. – kpv Feb 13 '18 at 16:58
• Lifting would use gravity. – CoryG Feb 13 '18 at 16:58
• Use centrifuge with a force transducer in the arm to measure the radial centrifugal force. Then m = F/ (w^2 r) – Charles Bretana Feb 13 '18 at 21:56

You could create an relative measure or scale of mass using springs. You could time the period of oscillation for a mass that's attached to a spring.

The time period of oscillation is proportional to $\sqrt{k\over m}$.

This is how they measure mass in microgravity environment.

• That seems really difficult to engineer over a horizontal axis without friction becoming the dominant force. – CoryG Feb 13 '18 at 16:39
• @CoryG you could do it using air tracks a lot of experiments of these nature are done using those . The air tracks have very little friction(essentially just air friction) which can be ignored .And they come with a very well machined surface. There are ones which are for 1 and 2 dimensions(it's sorta like a ice hockey table but much much more sophisticated) . Again these kind of equipment is Very common for these experiments . – Shirish Srivastava Feb 13 '18 at 16:52
• That still seems difficult to engineer (difficult to engineer and extremely high cost are interchangeable terms.) I'm trying to find something I can put together on a budget of a few hundred dollars for the whole thing. – CoryG Feb 13 '18 at 16:56
• @CoryG If you hang the spring vertically, the gravitational effects end up canceling out and not affecting the period of oscillation. – Acccumulation Feb 13 '18 at 17:20
• @Acccumulation This definitely isn't correct, the period of oscillation would change under different levels of gravity and mass relative to the tension of the spring, the purpose is to measure inertial mass independent of gravitational mass then compare, not to measure gravitational mass twice and call them the same from that. – CoryG Feb 13 '18 at 17:27

Simply rolling the spheres with the motive force applied at differing radii would allow you to compensate for differing moments of inertia. If you push through the center of mass, then the moment of inertia disappears as a factor. Also, suppose you put a known mass on one side of a rod, and an unknown mass on the other. If you can find the center of mass of the resulting system, then you can find the unknown mass.

• The moment of inertia does not disappear for rolling spheres with different density profiles. How would you determine the center of mass in the pendulum version without gravity? – CoryG Feb 13 '18 at 17:36

Not sure if I am understanding you correctly, it is not like you necessarily don't want to use gravity right? Measurement just needs to be independent of gravitational mass. You could first weigh the spheres so you know what the gravitational force is. And then measure the time it takes to fall a certain distance.

Then you will know the force, the time gives you acceleration and you can compute inertial mass.

• The issue there is that both measurements are done with gravitational mass. It is believed that gravitational mass and inertial mass are the same but the purpose of the experimental setup is to confirm or deny that (as a complementary experiment to a falling mass as you've described, actually.) Using the force of gravity (via a scale) to weigh the objects then using the force of gravity again (via acceleration of the objects dropping) would always equal the same thing regardless because both measurements rely on gravity for the result. Also, gravity is an acceleration, not a force. – CoryG Feb 13 '18 at 19:37
• I don't see why you are measuring both times the gravitational mass. – Daniel Feb 13 '18 at 19:47
• If you think that gravity is an acceleration then I see why you might think it is measuring the same thing twice. But it isn't. Gravity is a force (in Newtonian world). So if you know the force and you know the acceleration it is producing, it will tell you the inertial mass independently of the gravitational mass.And it is not that gravitational mass is the same as inertial mass, it is that they are proportional to each other. – Daniel Feb 13 '18 at 19:51
• The mass on the scale is interpreted from the force felt by gravity (acceleration) pulling the gravitational mass (e.g. F=ma then implicitly taking out a.) The time it takes them to fall a fixed distance is going to be the same regardless of mass (assuming negligible surface area and in turn air resistance) but is again determined by the acceleration of gravity. – CoryG Feb 13 '18 at 19:52
• Gravity as we measure it is an acceleration, gravity in theory is a force. Measurements trump theory in reality and in speech as far as I'm concerned. – CoryG Feb 13 '18 at 19:54

You can measure inertial mass by using newtons equation or equivalence principle. Here is an example that does not use gravity. About one minute and 20 seconds into the video you will see. https://www.youtube.com/watch?v=eU2hGIrBULA&sns=em

Similar to Charles Bretana's suggestion, you can use a circular plate with a wall around the edge. Perhaps use two spheres at once. Position the spheres against the wall, on opposite sides to provide balance, and restrict tangential movement of the spheres relative to the plate. Now rotate the plate and measure the force against the wall from the sphere. Also measure the rotation rate and radial distance of the center of the sphere. Perhaps a modified record turntable will work. I don't know the cheapest way to measure the force against wall.