I brought an old analytical balance into my kid's elementary school for "Science Day", and I explained the difference in using a scale (measures weight) and the balance (measures mass). I told them that although objects on a scale will weigh different amounts depending on if they are weighed on the Earth or Moon, the balance will work the same. Later that night, I was wondering: If the arms of the balance have unequal masses and the balance follows a geodesic, what will happen if I initially set the arms to be balanced? There is no gravitational force acting on the masses to move the balance arms, so will the balance continue to read incorrectly? I'm assuming I've forgotten something obvious....
If I understand your question correctly you are asking what would happen to a balance in free fall. Well if there are no other forces acting on the balance arm (friction, air resistance etc.) the arms will stay where you position them. This is because the masses are stationary wrt the balance frame of reference and not subject to any outside forces. Another way to look at it is both a scale and a balance require a gravitational field (or an acceleration) to operate. In other words a scale is directly measuring the force on the test mass. The balance is comparing the forces on the two masses. Make sense?
I hear what you are saying, and that makes sense. But what's the principle of the Watt balance then? Is accurate knowledge of 'g' always required to define the Kilogram? http://en.wikipedia.org/wiki/Watt_balance Is there any way to separate measurements of mass and gravity? Is the only correct solution to count atoms?
The error in the kilogram prototype is on the order of a few parts in 10^{8}. The uncertainty in the universal gravitational constant is a lot worse than that, one part in 10^{5}. It is hard to untangle G from mass. We do a lot better measuring the product of G and mass. For example we know GM_{sun} to one part in 10^{10} and GM_{earth} to one part in 50,000,000. No. The Watt balance uses a different approach. It links the kilogram to Plank's constant. From http://www.bipm.org/en/scientific/elec/watt_balance/:
I guess counting atoms would do the trick although I'd be pretty tired after the first 10^20 or so. :^} There's some pretty interesting stuff in the various links on the Watt balance.
Thanks for the link. But again, it seems that what is really measured is 'mg', or at least 'mg' is re-phrased in terms of Planck's constant. I'm not trying to be pedantic here- it's odd for me to think that mass and gravity cannot be easily untangled; no such confusion exists for voltage and charge, for example.
Paw your first response was spot on. Gravity g or G is an effect of masses interacting with space time. The two are not mutually exclusive. To separate the two would be to destroy the very universe itself. To measure a body's mass you have to accelerate it.(push it) Fortunately for us gravity does this constantly. Masses cannot hide from gravity or any acceleration due to any force acting on it and visca-versa. Its a conundrum much like the Uncertainty principal. Evidence is the continuously accelerating expansion of the universe itself, without which acceleration the matter that makes up the universe would not exist nor be measurable. Counting atoms eh? the number of atoms does not constitute a body's mass. Rhetorical questions: How do physicists determine the mass of an individual atom? How did physicists discover the existence of black holes? A very interesting balance is the Cavendish, used to determine the value of G. Incidentally I believe this balance would work in microgravity or free fall because the gravity used is enclosed in the system. I believe it could be adapted to measure the unknown mass of one of the four bodies given the other three.
I don't understand your original question, though it seems something very interesting is being asked, so my resppnse may be very tangential. Anyway, I believe Schutz's text comments that the Schwarzschild mass parameter is determined by "weighing", and there is no integration over any quantity of matter or counting of atoms which is a different "mass". I think Hawking and Ellis also distinguish these different masses in their discussion of gravitational collapse, and this may be one of the reasons their critical density for gravitational collapse is only approximate (a related one being the interpretation of the radial coordinate). I'm not sure I'm remembering or understanding these correctly, but maybe they will be useful references. Try the discussions around Eqn 8.58 and 10.41 http://books.google.com/books?id=qhDFuWbLlgQC
This makes sense: "..because inertial and gravitational masses of all physcial objects are proportional, any variety of test particles or masses will undergo identical accelerations in an ambient gravitational field. Differences in accelerations would be due to differences in the gravitational fieldat their respective locations..." quote(slightly modified) from Peter Bergmann, a student of Einstein, THE RIDDLE OF GRAVITATION
[pedantic] The balance directly compares the forces that the two masses exert on the balance. These forces equal the corresponding forces that the balance exerts on the masses, by Newton's Third Law. [/pedantic]