For anything we deal with on a day-to-day basis, mass will be pretty close to being proportional (within about 1%) to the total number of neutrons + protons. But that’s not inherently what mass is, for a few reasons. First, neutrons and protons have slightly different masses (the neutron is a little bit heavier). Second, there’s also binding energy, which differs from one nucleus to another, and which also accounts for some mass (this is the difference in mass that nuclear energy generation, controlled or in weapons, is based on). Third, not all matter is made up of protons and neutrons, and in fact, most of the mass in the Universe is made out of something else (though we don’t yet know what).
jnglmassiv, my off-the-shelf bathroom scale would work just fine in a vacuum. It contains no LCDs, no capacitors, nor anything electronic at all: It’s just a spring connected mechanically to a moving dial. Of course, I’m sure that it has other inaccuracies which dwarf those caused by buoyancy, but if you care about that much accuracy, you’re going to have to go with some sort of specialized instrument anyway. And the fact that there are easier ways to deal with the buoyancy problem (like assuming a density and calibrating the scale based on that) doesn’t change the fact that there’s also a way of dealing with it that is conceptually quite simple, and in practice still not all that hard.
The error due to buoyancy of feathers is less than one might think. For the purposes of assessing buoyancy, what is the density of feathers? Pretty high, actually. Feathers have a very loose structure which confers a lot of aerodynamic drag compared to their weight, but the gaps between feather elements are filled with air, which has a net zero effect on buoyancy.
So what is the density of the actual material from which feathers are made? Feathers are modified hairs; both are made of keratin. Couldn’t quickly find a direct reference for the density of keratin, but according to this PDF, wool fibers have a density of 1.3 grams per cc, which is probably a good estimate for the keratin from which feathers are made. Compare with lead at 11.34 g/cc, and sea-level air at 0.001225 g/cc.
So if we’re using a scale to measure the weight of a 1000-pound mass of lead in open sea-level atmosphere, buoyancy will cause the scale to indicate 999.89 pounds. If we’re using a scale to measure the weight of a 1000-pound mass of feathers, the scale will show 999.06 pounds.
To compensate for buoyancy: Put the object to be weighed in a sealed container and weigh. Take the object out of the sealed container and weigh the sealed (but now empty) container. Subtract the second measurement from the first and the result is the actual weight of the object (within the limits of accuracy and repeatability of your measuring system). No need to consider the volume or density of the object, and it even works for lighter-than-air objects!
Note that gravitation is also different around the earth and based on altitude.
The force calculated would also change based on the local acceleration due to gravity. For anything from about feathers and denser the local variation in gravity will actually exceed that of buoyancy. For stainless steel the local gravitational variance will be around 6-9 time greater than the displacement of air near sea level.
Once again, if you consider mass as the property that is the resistance to changes in movement and weight as the product of mass times the local acceleration of gravity this becomes less of an issue.
As force is a vector the tensor form would allow you to break out most of theses functions into separate elements as buoyancy acts on a different element than gravitational acceleration.
Where you do need to density is if your precision needs are to the point where the fact that gravitationally bound systems, or chemically bound systems have more negative potential energy than the sum of its parts. The definitions of standard gravity, rest mass and metric form of weight help you work with this.
Mass-energy equivalence shows that mass is concentrated energy, and even if you consider the percentage of the mass of the proton that we know
33% quark energy
37% glue field energy
9% u,d, and s quark scalar condensates.
Just consider rest mass as he sum of all energy bound in a system.
You can keep weight simple by just stating it is the product of mass and the acceleration due to gravity or you can complicate it by trying to go with what seems like a simpler definition.
But due to the quirks of Newtonian physics being a 3D bundle over a 1D manifold the energy from Gravity in the Newtonian model won’t be conserved in momentum so you will have to resort to tensors anyway if your needs are for this level of accuracy in a dynamical system. You will end up using the rest mass form for one of your vectors anyway.
Rest mass, fundamentally, a function of all energy, of all types contained in a closed system. Rest mass, fundamentally, is a measure of the property of inertia of an object. Adding or removing energy by either re-arranging the components or adding or removing any energy will change this measurement. A compressed spring has more mass than a relaxed one, a warm object has more mass than a cold one, and a group of objects tightly packed have less mass than when they are spread far apart if the “object” you are looking at contains all of those parts.
Don’t let superseded theories or non-technical use of the terms confuse this point which does match with our current understanding.
If you care about what mass is fundamentally only use the technical form; Mass is a property of a physical body and a measure of its resistance to acceleration. This property is due to the total energy content of the system.
It is only our assumptions, intuitions and artifacts of our educational system that make this hard to grasp for adults.
The entire concept of “Apparent Weight” is actually quite toxic to our educational efforts.
The operational vs gravitational definitions of weight will never be resolved, as teachers are unwilling to accept the gravitational version as a standard, abandoning the word is the only option.
This only works if you always have the same amount of air in the container, with and without the object (I’m guessing you meant none). That does mean that you don’t need to put the scale itself in a vacuum chamber, but you do still need a vacuum chamber.
Machine Elf, in practice, the error would be even smaller, and might be in the opposite direction, because the scale was probably calibrated in an atmosphere. If the scale’s response is linear, for instance, you could calibrate it by taking all weight off of it, marking the needle’s position as “0”, then putting a 1000-kg test mass on it, and marking the needle’s position as “1000”, and then dividing the interval between those two marks into 1000 equal parts. But that test mass that you used would also have some buoyancy. If the test mass happened to be made out of lead (or something with the same density as lead), then the scale would read exactly 1000 kg when it had 1000 kg of lead on it, because that’s what it was calibrated to read. And if the test mass used for calibration was less dense than lead, then the lead would actually read slightly more than 1000 kg on the scale’s display.
While it’s nice to understand that there is a small difference between weighing something in a vacuum and weighing it in air, in practice, 99.99999999999% of measurements are made in air. I don’t know of anyone who steps onto their bathroom scale in a vacuum.
As Chronos pointed out the precision is what matters, the precision you require will dictate the corrections you have to make.
As this thread is about what the properties are and not what convenient ways we have to approximate them, the answer is.
mass=Force/acceleration
Weight = mass * acceleration due to gravity.
The operational definition of weight is problematic because it hides those relations due to assumptions that may hold fine in our daily lives but which do not describe what these fundamental properties are.
Note that the simple forms of Newtonian mechanics only hold in a inertial frame. We simply do not exist in an inertial frame, and the proper way of dealing with this reality is just accepting that and not trying to change those relations to fit our non-inertial existence at the expense of the core concepts.
There is a direct and fundamental relationship between the above forms of both mass and weight. But this is because in this context Weight = mass acceleration due to gravity. It doesn’t matter if that same acceleration is provided by a rocket or gravity. That force vector in “F = ma” doesn’t depend on committing the act of weighing, and that force exists external of any observers decision to measure that force.
This force as only a measure of the mass times the intensity of acceleration and in some cases this is a gravity field. The fact that people don’t get that Weight = mass * acceleration is just Force = mass * acceleration with a named type of acceleration is why the concept is confusing.
The problem is not with the physics, it is merely an artifact of people refusing to accept a common definition.
If you want to account for the* buoyancy force* you simply need to add another vector defined off of that force. There is no value in co-opting the specially defined force for gravity called weight to try and make it fit in with observations so you can ignore the other components that combine into the overall force.
The strength of acceleration for gravity changes on the earth and on other bodies, but it is not useful as a precision measure of force if you need to account for other forces within a system.
You simply don’t change the core ideas to match up with observations in a frame that needs many other forces to be accounted for if the needed level of precision causes those forces to be significant to your answer.
To repeat again, these concepts are not ambiguous, it is only the changing scope that makes them so.
mass=Force/acceleration
Weight = mass * acceleration due to gravity.
Irrespective of some practical uses without high precision requirements, if those relationships aren’t concrete in your understanding it is probably best to revisit those concepts. Just remember to ignore your middle school teachers who were lying to you, because they thought you were not as smart as you actually are.
It’s not entirely theoretical though. Some years ago I attended a course in flow measurement at NEL’s Densitometer Calibration Facility in East Kilbride. When measuring weight, they not only compensated for buoyancy but also local gravity. They had had a man come up from London with an official gravity-measuring device (gravimeter).
The trainer admitted that he had been disappointed with the gravity measurement. He expected some big and elaborate oscillating mechanism that would need to be carefully set up and solemnly observed for a period of time before the result would be announced. Instead, the instrument was taken out of its box, a button was pressed and a small click was heard, and that was that. Very anticlimactic.
Your bathroom scale is badly obsolete. I’ve weighed myself on a scale accurate to 0.1 grams. This is useful for those days when you really want to see how big your lunch was or when you can’t remember if you had one aspirin tablet or two.
It actually wasn’t all that long ago that mechanical balances were dominant and there are still plenty of them in use. Fascinating contraptions, zilloins of moving parts, complicated mirror and lens projection patterns. Devilish to try to work on, though.
If we’re going to be snarky, at least be correctly snarky: It’s a service where communion is consecrated. On Good Friday, for instance, most Catholic churches will have a service where communion is distributed, but it’s all been consecrated on a previous day, and that service is hence not a Mass.
If one wants to open another can of worms, we once had to consider two forms of mass: inertial and gravitational. The first gives rise to matter’s resistance to change in motion, and the other caused things to attract one another. Everyone knew they always seemed to be the same, but careful thinkers tried to keep them separate.
But Einstein settled that when his theory showed them to not just be equal, but different effects of the same fundamental property. And that the property is also energy.
Kind of disappointed to not be reading some explanation about how mass is a property imbued by interaction of fundamental particles with the Higgs field … c’mon nerds! Edify!
Exactly. Near the surface of the Earth, weight is defined as: w = mg,
where: w = weight (i.e. the magnitude of the attractive force exerted by the Earth on the object, aka the force of gravity) m = mass of the object g = acceleration of gravity near the surface of the Earth (9.80 m/s[sup]2[/sup])
Buoyancy is irrelevant in this definition.
I disagree. When I was teaching this material, I never had too much difficulty simply defining [true] weight as w = mg, and only bringing up the idea of “apparent weight” when discussing what a bathroom scale would actually read depending on the situation at hand (e.g. taking buoyancy into account, or an object in an elevator that is accelerating up or down, or in free fall, etc.).
Nitpick…your scale may be precise to 0.1 grams, but you really don’t know anything about its accuracy unless it’s been calibrated recently with several known masses.
Except if you are sticking with pure Newtonian (which orbital mechanics doesn’t) w = mg even holds for people orbiting the earth in the ISS.
While the Newtonian model starts to fail in general here the “continuous falling” weight or force still works out. It is the operational or apparent weight model breaks down. In fact if you calculate the local acceleration to gravity and correct to standard gravity in the metric system their “weight” will be similar.
As we use the Post-Newtonian formalism to keep the math simple this is the whole reason astronauts and others try to explain “micro-gravity” in the first place. If we were working under a General Relativistic model where it is not a force but a result of two objects changing the geometry between to objects this model wouldn’t hold but it does under Newtonian or more correctly Hamiltonian mechanics.
This is the same issue with other fictional or pseudo forces, but with the earth as a frame and the common analogy with continuous free fall, I would argue your response demonstrated my claim.
It is hard enough to even consider bring T[sup]00[/sup] = ρ, into the discussion or even considering that gravity is not a force that interacts with objects, but the space between them is way past the simpler m = F/a model.
The Higgs field, Yukawa interaction, and spontaneous symmetry breaking are covered in some links I provided earlier in the thread.
As gravity is being treated as a force in this thread and by common every day use there isn’t a lot of value in discussing more realistic models.
