Does the airplane get lighter after the meal?

Cecil's Columns/Staff Reports
1

http://www.straightdope.com/classics/a3_144.html

So I’m sure this has come up already, but I think I must be missing something. Cecil concedes that some immeasurably small amount of mass would be lost on the plane citing the good old e=mc^2 equation, but this isn’t even applicable, is it? What I mean is, no new energy is being created, its being changed from one state (being stored) to another (heat and further chemical changes). Likewise, no mass is actually being destroyed; it’s being chemically broken apart (which is what is releasing energy) but no mass is actually dissapearing. I must be missing something, or perhaps energy really is being created (rather than transformed/transferred, whatever, through digestion) and I’m just uneducated in the intricacies of science.

2

As the quote in the article said:

“A loss of mass occurs whenever internal energy (nuclear, electrical, chemical, etc.) is converted into energy of motion. Only in the nuclear case is the amount of energy so large that [it results] in an observable change in mass, but in principle E=mc^2 is as descriptive of a chemical explosive, a gasoline engine, or a flying bird [or, I might add, a flying human] as it is of a nuclear explosion.”

Seems pretty conclusive to me.

3

Actually, ThatGuyOverThere is correct. Some of the mass of the food ( a very tiny amount) is indeed converted to other forms of energy (primarily heat), but so long as that heat remains aboard the airplane, it still contributes the same amount to the total mass of the plane. Of course, some heat would be lost through the skin of the plane, but as with Cecil’s air recirculation example, this would occur regardless of whether the meal was served or not. And the mass loss through recirculated air would easily dominate over the loss through heat loss.

4

Eh? Heat has mass? Also, **ThatGuyOverThere **is disputing that any mass is converted to energy at all, in which he *is *incorrect.

5

Yes, heat has mass!

Take N identical particles at 0 Kelvin. Ignoring some tiny quantum effects we can say, that their speed at this temperature is v(n, 0) = 0 and their mass would be m0.

Now we will heat these particles, e.g. through laser light, to some temperature T. This adds energy to those N particles, and their average speed will raise, which is synonym to say that their temperature rises. Therefore we can say v(n,T) > 0 for T > 0.

Now, we can use the formula for relativistic mass: m(v) = m0 / sqrt(1 - v^2/c0^2)

If we replace v with v(n,T), then we can calculate the mass of each particle: m(n, T) = m0 / sqrt(1 - v(n,T)^2/c0^2)

We don’t know the exact speed of each particle, instead we only know v(n,T) > 0 for T > 0, but this is sufficient to conclude: m(n,T) > m0 for T > 0.

And this is nothing else, but to say “heat has mass”.

“E = m c^2” is a fundamental conclusion from special relativity, and says “energy and mass are equivalent”. Hence, as long as no energy from the digestion of meals leaves the plane, the plane will also not loose any mass. (However, this is only true as long, as you do not change the frame of reference, the mass of that plane measured from the ground is larger, than the mass of the very same plane measured from another plane, that follows it.)

Actually the quote is misleading, I bolded my insertion to clarify: “A loss of rest mass occurs whenever internal energy (nuclear, electrical, chemical, etc.) is converted into energy of motion. Only in the nuclear case is the amount of energy so large that [it results] in an observable change in mass, but in principle E=mc^2 is as descriptive of a chemical explosive, a gasoline engine, or a flying bird [or, I might add, a flying human] as it is of a nuclear explosion.”

However, “rest mass” is only of significance if you use the frame of reference, in which the debated mass rests, otherwise you have to use the “relativistic mass”.

cu

6

Actually, it’s common practice nowadays in relativity to use the term “mass” synonymously with “rest mass”. If you ever need to use relativistic mass, you can just call that the total energy. (Rest) mass is specifically the part of the total energy which can’t be transformed away.

The interesting thing about this is that when a system of particles is heated up, no individual particle gains any mass, but the system as a whole does, since you can’t transform away the heat energy by going to a different frame. In fact, you can even have a mass if your system consists entirely of photons, despite the fact that photons themselves have no mass at all.

7

Do you have quote for this? It is my understanding, that it is a statistical effect, that you cannot transform the heat away by going to a different frame, because while you try to transform the heat/cinetic energy of a single particle away, you will actually add energy to other particles due to the very same transformation.

Yet, IANAPhysicist, and ready to learn more about the topic.

cu

8

Yes, this is correct. There is going to be some reference frame in which the energy is a minimum, and any other reference frame will result in a higher total energy for the system. The energy of the system in the minimal-energy frame is the mass.

9

You guys are all missing the obvious here. Of course the plane will be lighter after the meal is served, because all the time that people have been eating, the plane has been cruising along and burning up jet fuel, which is converted to energy and exhausted out of the engines.

10

True - and covered by Cecil.