Is a spinning top heavier?

Factual Questions
1

Not being a physicist, I’m a bit confused about the relation between mass and energy. Of course, we all know that E = mc^2, but does that imply an equality or a conversion?

The question re-arose in my mind due to another thread where someone who seemed to know what he was talking about said that a photon has zero mass, yet is still affected by gravity. Putting that bit aside, I still think that any kind of energy (including kinetic energy) has to have mass, or the equivalent of mass; otherwise we could in theory make a perpetual motion machine.

Back to the spinning top. Say we can easily turn energy into matter and vice versa, with negligible loss in the process. (Yeah, I know, questionable hypothesis. Bear with me for a moment.) Also assume we can raise and lower an object (relative to some massy object, such as the Earth) with negligible energy loss.

Spin a top. Raise it. Convert the kinetic energy into matter. Lower top and created matter, harvesting the energy from the drop.

Convert the matter back into kinetic energy, spinning the top. Zero sum gain there. Use the energy from lowering the top to raising it again. Another zero sum.

But we have the energy from lowering the matter left over. Of course, we also have lost due to any inefficiencies in the processes above, but if we really spin that top, we should be able to get enough energy from lowering the matter to more than compensate.

Of course I don’t believe this could work or that we can violate conservation, so obviously there’s a flaw. The simplest explanation is that the spinning top is heavier, by an amount equal to the kinetic energy of its spinning. But I was told by a physicist who should have known, that that wasn’t true. What is the explanation?

Note that this game can be played a number of different ways, such as beaming energy such as light to a higher altitude and converting it into matter and dropping it. If it’s free to move energy of any kind out of a gravity well, then conservation seems to be broken. (Just as it would be broken if we could raise matter, or spin a top, without expending energy.)

Thanks!

2

Unless there’s some particularly subtle effect that I’m missing, I think your physicist friend is mistaken. Of course, the amount of mass gain for a standard top is minuscule; the fractional gain in the mass-energy of the top is going to be on the order of magnitude of

∆M/M = ( (speed of edge of top)/(speed of light) )[sup]2[/sup]

which is going to be ridiculously small for any reasonable physical top; it would tear itself apart due to centrifugal force well before the edge got up to the speed of light.

3

Yes, in fact the spinning top is ‘heavier’ (for the right definition of ‘heavier’) than the non-spinning top, by an amount equal to its kinetic energy. It’s of course such a tiny amount heavier that it’s essentially undetectable, but it is technically heavier, just as a charged battery is ever so slightly heavier than an uncharged one, but an amount equal to the potential electric energy inside.

4

Light is affected by gravity, so any energy gains you get from dropping matter would be lost when you beam it back up to your orbital matter generators.

5

A spinning top is upright. The center of mass for an upright top is farther from the center of the earth vs. the same top laying on its side. Therefore wouldn’t the former be slightly lighter than the latter?

6

This reminds me of a good thread here once on the weight of the information on the Internet.

7

Since protons weigh so much more than their constituent quarks (Proton - Wikipedia), does that mean the proton is higher in energy? Why is it stable then?

8

Since it is all relative to one’s frame of reference, if one where a point on the spinning top, and the universe suddenly was whirling around one when it wasn’t before, does the whole universe gain mass, and do very distant objects, like stars, exceed the speed of light?

9

Actually, that’s permitted. The Bad Astronomer (bless his little protons!) has said that, yes, the equations can be solved in such a way as to describe a stationary earth, with the cosmos spinning about it…and everything behaves according to our observations. Geostationary satellites stay in orbit…held up by the tug of the mass of the vast whirling cosmos. Since space itself is moving faster than light, and no objects are ever moving faster than light through space, the equations don’t mind.

The same is true for a stationary Mars, or Jupiter, or any object, anywhere, including the little toy top.

(For the top, the experiment fails, because the top slows down, which a whirling cosmos could not. But that’s the “acceleration” thing that changes from SR to GR, and also solves the Twin Paradox.)

Messy!

10

Neither. Mass and energy are not things they’re properties of a system. Mass is equal to the energy of a system that cannot be transformed away, and mass cannot be converted to energy.

11

Thanks! I’ll have to wrap my mind around that one.

12

Wrong. Mass and energy are equivalent. And the formula that is used to show this is E=m*c^2. If you use the right units, so that c=1, then you get E=m.

From PhysLink:
By the equation E = mc2, it is stated that energy and mass are related . We have seen the conversion of mass to energy in an atomic bomb and yes, energy can be converted back to mass. Such a process was first observed by C.D Anderson in 1932 where he discovered the creation of an electron-positron pair from the energy of a gamma ray photon.

13

Thanks everyone. Either my friend was wrong, or we miscommunicated. Ignorance fought.

This has always blown my mind, and I can’t quite make sense of it. I don’t see why other stuff whirling around out there would pull satellites up, any more than if that stuff wasn’t whirling around.

Other conundrums on this include, if there’s only one planet in the universe, by definition it wouldn’t be rotating. Yet if you applied force (like Hero’s steam engine) to make it rotate … would it then rotate? Only in relation to that steam you jetted to make it rotate?

If there are only two bodies, rotating about their barycenter … wait … without the rest of that stuff out there, who’s to say they’re rotating? Would they fall together?

I should stick to the simple stuff! But people who know waaaaay more than I ever will make this claim about angular motion being relative, just like linear motion.

I guess linear acceleration would be relative, too. Ouch.

14

In both of these cases, you have a rotating reference frame, in which you can measure the Coriolis Effect.

On a solo rotating planet, a ball thrown a sufficiently long distance will trace a curved path from the POV of an observer on the ground. Exceptions would be a ball thrown straight up while standing at either pole, or a ball thrown due east or west while standing at the equator.

In the case of two bodies rotating about their barycenter, a ball thrown from one to the other will also follow a curved path.

15

That’s what I would have thought, newme, but evidently not: a solo planet is just not rotating because there’s no frame of reference (or something like that). See Trinopus’s post. If you’re not contradicting his post, then I’m completely confused. I don’t understand Trinopus’s statement, but I do believe it’s the concensus opinion among those who understand this stuff.

16

In Newtonian mechanics, a rotating reference frame is fundamentally different from a stationary or constant-velocity reference frame, in that observers in the rotating reference frame experience centrifugal and Coriolis acceleration, while observers in other reference frames do not. This means that you can tell if you are in a rotating reference frame without looking at any non-local object. If you’re feeling centrifugal forces and Coriolis forces, you are in a rotating reference frame, without being relative to anything.

In that picture, if you take the Earth as non-rotating, the entire universe experiences centrifugal force, and that is the force that counteracts gravity to keep geosynchronous satellites aloft.

This brings up an amazing coincidence that if you lived in an opaque, closed laboratory floating in space, you could, using only stuff inside the laboratory, define “non-rotating reference frame” as "reference frame which experiences no centrifugal or Coriolis force. Then if you escaped the laboratory and observed the external universe, you would notice that those "non-rotating reference frame"s that you defined inside the lab, are exactly those frames in which the distant galaxies are not rotating.

This gives rise to Mach’s Principle, the idea that the bulk motion of distant matter in the universe somehow “causes” the distinction in local frames, which is probably what Trinopus’s reference is referencing.

17

That’s the part I don’t understand.

First, centrifugal force is only an apparent force, not a real force. To see a (apparent) centrifugal force, there has to be an equal and opposite (real) centripital force.

Second, how would all the universe spinning around exert a centrifugal force?

Now if someone said, with lots of technical relativistic jargon, that the spinning stuff warps spacetime and … well, my eyes would glass over a bit, but I’d have to swallow it. It’s over my head to think about critically.

18

Thanks for that; I can look that up and maybe understand it a little (or at least, come to terms with it).

19

I don’t have time to explain why you are incorrect, but you might want to ponder this:

A vault that completely contains a nuclear detonation weighs the same both before and after the explosion. No mass is converted to energy - potential energy has been converted to kinetic and electromagnetic energy; m and E remain the same.

m[sup]2[/sup] = E[sup]2[/sup] - p[sup]2[/sup] is an equation not an identity.
(c = 1)

20

Unless you’re Bond.