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Is the reason we don't have any perpetual motion machines because of the first law of thermodynamics? Would it be possible to create one if (for example) there was a wheel, and it had a push– if it was frictionless, and in a vacuum, would it spin forever? Also, would gravity matter if friction wasn't in the equation?

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    $\begingroup$ Note: A perpetual motion machine is not the same thing as an infinite-motion machine. To call it a perpetual motion machine, you have to not only make it run forever but you also must be able to extract energy from it continuously. In other words, for an infinite-motion machine there must be no energy losses, whereas for a perpetual motion machine there must also be energy generation (or energy replenishment). $\endgroup$ – Steeven Dec 17 '19 at 15:31
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Is the reason we don't have any perpetual motion machines because of the first law of thermodynamics?

That's one of the reasons, and it applies to a perpetual motion machine of the first kind, a machine that can produce work without any energy input in violation of the first law (conservation of energy).

There is also a perpetual motion machine of the second type, which is a machine that can operate in a cycle exchanging heat with only one temperature reservoir completely converting 100% of the heat into work. This machine violates the second law of thermodynamics.

Would it be possible to create one if (for example) there was a wheel, and it had a push– if it was frictionless, and in a vacuum, would it spin forever?

What you are describing is sometimes call a perpetual motion machine of the third kind. It is based on the idea that motion, once started, can continue forever. This requires the complete elimination of any and all forms of friction, which is not possible, even when operating in a vacuum (which in practice is never perfect). Even space is not completely free of particles.

Also, would gravity matter if friction wasn't in the equation?

Not quite sure what you mean by "would gravity matter". As the range of gravity is essentially infinite, it is impossible to completely isolate the machine from gravity and gravitational forces (Newtonian gravity).

Hope this helps.

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    $\begingroup$ Gravity could cause friction through tidal effects, even if nothing else was touching the wheel. $\endgroup$ – rghome Dec 17 '19 at 15:45
  • $\begingroup$ @rghome Good point, though the effect would be small depending on the size of the wheel. I'll add it to may answer. Thanks. $\endgroup$ – Bob D Dec 17 '19 at 15:49
  • $\begingroup$ @rghome 3 but I wonder would the friction associated with stretching and squeezing affect rotation? $\endgroup$ – Bob D Dec 17 '19 at 18:21
  • $\begingroup$ @BobD It should, from my understanding. That's how satellites can become tidally locked to their host, like the moon. The tidal forces brought its rotation to a stop. $\endgroup$ – thanby Dec 17 '19 at 23:40
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    $\begingroup$ @MooingDuck, an object doesn't need to be in orbit to experience tidal friction, it just needs to be spinning. The spin is what breaks the symmetry of force: gravity elongates the object in the direction of the gravity gradient, but then the object rotates slightly, bringing the elongation out of alignment, so gravity pulls it backwards (and warps the object to change the elongation axis). The overall result is the elongation axis isn't quite aligned with the gravitational gradient, producing a net torque, while the constant re-shaping dissipates the rotational energy as heat. $\endgroup$ – Mark Dec 18 '19 at 2:10
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A wheel frictionlessly rotating in a vacuum, although it would do so indefinitely, isn't a perpetual motion machine. It doesn't perform any work: e.g. you can't make a light bulb give off light using the energy of this wheel for unlimited amount of time. As soon as you try, you'll slow the rotation down, and the motion will cease being perpetual.

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    $\begingroup$ It won't rotate forever, actually, due to the Doppler effect. The wheel is emitting thermal radiation in all directions. Any photon leaving near-tangential to the rim will add or subtract angular momentum. Because of the Doppler effect, photons leaving in the same direction as the rotation will have slightly more energy than photons leaving in the opposite direction, slowing the rotation. It'll take a LONG time, but the wheel will stop eventually. $\endgroup$ – Ryan_L Dec 18 '19 at 0:47
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    $\begingroup$ @Ryan_L that's an interesting way to think of the problem, but it might not stop the wheel completely. E.g. the wheel might be cold enough that after it emits all its heat energy as thermal radiation (reaching $0\,\mathrm K$, at $t\to\infty$) and gives away corresponding amount of angular momentum, it'll still rotate. $\endgroup$ – Ruslan Dec 18 '19 at 9:01
  • $\begingroup$ But it'll never actually reach absolute zero, while there's nothing stopping it from reaching zero angular momentum. I guess maybe as it asymptotically approaches absolute zero it may also asymptotically approach zero angular momentum. $\endgroup$ – Ryan_L Dec 18 '19 at 16:55
  • $\begingroup$ @Ryan_L there is something stopping it from reaching zero angular momentum: the rate of emission. If it's so slow as to only ever emit e.g. half of available angular momentum, then the remaining angular momentum will be in the rotation. That's an interesting question though on its own, what fraction of initial angular momentum can be emitted given initial temperature. $\endgroup$ – Ruslan Dec 18 '19 at 17:00
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"Friction" is the simplest one-word answer, though you seem eager to simply discard it from your math without addressing how this would ever be done in reality. Pretending that there isn't a flawgic gap there, another broader version of the same one-word answer is "entropy."

That being said, these issues are all elements of the material space side of the whole spacetime situation. I began to wonder if it might be possible to find a theoretical opportunity for a machine to be run for a timespan that is approaching/functionally equivalent to infinite... not by perfecting the machine, but by situating it in circumstances different enough that "the machine's perception of time" and/or our perception of its relationship to time could be manipulated or distorted to meet the letter of the request, if a bit disingenuously.

What if a cheaply hacked together RunsALongTime9000 device was activated in the neighbourhood of a black hole or other gravitational anomaly? The seemingly obliterative punchline of a black hole complicates this, as does our difficulty in mapping out the play-by-play experience of being rended in one of these freaky space-blenders. But orbiting satellites and International Space Station long-stays both involve significant enough deviance from our standard Earth's Crust Gravity that computed times have to be corrected after being determined properly, and empirical human experience can began to perceive the different rate of the passage of time in orbit vs on Earth. It may not take anything near the gravitational scale of a black hole to allow Earth-lubbers observing from home to witness a machine operating at basically the same level of functionality for longer than any human lifespan, and so in pragmatic terms, "forever".

Another question for the question worth exploring: Does a machine have to have moving parts (or the electronic equivalent) in order to be deemed a machine or in order to be deemed in operation? I have heard Stonehenge described (in line with certain theories of its purpose and function) as an astronomical computer. But friction is only a very, very long-timeframe factor there. Although a factor it remains, and again the problem of erosion and collapse can be more wholely identified as the tendency towards entropy as time progresses.

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