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Let's consider two simple effects for which equivalent everyday examples are easy to find:

  1. A helium balloon lifts a piece of rock from the ground into the air.
  2. A permanent magnet lifts a nail off from a table.

In both cases, we have an object that is not receiving or producing any energy (at least I'm unable to see any obvious source of it) but that is nevertheless able to do work – thus necessarily expending energy – by lifting the weights against the gravitational pull of the Earth.

Where does that energy come from?

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The existing answers as of time of posting are incorrect as regards balloons, except in some cosmic sense tracking the energy of every particle through all of time.

For example: We expend 286 kJ to electrolyze 1 mole of water, put the mole of hydrogen gas in a massless 22L balloon, and float the balloon to a height of 5km. We have done just 100 J of work ($0.002 kg/mol \times 1 mol \times 5000m \times 10 m/s^2$). We then light it on fire. We'll get exactly 286 kJ of heat back, even though we've done 100 J of work. The hydrogen, now bound with new oxygen molecules, will eventually fall to the ground as rain, depositing the 100 J as extra heat. We can repeat this process as often as we want, gaining 100 J each time. Or we can truncate the process by igniting the balloon at 500m instead of 5000m, and only gain 10J each time - there's nothing special about the 100 J figure. If we had a magical indestructible infinitely stretchable massless balloon, the hydrogen would leave the planet and never come back, eventually reaching such distance that the gravitational potential energy is nearly that corresponding to infinite distance, because of the inverse square dependence of gravitational force. This would be just 125 kJ of gravitational potential energy ($0.5 \times .002 kg \times {v_e}^2$). If we remove the balloon envelope in the depths of interstellar space and wait a few billion years, the gas will disperse and will eventually combine with oxygen molecules, and we'll get our 286 kJ back.

This entirely variable amount of extra energy comes from the gravitational potential energy of the surrounding fluid, which descends, filling the volume left by the ascending buoyant object. This is similar to how the counter-weight on an elevator cable provides most of the energy required to lift the elevator car. It has nothing to do with the energy required to gather the material together.

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    $\begingroup$ "extra energy comes from the gravitational potential energy of the surrounding fluid, which descends" yes, this +1. As the balloon goes up air goes down. The mass of the air that goes down is greater than the mass of the balloon plus the rock that goes up. There really is no need to track things back to inflating the balloon. $\endgroup$
    – Dale
    Aug 30, 2022 at 16:47
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To fill a helium ballon you have to do work. either to extend the rubber or to expel the air from a container, then you can lift a weight say 10 m. and put it down there, to get the ballon down the 10 m to get the next weight you need as much force at least all along the 10 m as for getting the weight up. With the magnet it is a little more complicated, the magnetic field contains the energy, and it is weakened, you can see this, since the magnet which can just lift 1kg can not lift a second one. again if you lift it some cm you will need a force to separate magnet and weight to use it a second time. with hot air it is the same, the energy to heat it and get part of the air out is what lets you get the weight up. even if you would not loose heat, you need energy to bring it down for the next weight.

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Both are examples of what can be thought of as potential energy.

For the ballon, it cost energy to gather that helium and put it in a ballon. Just as it will cost energy to reel that ballon back down once it has risen to some altitude.

For the magnet, it cost energy to magnetize that lump of iron. All these atomic magnets that make up the magnet are pointing the same way, and they might rather not do so (the details of what magnetic configuration of a material is energetically favorable is actually a rabbit hole I'd rather not go down now... Consider this a hand-wavy example).

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