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A classical pendulum clock is powered by gravitational potential energy by weights. While a hybrid pendulum clock is somehow propelled by electric current. Both have the same pendulum swing as the regulator.

A pendulum's frequency is dependent both on its length as on the value of small g according to Huygens formula. Considering the length stays the same, but only small g varies due to height differences between locations: let's say first position is at sea level, and the next position is at 20 km above sea level. Considering the value of g drops significantly there, the frequency of ticking should become lower at that altitude. Time dilation should also be taken into account, but this is only a very small effect.

However: one of the pendulums is powered by a weight (which is also dependent on the value of g), while the other is powered by an electric battery (which is sensitive to time dilation). What effect would this difference have on their mutual frequencies? Would they start to run out of phase, as one is dependent on a gravitational potential energy power source and the other on a quantum-mechanical potential energy power source. Of which only the latter is intrinsically sensitive to time dilation!

Would love to hear your thoughts on this conundrum. Do the clocks tick in phase with eachother at both locations, or will they start to differ due to the time dilation in the electric power source? Is there some way to do this calculation?

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  • $\begingroup$ Why would they run out of phase? "Both have the same pendulum swing as the regulator." Also, gravitational time dilation affects everything, both the electronics and the falling weight. $\endgroup$
    – PM 2Ring
    Commented Jan 29 at 2:28
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    $\begingroup$ IDK if there is any difference, but if there is any difference, I bet the difference is too small to be measured by any practical pendulum clock. $\endgroup$
    – Solomon Slow
    Commented Jan 29 at 2:28
  • $\begingroup$ FWIW, the time dilation due to Earth's gravity (at the surface) is ~21.9 milliseconds per year. Good luck measuring at that precision using a pendulum. $\endgroup$
    – PM 2Ring
    Commented Jan 29 at 2:38
  • $\begingroup$ @PM2Ring That's actually an interesting question for an experimentalist. LIGO reports measured violin mode Q factors of 1e9 (!) for their mirror suspensions (repository.lsu.edu/cgi/…). If I am not mistaken that just about gets us to the required level of phase noise. Maybe I am overlooking something, but I wouldn't be completely dismayed about making relativistic measurements with a "grandfather clock". $\endgroup$
    – FlatterMann
    Commented Jan 29 at 5:20
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    $\begingroup$ @Apsteronaldo Yes, we're definitely getting into Chat territory. ;) See John Rennie's excellent post on time: physics.stackexchange.com/q/235511/123208 I agree with Einstein: time is fundamentally geometrical: it's a component of spacetime geometry. So time is like a spatial direction in some respects, but there are also hugely important differences. $\endgroup$
    – PM 2Ring
    Commented Feb 3 at 22:06

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Clocks are designed with some sort of a frequency standard. Often the frequency standard requires energy input, but the frequency standard is designed to be rather insensitive to the energy input. The energy input typically does not need to be carefully regulated and, by design, even fairly large variations will cause little variation in the frequency.

In your case, the frequency standard is the pendulum. Whether the power is provided by a battery or by weights is not particularly important. The gravitational potential and gravitational acceleration affect the frequency, but the details of the power source do not. So they tick in phase, within the limits of their stability.

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  • $\begingroup$ Thank you for your comment. I agree with your analysis, as indeed a battery powered quartz watch in its ticking doesn't depend on how full its battery is charged. I wonder though if there might be an edge case, with very little energy, when the potential energy of the hybrid becomes determined by quantum mechanics and the behaviour starts to change though compared to the classical pendulum clock with the gravitational potential as the energy source. So on the micro energy scales gravity might differ from quantum mechanics as the energy source. Just thinking out loud here, wonder what you think $\endgroup$
    – Apsteronaldo
    Commented Feb 2 at 20:32

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