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Consider a weightless vertical rod hinged at its center. Two identical inflated balloons are tied at each end of the rod. Entire system is under influence of gravity and is in air, so there are buoyancy forces on the balloon.(The system is in equilibrium, but is unstable because slightest push will cause the rod to rotate and become horizontal.) Now I heat the bottom balloon by supplying thermal energy equal to Q, and cool the top balloon at the same time by removing thermal energy equal to Q. So bottom balloon expands and the top one contracts. Overall net transfer of energy to this system (rod+balloon) is zero, but the potential energy of the system has increased, because density of air inside the balloon at the top is now greater than that at the bottom, compared to what it was before.

I know energy conservation cannot be violated, so where is the flaw in the argument?

Edit: Say initially, the density of air inside the balloons is the same as that of ambient air, so initially there are no buoyancy forces on the balloon.

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    $\begingroup$ The density of the balloons may change but their masses don't... $\endgroup$
    – lemon
    Commented Jul 7, 2016 at 10:46
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    $\begingroup$ How does density affect energy in this case? Can you explain that? $\endgroup$
    – Lelouch
    Commented Jul 7, 2016 at 12:42
  • $\begingroup$ @LeLouch You obtain a top-heavy configuration in the latter case. $\endgroup$
    – Deep
    Commented Jul 8, 2016 at 5:08
  • $\begingroup$ @lemon Mass doesn't change but effective gravity acting on the two balloons changes when their densities change (think buoyancy). If it is any help, you may think that initially, density of air in both the balloons was identical to that of atmospheric air and so there are no buoyancy forces on them (equivalently effective gravity on them is zero). $\endgroup$
    – Deep
    Commented Jul 8, 2016 at 5:12
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    $\begingroup$ Both the initial and final configurations have PE(same) there is no net change in that. in the second case, when we slightly tip over the system, external forces do work, so energy is not conserved. It is not a violation. i hope you see that. $\endgroup$
    – Lelouch
    Commented Jul 8, 2016 at 7:16

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