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When the string (of length $l$) connecting the bob (of mass $m$) of a pendulum makes an angle $\theta$ w.r.t the vertical, its total energy is given by $$E=\frac{1}{2}ml^2\dot{\theta}^2+mgl(1-\cos\theta)=\text{constant}$$ Now, imagine a situation, a thought experiment, in which gravity is instantaneously switched off. Here are the following questions.

  1. What happens to its potential energy? I mean, does it become zero because $g=0$? If yes, what happens to the conservation of energy?

  2. Although, now there is no force due to gravity on the bob, there is an inward string tension. How will the motion of the bob change?

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  1. If you "switch off" gravity, conservation of energy does not hold, since it is derived from the time translation invariance of the Lagrangian, and a Lagrangian that suddenly shuts down a force is not time translation invariant.

  2. After switching the force off, the only force acting on the bob is that of the string. What exactly happens is dependent on how fast the bob is at the time you turn gravity off. Your former simple system with one degree of freedom (the angle) turns into a two-dimensional system (distance from the anchor point and angle) subject to the constraint that the radial coordinate cannot be larger than the string length, since the bob is now no longer constraint to only move such that the string is taut.

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  • $\begingroup$ I understand your first point based on symmetry argument. But physically, how and where does the missing potential energy dissipate to? I don't see any "friction agent" here. $\endgroup$ – SRS Nov 8 '16 at 14:49
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    $\begingroup$ @SRS Physically, it is impossible to "turn off gravity". $\endgroup$ – ACuriousMind Nov 8 '16 at 14:52
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    $\begingroup$ If it is on a platform in a vacuum and the platform falls, then there won't be weight. $\endgroup$ – Orion Yeung Jul 15 '17 at 4:38
  • $\begingroup$ @ACuriousMind Okay. The acceleration of the bob cannot be made zero even by letting it fall free. Right? $\endgroup$ – SRS Aug 28 '17 at 20:47
  • $\begingroup$ @SRS: Well, you can - but in the sense that then you have switched frames, and since energy is not invariant between frames your question about energy conservation is then the same as physics.stackexchange.com/q/154842/50583 $\endgroup$ – ACuriousMind Aug 29 '17 at 14:55

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