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This is a basic question about energy conservation and classical mechanics:

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Question: Under what situations can this motion be perpetual?

  1. Without gravity and without frictions.

  2. Without gravity and with frictions.

  3. With gravity and without frictions.

  4. With gravity and with frictions.

  5. Others setup (please state the setup)

  6. Impossible to be perpetual

Gravity (say) is along the vertical $y$ direction, with a constant gravitational force and a linear gravitational potential $V(y)=mgy$.

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  • $\begingroup$ An important thing to note is that the radius of the outer wheel is twice that of the inner one. $\endgroup$ – Shubham Kumar Oct 5 '20 at 2:06
  • $\begingroup$ Is a bar freely rotating in space in perpetual motion according to your definition? $\endgroup$ – Wolphram jonny Oct 5 '20 at 2:34
  • $\begingroup$ Can we model the motion of the inner wheel on the outer wheel as a pure rolling motion? $\endgroup$ – Vivek Pandey Oct 5 '20 at 2:42
  • $\begingroup$ p.s. I do not downvote -- for those who receive down votes below -- it is not due to me. ;p $\endgroup$ – ann marie cœur Oct 5 '20 at 13:23
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Neglecting GR effects, the answer is clearly meant to be (1) for all speeds and (3) provided the inner gear is moving fast enough, because gravity is a conservative force.

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Despite the friction, body should move without any external forces, you can use definition to find answer: https://en.m.wikipedia.org/wiki/Perpetual_motion_(disambiguation)

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  • $\begingroup$ Please provide these remarks in comment box. The answer box is for proper answers. $\endgroup$ – Shubham Kumar Oct 5 '20 at 2:32
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Look , the object in question is undergoing accelerated motion so it will create ripple in spacetime fabric . The energy will be released in the form of gravitational waves which will ultimately make it slow down .

Had there been friction the ceasing of motion would be clearly apparent. But in case of energy released due to G - waves it may take a billion years to observe significant change in its motion.

G - waves are one of the reasons black holes which were initially a member of binary system rotating about their common centre of mass cease to rotate and merge together.

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Yup, perpetual motion. The playground for physics hobbiests. Inside a proton, perhaps..But nowhere else. So your question, to spite the conditions you established, is moot. There is no perpetual motion.

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