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SHM · Lecture 25 of 26

A particle on the axis of a ring does SHM under gravity

By Lalit, IIT Madras

A particle on the axis of a ring, JEE Advanced SHM: small oscillations under the ring's gravity, and the new T when the ring is free to move.

Watch lecture 25

A chalkboard titled 'The ring moves too' shows a ring edge-on as a thin vertical bar on a horizontal axis, a small circle for the particle to its right, a dashed vertical line marking the centre of mass between them, and the boxed formula T = 2π√(R³/(G(M + m))) beside the note that M + m is about M when M is much bigger than m.

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What this lecture covers

Let go on a uniform ring's axis, a particle is pulled towards the centre O and swings through it. For x much smaller than R, x² drops out against R², so small oscillations are SHM, with T independent of the particle's mass. Next, a swing from 2R is put to the same test. Freeing the ring keeps the centre of mass fixed, and with M = m the period becomes 1/root 2 of the fixed-ring value.

In this lecture

  1. Play from A particle released on a ring's axis
  2. Play from The ring's pull on its axis, passing O
  3. Play from What SHM needs: x much less than R
  4. Play from Small oscillations: ω and T
  5. Play from SHM only near O; release at 2R
  6. Play from The ring free to move too
  7. Play from Centre of mass: M x1 + m x2 = 0
  8. Play from SHM about the centre of mass
  9. Play from T for the free ring, and T when M = m
  10. Play from Recap: particle on a ring's axis