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SHM · Lecture 10 of 26

Energy in SHM: why KE and PE repeat at twice the frequency

By Lalit, IIT Madras

Energy in SHM for JEE: where U = 1/2 kx² comes from, and why the total energy 1/2 kA² stays fixed as the block moves.

Watch lecture 10

A chalkboard titled One oscillation, KE and U shows a block of mass m on a spring with an axis marked −A, 0 and A and KE, U and E bars beneath it, and on the right an x against t sine wave above KE and U curves that swing twice per oscillation under a flat dashed E line, with the note that one oscillation takes T.

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What this lecture covers

Potential energy comes from the work of a conservative force: adding the SHM force's work over small steps gives U = 1/2 kx², largest at the extremes. Kinetic energy is 1/2 k(A² − x²), and the total 1/2 kA² never changes. Writing sin² in terms of cos 2 θ shows how many times KE and U repeat per period. Against x the energy graphs are parabolas, and an energy averaged over time differs from one averaged over position.

In this lecture

  1. Play from Energy in SHM and conservative forces
  2. Play from PE from the work of the SHM force
  3. Play from Potential energy U = 1/2 kx²
  4. Play from Kinetic energy: KE = 1/2 k(A² − x²)
  5. Play from Total energy E = 1/2 kA² is constant
  6. Play from KE and U repeat at twice the frequency
  7. Play from The sin² to cos 2 θ identity
  8. Play from Energy graphs against x
  9. Play from Average energy: over time or position
  10. Play from Recap: energy in SHM