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SHM · Lecture 3 of 26

SHM is the shadow of circular motion: the reference circle

By Lalit, IIT Madras

SHM as the shadow of uniform circular motion, for JEE: the reference circle gives ω = 2 π f and a = −ω² x.

Watch lecture 3

A reference circle with a particle and its blue shadow on the horizontal diameter, the acceleration ω²A marked on the particle pointing to the centre, a dashed line down to a block on a spring with acceleration a, and the formulas x = A sin ωt and a = −ω²x on the right.

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What this lecture covers

T = 2 π/ω comes first, and the circle explains it. A point moving uniformly round a circle casts a shadow on a diameter, and that shadow does SHM. Radians and angular speed give ω, the circle's own angular speed, and ω = 2 π f. Acceleration in SHM is proportional to x and opposite to it. The angle ω t gives x = A sin ω t, and the centripetal acceleration, projected, gives a = −ω² x.

In this lecture

  1. Play from SHM so far, and T = 2 π/ω
  2. Play from SHM as the shadow of circular motion
  3. Play from The shadow's speed, why a circle helps
  4. Play from One round is one oscillation
  5. Play from Radian and angular speed
  6. Play from Frequency f from the time period T
  7. Play from Angular frequency: ω = 2 π f
  8. Play from In SHM, a is proportional to −x
  9. Play from Position: x = A sin ω t
  10. Play from Shadow's acceleration: a = −ω² x