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SHM · Lecture 5 of 26

x = A sin(ωt + φ): the SHM equation, term by term

By Lalit, IIT Madras

x = A sin(ω t + φ), the SHM equation, derived for JEE from v = dx/dt, with φ as the constant of integration that fixes the start.

Watch lecture 5

Two numbered spring-block systems with the blocks at different positions about a dashed vertical line, above an x against time graph of two sine curves of different heights, one of them dashed, that share the same period, beside notes saying A is the energy given at the start, φ is where it was at t = 0 and which way it went, and ω is the spring and the mass, not the push.

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What this lecture covers

Getting x as a function of t starts from v = dx/dt. Separating x and t and using the standard integral leaves a constant of integration, the phase constant φ in x = A sin(ω t + φ). Phi records where the block was at t = 0 and which way it was moving. Differentiating twice gives back d²x/dt² = −ω² x. The amplitude is fixed by the energy given at the start, and ω t + φ is the phase.

In this lecture

  1. Play from What we know so far about SHM
  2. Play from The question: x at a time t
  3. Play from Separating x and t, standard integral
  4. Play from Phi, the constant of integration
  5. Play from Result: x = A sin(ω t + φ)
  6. Play from Phi: start position and direction
  7. Play from The equation d²x/dt² = −ω² x
  8. Play from Infinitely many SHMs, one force law
  9. Play from A is set by the starting energy
  10. Play from The phase ω t + φ, and recap