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SHM · Lecture 4 of 26

Speed at any position in SHM, and the time period of a spring-block

By Lalit, IIT Madras

Speed at any position in SHM for JEE Main and Advanced: v = ω root(A² − x²) from a = v dv/dx, and the spring-block period T = 2 π root(m/k).

Watch lecture 4

A block of mass m on a spring of constant k with arrows marking a, v and x beside a dashed mean-position line, an ellipse of v against x with ωA and −ωA on the v axis and −A and A on the x axis, its upper half labelled moving right and its lower half moving left, and the boxed formulas v = ±ω√(A² − x²) and a = −ω²x.

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What this lecture covers

Since a = −kx/m gives acceleration in terms of x, the chain rule a = v dv/dx brings in speed. Separating variables and integrating, with limits at the extreme where v = 0, gives v = ω root(A² − x²). The spring-block period T = 2 π root(m/k) comes out alongside. Then follow v max = ω A and a max = ω² A, and the v-x graph turns out to be an ellipse.

In this lecture

  1. Play from SHM so far: a = −kx/m, ω, T, f
  2. Play from Speed at any position via a = v dv/dx
  3. Play from Integral of a power, and the limits
  4. Play from Result: v as a function of x
  5. Play from Time period T = 2 π √(m/k)
  6. Play from Speed: v = ω √(A² − x²)
  7. Play from Max speed ω A, max a ω² A
  8. Play from Using v and a at any position
  9. Play from Acceleration: a = −ω² x
  10. Play from Recap: speed and time period