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Simple Harmonic Motion revision notes

Every idea in the Simple Harmonic Motion chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 35 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Periodic motion and the SHM equation

Period, frequency, angular frequency
5× asked, last 2026
T=1f=2πωT=\frac{1}{f}=\frac{2\pi}{\omega}
Periodic: repeats after TT. Oscillatory: to and fro about a mean. Every SHM is periodic; not every periodic motion is SHM.
  • Test a function: rewrite powers with double-angle formulas. One sinusoid (plus a constant) is SHM; two different frequencies are periodic but not SHM; an irrational frequency ratio is not periodic.
  • Not period 2π/ω2\pi/\omega for sin⁡2ωt\sin^2\omega t. It equals (1−cos⁡2ωt)/2(1-\cos2\omega t)/2, so it is SHM at 2ω2\omega with period π/ω\pi/\omega about a shifted mean.
Condition for SHM
6× asked, last 2024
F=−kx,a=−ω2x,ω=kmF=-kx,\qquad a=-\omega^2x,\qquad \omega=\sqrt{\frac{k}{m}}
xx from the mean (equilibrium) position. The restoring force always points towards the mean.
  • Restoring force is proportional to xx and opposite to it.
Displacement equation
8× asked, last 2026
x=Asin⁡(ωt+ϕ)x=A\sin(\omega t+\phi)
AA = amplitude, ωt+ϕ\omega t+\phi = phase, ϕ\phi = initial phase. A cosine form is the sine form with ϕ\phi shifted by π/2\pi/2.
  • Not just sin⁡ϕ=x0/A\sin\phi=x_0/A. Two angles fit. Pick the one where cos⁡ϕ\cos\phi has the sign of the start velocity: towards −x-x gives the 2nd quadrant if x0>0x_0>0, the 3rd if x0<0x_0<0.
  • Not ωt=π/2\omega t=\pi/2 straight away. Set the full phase ωt+ϕ\omega t+\phi equal to the target phase, then subtract the initial phase before dividing by ω\omega.
Amplitude and phase from start values
3× asked, last 2025
A=x02+v02ω2,tan⁡ϕ=ωx0v0A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}},\qquad \tan\phi=\frac{\omega x_0}{v_0}
For x=Asin⁡(ωt+ϕ)x=A\sin(\omega t+\phi) with x0,v0x_0,v_0 at t=0t=0. So x0x_0, v0v_0 and ω\omega fix the whole motion.
  • In cosine form x=Ccos⁡(ωt−ϕ)x=C\cos(\omega t-\phi) the amplitude CC is the same, but tan⁡ϕ=v0/ωx0\tan\phi=v_0/\omega x_0.
Reference circle
2× asked, last 2023
x=rcos⁡(ωt+ϕ0)x=r\cos(\omega t+\phi_0)
Uniform circular motion of radius rr projected on a diameter is SHM with A=rA=r and the same ω\omega (angle per second).
ωt\omega tx=Acos⁡ωtx=A\cos\omega t
A point going round at ω\omega: its shadow on a diameter does SHM
Also remember
  • Distance travelled: 4A4A per period and 2A2A per half period from any start. A quarter period covers AA only when it starts at the mean or an extreme.

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Practise Simple Harmonic Motion

30 more concepts in 6 topics

Velocity, acceleration and timing; Energy in SHM; Spring–mass and other linear oscillators; Pendulums; and Superposition and damping, plus ideas from gravitation. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Velocity, acceleration and timing

  • Velocity11×
  • Velocity–displacement ellipse5×
  • Speeds at two positions1×
  • Acceleration and force6×
  • Graphs of x, v, a against time1×
  • Time between two positions10×

Energy in SHM

  • Kinetic, potential and total energy14×
  • Energy graphs9×
  • Average energies1×

Spring–mass and other linear oscillators

  • Spring–mass system9×
  • Vertical spring2×
  • Hooke's law and a wire as a spring2×
  • Springs in series and parallel10×
  • Cut spring2×
  • Two bodies on one spring3×
  • Sudden change of mass or speed5×
  • Block on a block1×
  • Force method for any small oscillation6×
  • Energy method2×
  • Floating body2×
  • Liquid in a U-tube

Pendulums

  • Simple pendulum7×
  • Seconds pendulum3×
  • Effective gravity8×
  • Pendulum at a large angle4×
  • Physical (compound) pendulum2×
  • Angular SHM2×

Superposition and damping

  • Two SHMs along one line6×
  • Two perpendicular SHMs1×
  • Damped pendulum (experiment)2×

Also asked here: gravitation

  • Key points1

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About 130 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Simple Harmonic Motion

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