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Simple Harmonic Motion formula sheet

Every formula in the Simple Harmonic Motion chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 38 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 formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Periodic motion and the SHM equation

Period, frequency, angular frequency
5× asked, last 2026
T=1f=2πωT=\frac1f=\frac{2\pi}\omega
Every SHM is periodic; not every periodic motion is SHM.
Condition for SHM
6× asked, last 2024
F=−kx,a=−ω2x,ω=kmF=-kx,\qquad a=-\omega^2x,\qquad \omega=\sqrt{\frac km}
xx from the mean position; restoring force always towards the mean.
Displacement equation
8× asked, last 2026
x=Asin⁡(ωt+ϕ)x=A\sin(\omega t+\phi)
AA = amplitude, ωt+ϕ\omega t+\phi = phase. Distance: 4A4A per period, 2A2A per half period.
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); quadrant from the sign of v0v_0. Cosine form Acos⁡(ωt+ϕ)A\cos(\omega t+\phi): tan⁡ϕ=−v0/ωx0\tan\phi=-v_0/\omega x_0.
Sum of sine and cosine
asin⁡ωt+bcos⁡ωt=a2+b2 sin⁡(ωt+δ),tan⁡δ=baa\sin\omega t+b\cos\omega t=\sqrt{a^2+b^2}\,\sin(\omega t+\delta),\quad \tan\delta=\frac ba
A sum like this is SHM with amplitude a2+b2\sqrt{a^2+b^2}.
Reference circle
2× asked, last 2023
x=rcos⁡(ωt+ϕ0)x=r\cos(\omega t+\phi_0)
Uniform circular motion projected on a diameter: SHM, A=rA=r, same ω\omega.

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Practise Simple Harmonic Motion

32 more formulas in 5 topics

Velocity, acceleration and timing; Energy in SHM; Spring–mass and other linear oscillators; Pendulums; and Superposition and damping. 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×
  • Phase relations1×
  • Time between two positions10×

Energy in SHM

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

Spring–mass and other linear oscillators

  • Spring–mass system9×
  • Vertical spring2×
  • 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 method6×
  • Energy method2×
  • Floating body; U-tube2×

Pendulums

  • Simple pendulum7×
  • Seconds pendulum3×
  • Effective gravity8×
  • Bob in a liquid or a field
  • Pendulum clock error
  • Pendulum at a large angle4×
  • Physical pendulum2×
  • Angular SHM2×

Superposition and damping

  • Two SHMs along one line
  • Two perpendicular SHMs1×
  • Damped oscillation2×

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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.

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