Simple Harmonic Motion
x(t) = A cos(ωt + φ) — the canonical oscillation.
Key Notes
Simple Harmonic Motion (SHM): periodic motion where restoring force is proportional to displacement and opposite in direction. F = −kx.
Equation of motion: m·d²x/dt² = −kx ⇒ d²x/dt² = −ω²·x with ω = √(k/m).
General solution: x(t) = A·cos(ωt + φ). A = amplitude, ω = angular frequency, φ = initial phase.
Period T = 2π/ω = 2π·√(m/k). Frequency f = 1/T.
SHM is the simplest periodic motion — many oscillators (pendulum, spring, sound, LC circuit) reduce to SHM at small amplitudes.
Two SHMs combine to give beats, interference, Lissajous figures.
Phase angle φ: shifts the wave in time; ωt + φ = 0 at maximum displacement.
Mathematical basis: any small oscillation around a stable equilibrium is SHM to leading order.
Formulas
Equation of motion
Defining differential equation of SHM.
Position
Sinusoidal — A = amplitude, ω = angular frequency, φ = phase.
Angular frequency (spring-mass)
Determined by stiffness and mass.
Period and frequency
Period is independent of amplitude (linear SHM).
Important Points
Period T does NOT depend on amplitude — that's the unique feature of SHM.
x, v, a in SHM are all sinusoids of the SAME ω.
If a force is restoring AND linear in displacement ⇒ motion is SHM.
Real systems are SHM only at small amplitudes. Large-amplitude pendulum, large-amplitude spring (Hookes' law violated) lose linearity.
Energy in SHM oscillates between KE and PE but total is constant.
Frequency depends on PHYSICAL parameters (k, m, g, L) — NOT on how hard you start it.
Simple Harmonic Motion notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.