🪤 Oscillations & Simple Harmonic Motion (SHM)

x = A sin(ωt + φ), v, a, energy in SHM, time period of spring–mass and pendulum systems

x = A sin(ωt + φ) v, a in SHM E = ½kA² T = 2π√(m/k), 2π√(L/g)
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💡 Simple Harmonic Motion at a glance

In SHM, restoring force is proportional to displacement and opposite in direction: F = −kx. Displacement can be written as x = A sin(ωt + φ) or x = A cos(ωt + φ) with angular frequency ω = 2π/T. Velocity is v = Aω cos(ωt + φ), acceleration is a = −ω²x, and total mechanical energy E = ½kA² remains constant in ideal SHM.

SHM calculators

Displacement, velocity, acceleration, energy, and time period

Amplitude (A)
m
Angular frequency (ω)
rad/s
Time (t) and phase (φ)
Using x = A sin(ωt + φ)
x = —, v = —, a = —
Mass–spring system (mass m, spring constant k)
Amplitude (A) and current displacement (x)
E = —, KE = —, PE = —
Ideal SHM: E = ½kA², PE = ½kx², KE = E − PE
Mass–spring system
Simple pendulum (small angle)
T_spring = —, T_pendulum = —
T_spring = 2π√(m/k), T_pendulum = 2π√(L/g) (small angle)
Mass–spring with damping (m, k, b)
ω₀ = —, ω' = —, Q = —
Classifies motion (under/critical/over-damped) and gives damped frequency & quality factor.
Driven oscillator (m, k, b, F₀, ω_d)
A_ss = —
Uses A = (F₀/m) / √((ω₀² − ω_d²)² + (2β ω_d)²) and compares ω_d with ω₀ for resonance.

📈 SHM visualization & graphs

🧮Step-by-Step Solution▼ Show

Key SHM formulas & systems

Concept / System Formula Notes
Displacement in SHM x = A sin(ωt + φ) = A cos(ωt + φ') A = amplitude, φ = phase constant
Velocity & acceleration v = Aω cos(ωt + φ), a = −ω²x Maximum v = Aω at x = 0, maximum |a| = Aω² at x = ±A
Angular frequency & time period ω = 2π/T, T = 2π/ω, f = 1/T f in hertz (Hz), ω in rad/s
Total energy in SHM E = ½kA² = ½mω²A² Constant for ideal SHM
Kinetic & potential energy KE = ½mω²(A² − x²), PE = ½kx² ⟨KE⟩ = ⟨PE⟩ = ¼kA² over a cycle
Mass–spring system T = 2π√(m/k) Horizontal or vertical (small oscillations)
Simple pendulum T = 2π√(L/g) Small-angle approximation (θ ≲ 10°)
Physical pendulum T = 2π√(I / m g d) I: moment of inertia about pivot, d: COM distance from pivot
Torsional pendulum T = 2π√(I / κ) κ: torsional constant of wire/rod
Floating cylinder T = 2π√(L / g) L: length of immersed part of cylinder
Liquid in U-tube T = 2π√(L / 2g) L: total length of liquid column
Springs in series 1/k_eff = 1/k₁ + 1/k₂ + … Use k_eff in T = 2π√(m/k_eff)
Damped SHM (under-damped) x = A e^(−bt/2m) sin(ω' t + φ) ω' = √(ω₀² − β²), β = b/(2m)
Resonance (driven SHM) A(ω_d) ∝ 1 / √((ω₀² − ω_d²)² + (2βω_d)²) Amplitude peaks near ω_d ≈ ω₀ for light damping

About this SHM tool

This page focuses on core oscillation formulas used in school physics, JEE, and NEET: displacement, velocity, acceleration, energy, and time period of standard SHM systems. For energy storage (KE, gravitational and elastic PE) see the Energy Calculator.

Visual SHM graphs & next steps

This tool already plots x(t), v(t), and a(t) for your chosen SHM parameters and shows step-by-step solutions for each calculator. In a later batch we will add Matter.js visualizations of a mass–spring system and a simple pendulum to make the motion even more intuitive.