The Amplitude Calculator determines the amplitude of simple harmonic motion from initial displacement, initial velocity, and angular frequency. Returns maximum displacement, velocity, acceleration, period, and frequency — essential for analyzing oscillating systems from springs to pendulums.
0.05831
m
5.831
cm
0.5831
m/s
5.831
m/s²
0.17
J/kg
0.05831
m
5.831
cm
0.5831
m/s
5.831
m/s²
0.17
J/kg
The calculator for amplitude of simple harmonic motion determines the maximum displacement from equilibrium of an oscillating system from its initial conditions and the system's angular frequency. Amplitude is the defining characteristic of oscillation magnitude, governing the energy stored and the peak forces experienced in any oscillating system.
For simple harmonic motion (SHM) with angular frequency ω, amplitude A is found from initial displacement x₀ and initial velocity v₀ using conservation of energy:
A = √[x₀² + (v₀/ω)²]
This follows from the SHM total energy: E = ½mω²A² = ½mω²x₀² + ½mv₀². The amplitude is completely determined by the initial state — no additional parameters are needed. The simple harmonic motion calculator provides the full time-domain solution x(t) = A cos(ωt + φ) from these same initial conditions.
Once amplitude is known, all other SHM extremes follow directly:
Maximum velocity and maximum displacement are 90° out of phase — when displacement is maximum (turning point), velocity is zero; when displacement is zero (equilibrium), velocity is maximum. Use this online calculator for any SHM system with known initial conditions.
The amplitude calculator applies to any system obeying ẍ + ω²x = 0:
The spring calculator, pendulum calculator, and oscillations and waves calculators provide system-specific tools applying these amplitude relationships.
Real oscillating systems lose energy to friction and drag, causing amplitude to decay: A(t) = A₀ × e^(−γt/2). Lightly damped systems oscillate many cycles before losing significant amplitude; critically damped systems return to equilibrium without oscillating in minimum time. This calculator computes the initial amplitude A₀ for the undamped case — the starting amplitude for any damped system. The Hooke's law calculator connects force and displacement for the spring systems underlying most SHM examples.
The general solution for simple harmonic motion is:
$$x(t) = A\cos(\omega t + \varphi)$$
At t = 0: x(0) = A cos φ = x₀ and ẋ(0) = −Aω sin φ = v₀.
Amplitude from initial conditions:
$$A = \sqrt{x_0^2 + \left(\frac{v_0}{\omega}\right)^2}$$
Derived by squaring and adding the initial conditions: x₀² + (v₀/ω)² = A²cos²φ + A²sin²φ = A².
Maximum velocity and acceleration:
$$v_{max} = A\omega, \quad a_{max} = A\omega^2$$
Energy per unit mass:
$$\frac{E}{m} = \frac{1}{2}\omega^2 A^2$$
The amplitude tells you the maximum displacement the oscillator reaches from its equilibrium position. A larger initial displacement or a faster initial kick (velocity) produces a larger amplitude. The angular frequency determines how the initial velocity contributes: at high ω, even a large v₀ adds relatively little to the amplitude because the oscillator quickly reverses direction. The maximum velocity and acceleration scale linearly and quadratically with ω respectively, explaining why high-frequency vibrations can produce enormous accelerations even with tiny amplitudes.
Inputs
Results
With x₀ = 5 cm and v₀ = 0.3 m/s at ω = 10 rad/s, the amplitude is about 5.83 cm — larger than the initial displacement alone because the initial velocity adds energy. The maximum velocity is 0.58 m/s.
Inputs
Results
A mass struck with v₀ = 2 m/s from equilibrium on a ω = 50 rad/s oscillator achieves A = 4 cm. Despite the small amplitude, the maximum acceleration is 100 m/s² (~10g) due to the high angular frequency.
The amplitude is determined by the initial conditions — specifically the initial displacement x₀ and initial velocity v₀ — together with the angular frequency ω. It is calculated as A = √(x₀² + (v₀/ω)²). Once set, the amplitude remains constant for ideal undamped oscillation. In real systems, damping gradually reduces the amplitude over time.
If v₀ = 0, the formula simplifies to A = |x₀|. The amplitude equals the initial displacement, and the mass oscillates symmetrically between +x₀ and −x₀. This is the most common textbook scenario — pull the mass to some displacement and let go.
If x₀ = 0, the formula gives A = |v₀|/ω. The amplitude is determined entirely by the initial velocity and the oscillator's natural frequency. A faster push or a lower natural frequency produces a larger amplitude.
The total mechanical energy of a simple harmonic oscillator is E = ½kA² = ½mω²A². At maximum displacement (x = A), all energy is potential; at equilibrium (x = 0), all energy is kinetic. The amplitude encodes this total energy, which remains constant throughout the motion for an undamped system.
The maximum velocity is vmax = Aω, occurring as the mass passes through the equilibrium position where all energy is kinetic. This relationship means that high-frequency oscillations with even small amplitudes can have surprisingly large maximum velocities.
No. For simple harmonic motion (linear restoring force), the period is completely independent of amplitude. This is the isochronous property discovered by Galileo. However, for nonlinear oscillators (like a pendulum at large angles), the period does depend on amplitude.
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