Natural Frequency Calculator
Find the natural frequency f₀, angular frequency ω₀, and oscillation period T for a spring-mass system, simple pendulum, or LC resonant circuit.
System type
N/m
kg
Cycles per second the system oscillates freely when undisturbed
- 1
Angular frequency ω₀
√(k ÷ m) = √(100 ÷ 1) = 10 - 2
Natural frequency f
ω₀ ÷ (2π) = 10 ÷ 6.2832 = 1.5915
How does this calculator work?
Natural frequency is how fast a system oscillates freely. For a spring-mass: f = (1/2π)√(k/m). For a pendulum: f = (1/2π)√(g/L). For an LC circuit: f = 1/(2π√(LC)). Angular frequency ω₀ = 2πf; period T = 1/f. All three formulas share a stiffness-to-inertia ratio structure.
Formula
How this is calculated
Every oscillating system has a natural frequency — the rate at which it vibrates freely once disturbed, with no sustained driving force. For a spring-mass system the restoring force follows Hooke's law, giving ω₀ = √(k/m) where k is spring stiffness in N/m and m is mass in kg. For a simple pendulum the small-angle approximation yields ω₀ = √(g/L), where g is gravitational acceleration and L is pendulum length; the approximation holds well for swings under roughly 15°. For an LC circuit the energy alternates between inductor and capacitor with ω₀ = 1/√(LC). In all three cases f = ω₀/(2π) in hertz, and period T = 1/f in seconds.
The three formulas share the same mathematical structure — a ratio of restoring stiffness to inertia — which is why mechanical and electrical resonators are mathematically equivalent. Engineers exploit this analogy to model vibrating structures, filter circuits, and antenna tuning. The a 1 m pendulum on Earth has T ≈ 2 s (the classic "seconds pendulum"), and an LC circuit with L = 10 mH and C = 100 µF resonates at about 159 Hz.
These results assume ideal, undamped (lossless) conditions. Real systems have friction, resistance, or air drag that gradually reduces amplitude and shifts the damped resonant frequency slightly below ω₀ to ω_d = √(ω₀² − γ²) where γ is the damping coefficient. For engineering applications where damping is significant, use the damped natural frequency instead.
Frequently asked questions
Natural frequency is the rate at which a system oscillates freely after being disturbed, with no applied driving force. It depends entirely on the system's physical properties — stiffness and mass for mechanical systems, or inductance and capacitance for LC circuits — not on how hard it was pushed.
A stiffer spring (larger k) increases natural frequency, while a heavier mass decreases it. Because f ∝ √(k/m), quadrupling k doubles f, and quadrupling m halves f. For a pendulum, frequency is independent of the bob's mass — only length and gravity matter.
For an undamped system they are the same. When damping is present, the resonant frequency (where a driving force produces maximum amplitude) falls slightly below the natural frequency. For lightly damped systems the difference is negligible; for heavily damped ones it can be significant.
Also known as
TG we-Calculate Editorial Team. (2026). Natural Frequency Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/natural-frequency-calculator
TG we-Calculate Editorial Team. "Natural Frequency Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/natural-frequency-calculator.
TG we-Calculate Editorial Team, "Natural Frequency Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/natural-frequency-calculator
@misc{wecalculate_natural_frequency_calculator, title = {Natural Frequency Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/natural-frequency-calculator}}, year = {2026}, note = {TG we-Calculate} }
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