Natural Frequency
Calculate the natural frequency of spring-mass systems, pendulums, and structures using f = (1/2π) × √(k/m). Free online physics calculator with charts and breakdowns for students and engineers.
About This Calculator
The Natural Frequency Calculator helps you find the frequency at which a system vibrates when disturbed from equilibrium. Natural frequency is a fundamental property of any physical object or structure, determined by its stiffness and mass distribution. Understanding natural frequencies is essential in mechanical engineering, civil engineering, acoustics, and physics to avoid catastrophic resonance failures.
The calculator supports three common oscillatory systems. For a spring-mass system, the natural frequency is given by f = (1/2π) × √(k/m) where k is the spring constant and m is the mass. For a simple pendulum, f = (1/2π) × √(g/L) where g = 9.81 m/s² and L is the pendulum length. For a structure with concentrated mass, f = (1/2π) × √(g/δ) where δ is the static deflection under the load. All results include natural frequency in Hz, angular frequency in rad/s, and oscillation period in seconds.
Applications
Engineering: Design machine foundations, vehicle suspensions, and building structures to avoid resonance with external vibrations. Physics education: Understand harmonic oscillators, wave mechanics, and vibrational analysis. Product design: Tune natural frequencies of consumer products to avoid unwanted vibrations or noise.
Assumptions and Limitations
This calculator assumes ideal conditions: no damping or friction, linear elastic behavior for springs, small-angle approximation for pendulums (θ < 15°), and rigid supports. Real-world systems may have slightly different natural frequencies due to damping, nonlinearity, and material imperfections.
Frequently Asked Questions
What is natural frequency?
Natural frequency is the frequency at which a system oscillates when disturbed from its equilibrium position without any external driving force. Every physical object has at least one natural frequency determined by its stiffness, mass distribution, and geometry.
How do you calculate natural frequency of a spring-mass system?
For a spring-mass system, the natural frequency is calculated using the formula f = (1/2π) × √(k/m), where k is the spring constant in N/m and m is the mass in kg. The angular frequency is ω = √(k/m) in rad/s.
What is the natural frequency of a simple pendulum?
For a simple pendulum, the natural frequency depends only on its length and gravity: f = (1/2π) × √(g/L), where g = 9.81 m/s² and L is the pendulum length. The period T = 2π × √(L/g).
What is the difference between natural frequency and resonant frequency?
Natural frequency is the frequency at which a system oscillates freely without external forces. Resonant frequency is the frequency of an external periodic force that causes maximum amplitude vibration. For most systems these are nearly equal, and prolonged resonance can lead to structural failure, such as the Tacoma Narrows bridge collapse.
Is this calculator free to use?
Yes, this natural frequency calculator is completely free to use with no registration or downloads required. You can share results via URL.
How accurate are the results?
Results are computed using standard physics formulas (f = (1/2π)√(k/m)) with calculations rounded to 4 decimal places. The calculator assumes ideal conditions such as no damping, linear springs, small-angle approximation for pendulums, and rigid connections.
What units does the calculator use?
The calculator uses SI units: spring constant in N/m, mass in kg, length in m, deflection in m. Results are displayed in Hz (natural frequency), rad/s (angular frequency), and seconds (period).