Spring Oscillation Calculator
Find the period, frequency, maximum velocity, or restoring force of a mass-spring system. Ideal for physics students and engineers analyzing oscillatory motion.
About this calculator
A mass-spring system undergoes simple harmonic motion (SHM) when a mass attached to a spring is displaced from equilibrium. The period of oscillation is given by T = 2π√(m/k), where m is the mass in kilograms and k is the spring constant in N/m. Frequency is the reciprocal relationship: f = (1/2π)√(k/m). The maximum velocity the mass reaches as it passes through equilibrium is v_max = A√(k/m), where A is the amplitude of oscillation. The maximum restoring force (from Hooke's Law) is F = kA. Together, these four formulas fully describe ideal undamped SHM, letting you predict how fast a spring oscillates and how forcefully it acts on the attached mass.
How to use
Suppose you have a 0.5 kg mass on a spring with constant k = 200 N/m and amplitude A = 0.1 m. To find the period: T = 2π√(0.5/200) = 2π√(0.0025) = 2π × 0.05 ≈ 0.314 s. For frequency: f = 1/T ≈ 3.18 Hz. For maximum velocity: v_max = 0.1 × √(200/0.5) = 0.1 × √400 = 0.1 × 20 = 2.0 m/s. For maximum force: F = 200 × 0.1 = 20 N. Enter your own values and select the desired output to instantly see the result.
Frequently asked questions
How does mass affect the period of a spring oscillation?
A heavier mass increases the period of oscillation because the spring must move more inertia back and forth. The relationship is T = 2π√(m/k), so period grows with the square root of mass. Doubling the mass multiplies the period by approximately 1.41 (√2). This means heavier objects oscillate more slowly on the same spring, which is why car suspension springs behave differently under different loads.
What is the difference between spring frequency and angular frequency?
Frequency (f) measures how many complete oscillations occur per second, expressed in Hertz (Hz), and is calculated as f = (1/2π)√(k/m). Angular frequency (ω) measures oscillation in radians per second: ω = √(k/m). The two are related by ω = 2πf. Angular frequency is often more convenient in mathematical derivations of SHM, while regular frequency is more intuitive for practical applications.
When should I include a damping coefficient in spring oscillation calculations?
Damping should be included whenever friction, air resistance, or a shock absorber acts on the system, gradually reducing the amplitude over time. In a damped system the effective frequency shifts slightly lower than the undamped natural frequency. Real-world systems such as vehicle suspensions, door closers, and seismic isolators all exhibit damping. For purely theoretical or lightly damped scenarios, the undamped formulas provide a very good approximation.