Spring
Calculate spring force using Hooke's Law F = kx, elastic potential energy EPE = ½kx², and SHM parameters — angular frequency, frequency, period, max velocity, and max acceleration for any spring-mass system.
About This Calculator
The Spring Calculator helps you analyze any spring-mass system using fundamental physics principles. Whether you are a physics student learning about Hooke's Law, an engineer designing spring mechanisms, or a hobbyist working on mechanical projects, this calculator provides instant results for spring force, elastic potential energy, and simple harmonic motion (SHM) parameters.
The calculator implements three core physics formulas. Hooke's Law (F = kx) relates the spring force to the spring constant and displacement. Elastic Potential Energy (EPE = ½kx²) calculates the energy stored in the deformed spring. Simple Harmonic Motion (ω = √(k/m)) determines how the spring-mass system oscillates, giving you angular frequency, frequency in Hz, period in seconds, maximum velocity, and maximum acceleration. The restoring force always acts opposite to the displacement direction (F = -kx in vector form), causing the characteristic back-and-forth oscillation.
The spring constant k is a measure of spring stiffness — higher values mean stiffer springs requiring more force for the same displacement. For helical coil springs, the spring constant depends on the shear modulus of the material, wire diameter, coil diameter, and number of active coils. Common materials include spring steel (G ≈ 79.3 GPa), stainless steel (G ≈ 69 GPa), and phosphor bronze (G ≈ 43 GPa).
Usage Notes
Spring constant k: Enter in N/m. Typical compression springs range from 10 to 500 N/m. Use the spring rate calculator on this site to compute k from coil geometry.
Displacement x: Enter in meters relative to the spring's equilibrium (unstretched) position. Positive values represent stretching, negative represent compression. The force magnitude is the same for equal displacements in either direction.
Mass m: Enter in kilograms. This determines the oscillation frequency. Larger masses produce slower oscillations with longer periods.
Frequently Asked Questions
What is Hooke's Law?
Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position: F = kx, where k is the spring constant in N/m and x is the displacement in meters. The law is named after the 17th-century physicist Robert Hooke and applies to any elastic material within its elastic limit.
How do you calculate spring force?
Spring force is calculated using Hooke's Law: F = k × x, where k is the spring constant (N/m) and x is the displacement from equilibrium (m). For example, a spring with k = 100 N/m stretched by 0.1 m produces a force of 10 N. The force always acts opposite to the direction of displacement, trying to restore the spring to its original length.
What is elastic potential energy in a spring?
Elastic potential energy stored in a spring is given by EPE = ½kx², where k is the spring constant in N/m and x is the displacement in meters. This energy represents the work done to stretch or compress the spring. When released, this stored energy converts to kinetic energy, causing the spring to oscillate.
How do I find the frequency of a spring-mass system?
The frequency of a spring-mass system depends on the spring constant k and the attached mass m. Angular frequency ω = √(k/m) in rad/s, the oscillation frequency f = ω/(2π) in Hz, and the period T = 1/f in seconds. A stiffer spring or lighter mass produces higher frequency oscillations.
What is the maximum velocity of a spring-mass system?
The maximum velocity of a spring-mass system occurs at the equilibrium position (x = 0) and is calculated as v_max = ω × x_max, where ω = √(k/m) is the angular frequency and x_max is the amplitude (maximum displacement). All energy is kinetic at this point as elastic potential energy is zero.
What is the spring constant and how is it determined?
The spring constant k measures the stiffness of a spring in N/m. It can be determined experimentally by applying a known force F and measuring the displacement x, then using k = F/x. Alternatively, for coil springs, k depends on the material (shear modulus G), wire diameter d, coil diameter D, and number of active coils N: k = Gd⁴/(8D³N).
How does mass affect spring oscillation?
Mass affects the oscillation frequency of a spring-mass system according to ω = √(k/m). Heavier masses oscillate more slowly (lower frequency, longer period) while lighter masses oscillate faster. For the same spring, doubling the mass reduces the frequency by a factor of √2. The amplitude can be set independently of the mass.
What is simple harmonic motion in springs?
Simple harmonic motion (SHM) occurs when a spring-mass system oscillates around its equilibrium position with a restoring force proportional to displacement (Hooke's Law). The motion is sinusoidal with constant amplitude and frequency. In SHM, maximum acceleration occurs at the amplitude extremes and maximum velocity at equilibrium.