Hooke's law and simple harmonic motion
A spring pulls back with a force proportional to how far it is stretched or compressed: F = −k x (Hooke's law, Robert Hooke, 1678). A mass m on such a spring undergoes simple harmonic motion with angular frequency ω = √(k/m), period T = 2π √(m/k) and frequency f = 1/T. Remarkably, for an ideal spring the period does not depend on the amplitude.
Energy in the oscillator
Energy continuously trades between kinetic energy ½ m v² and elastic potential energy ½ k x²; without friction their sum is constant, so the oscillation never dies out. Add damping and the amplitude decays over time, while the frequency shifts only slightly until the damping is strong.
How to use the calculator
Set the mass m, the spring constant k and the starting amplitude. The tool reports ω, T and f, and lets you invert the relationship — for example, the stiffness needed for a target period.
Note: the ideal model assumes a massless, linear (Hookean) spring and no friction. Real springs have mass, go non-linear at large stretch, and lose energy to damping, all of which the simple formula ignores.
Frequently asked questions
What is the period of a spring-mass system?
The period is T = 2*pi*sqrt(m/k), where m is the mass and k is the spring constant. A heavier mass or a softer spring gives a longer period.
What is Hooke's law?
Hooke's law says the restoring force of a spring is proportional to its displacement: F = -k*x, where k is the spring constant and x is the stretch or compression. The minus sign shows the force opposes the displacement.
Does amplitude affect the period of a spring-mass system?
No, for an ideal Hookean spring the period depends only on the mass and the spring constant, not on the amplitude. This is a defining feature of simple harmonic motion.
What is the angular frequency of a spring-mass system?
The angular frequency is omega = sqrt(k/m), in radians per second. It relates to the period by omega = 2*pi/T and to the ordinary frequency by omega = 2*pi*f.
References
- Hooke's law: R. Hooke (1678), Lectures de Potentia Restitutiva ("Of Spring"), F = -k x.
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics (Wiley) — simple harmonic motion.
- J. B. Marion & S. T. Thornton, Classical Dynamics of Particles and Systems (Cengage).