Spring Period Calculator
Spring oscillation period, frequency, and sag.
What it does
Mass on a spring keeps its own time. Stiffness pulls back while mass resists, and the exchange repeats at a period set by exactly two numbers: T equals 2 PI times root(m/k). One kilogram on 10 N/m swings every 1.987 s near 0.503 Hz with nearly a meter of static sag.
The tool above runs the ideal oscillator. Enter mass and stiffness, and it prints period, frequency, and static stretch. Defaults show the 1 kg case; practice spans half-kilo quick springs and 2 kg slow ones with matching sag figures.
Course texts agree verbatim. OpenStax prints T and f with mass and stiffness as the only factors, HyperPhysics adds omega with the mg/k static term, and MIT 8.03 frames Hooke oscillators in its vibrations unit. Damping stays outside everywhere here.
Sag grounds the abstraction. The same stiffness holding the mass statically stretches 0.981 m, which sizes mounts and travel before dynamics start. Real springs add mass and damping corrections; ideal math still rules first estimates.
How to use
- Enter the mass in kg.
- Enter the stiffness in N/m.
- Read period first, then frequency.
- Read static sag for mounting.
- Compare rates across springs.
- Add damping analysis separately.
Formula
- m
- Hanging mass in kg.
- k
- Spring rate in N/m.
OpenStax prints T and f exactly so; HyperPhysics adds omega = sqrt(k/m) with the mg/k static term; MIT 8.03 frames Hooke oscillators.
Worked example
Example: 1 kg, 10 N/m
Defaults: 1 kg on a 10 N/m spring.
- Inputs: 1 kg mass, 10 N/m spring.
- Ratio: 1 / 10 = 0.1, root 0.31623.
- Period: 2 x PI x 0.31623 = 1.987 s.
- Frequency: 1 / 1.987 = 0.503 Hz.
- Sag: 1 x 9.81 / 10 = 0.981 m. Tool shows 1.987 s, 0.503 Hz, 0.981 m sag.
Reference table
| Mass / k | Period | Frequency |
|---|---|---|
| 0.5 kg / 20 | 0.993 s | 1.007 Hz |
| 1 kg / 10 | 1.987 s | 0.503 Hz |
| 2 kg / 50 | 1.257 s | 0.796 Hz |
| 1 kg / 20 | 1.405 s | 0.712 Hz |
Limits
- Ideal only. Damping and spring mass excluded.
- Linear Hooke range; overstretch deviates.
- Small angles N/A; axial motion assumed.
- g fixed at 9.81.
Common mistakes
- Mixing grams with kg in m.
- Reading k in lb/in without converting.
- Forgetting sag in mount design.
- Ignoring damping on real rigs.
- Expecting amplitude to set rate.
Try it yourself
Quick checks. Answers sit tucked under each question.
0.5 kg on 20 N/m?
m/k = 0.025, root 0.15811. T = 0.993 s. f = 1.007 Hz. Sag 0.5 x 9.81 / 20 = 0.245 m.
2 kg on 50 N/m?
m/k = 0.04, root 0.2. T = 1.257 s. f = 0.796 Hz. Sag 2 x 9.81 / 50 = 0.392 m.
Questions
How is the period figured?
2 PI times root(m/k). Here 6.2832 x 0.31623 = 1.987 s.
How is frequency figured?
One over period. Here 1/1.987 = 0.503 Hz.
What is static sag?
Weight over stiffness: 1 x 9.81 / 10 = 0.981 m of static stretch.
What changes the rate?
Heavier mass slows it (root m); stiffer spring quickens it (root k). Only m and k matter.
Does damping matter?
Yes. Damping bleeds energy and stretches real periods slightly past ideal.
Does spring mass count?
No. Ideal massless spring assumed; heavy springs need effective-mass corrections.
Sources
- HyperPhysics: Simple Harmonic Motion
- HyperPhysics: SHM Frequency (omega, mg/k term)
- MIT OCW 8.03SC: Vibrations Lecture 1 (Hooke oscillators)