Simple harmonic motion for a mass and spring
An ideal mass attached to a linear spring undergoes simple harmonic motion with angular frequency determined by spring constant and mass.
Amplitude does not change ideal frequency, but it sets maximum speed, maximum acceleration, and total mechanical energy.
How to use the SHM calculator
- Choose a calculation: Calculate all properties, mass, or spring constant.
- Enter known values: Use positive mass, stiffness, frequency, and amplitude where requested.
- Calculate: Review the default or updated oscillator results.
- Check assumptions: Confirm the spring behaves linearly and damping is negligible.
Formula and variables
The formulas assume an ideal linear spring with no damping.
ω = √(k/m); T = 2π/ω; f = ω/(2π); vmax = ωA; amax = ω²A; E = ½kA²- m — Mass
- Oscillating mass (kg)
- k — Spring constant
- Spring stiffness (N/m)
- A — Amplitude
- Maximum displacement from equilibrium (m)
- ω — Angular frequency
- Angular oscillation rate (rad/s)
One-kilogram oscillator
A 1 kg mass is attached to a 100 N/m spring with 0.1 m amplitude.
- Mass
- 1 kg
- Spring constant
- 100 N/m
- Amplitude
- 0.1 m
- ω = √(100/1) = 10 rad/s
- T = 2π/10
Result: T ≈ 0.6283 s, f ≈ 1.5915 Hz, and E = 0.5 J.
The mass completes about 1.59 cycles each second.
Understanding your results
Stiffness raises frequency
Frequency grows with the square root of k/m.
- Increasing mass lowers frequency.
- Amplitude does not change ideal period.
- Maximum speed is ωA.
- Maximum acceleration is ω²A.
Assumptions
- The spring obeys Hooke’s law.
- The spring is massless or its effective mass is negligible.
- Damping, friction, and external driving are negligible.
Limitations
- Does not model damping, forcing, nonlinear springs, or spring mass.
- Energy output is the ideal conserved mechanical energy.
- Real measurements require uncertainty analysis.
Common mistakes
- Using grams instead of kilograms.
- Confusing angular frequency with frequency in hertz.
- Using peak-to-peak displacement as amplitude.
- Applying the model to a nonlinear spring.
Practical use cases
Physics homework
Calculate connected SHM quantities from mass, stiffness, and amplitude.
Laboratory checks
Estimate mass or spring constant from oscillation frequency.
Frequently asked questions
Does amplitude affect ideal SHM frequency?
No. In the ideal linear model, frequency depends on k and m.
What is the difference between ω and f?
Angular frequency is in radians per second and equals 2π times frequency in hertz.
Sources and review
- Simple Harmonic Motion — OpenStax University Physics Volume 1. Accessed 2026-07-14.
Reviewed 2026-07-14.