What this calculator does
A mass bouncing on a spring oscillates at a rate set by two things only: how heavy the mass is and how stiff the spring is. Amplitude does not appear in the period formula at all, so a big swing and a small one take exactly the same time.
That independence is what makes the system useful as a timekeeper and as the standard model for simple harmonic motion. It holds as long as the spring obeys Hooke's law, which real springs do until they are stretched too far.
The formula
The period is 2π times the square root of mass divided by spring constant. Frequency is its reciprocal. Maximum speed is amplitude times angular frequency, and maximum acceleration is amplitude times angular frequency squared.
| Term | Meaning |
|---|---|
| Spring constant (k) | Stiffness, in newtons per metre. A higher value means a stiffer spring and a faster oscillation. |
| Amplitude (A) | The maximum displacement from the rest position. It sets the speed and energy but not the period. |
| Angular frequency (ω) | The square root of k over m, in radians per second. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The oscillating mass in kilograms. |
| Spring constant (N/m) | The spring constant in newtons per metre, which is the force needed per metre of stretch. |
| Amplitude (m) | The amplitude in metres. It affects the speed and acceleration figures but not the period or frequency. |
When to use it
Designing a suspension or mount
The natural frequency of a mass on a spring determines what vibrations it will amplify, which is the first thing to check when isolating equipment.
Predicting an oscillation rate
Any system that can be modelled as a mass and a restoring force follows this relationship, which makes it far more general than springs alone.
Working a physics problem
Simple harmonic motion questions almost always start by finding the angular frequency from mass and stiffness.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does spring stiffness change the period?
The same mass on springs of increasing stiffness.
| Spring constant | Period | Maximum speed |
|---|---|---|
| 5 N/m | 1.987 s | 0.1581 m/s |
| 10 N/m | 1.405 s | 0.2236 m/s |
| 20 N/m | 0.9935 s | 0.3162 m/s |
| 40 N/m | 0.7025 s | 0.4472 m/s |
Questions
Why does amplitude not affect the period?
Because a larger displacement produces a proportionally larger restoring force. The mass has further to travel but it is pushed harder the whole way, and the two effects cancel exactly.
Does gravity change the period?
No, for a vertical spring it does not. Gravity shifts the equilibrium position downward but leaves the oscillation about that new position unchanged, which is why the formula contains no g.
What happens if the spring is stretched too far?
It stops obeying Hooke's law and the motion is no longer simple harmonic. The period then depends on amplitude after all, and the formula no longer applies.
How does this compare with a pendulum?
Both are simple harmonic oscillators with the same mathematical structure, but a pendulum's period depends on length and gravity while a spring's depends on mass and stiffness. A spring works identically in zero gravity; a pendulum does not work at all.
For the energy stored in the same oscillation, see the simple harmonic motion energy calculator. For measuring the spring constant itself, see the spring constant calculator.