Spring Mass System Calculator

Hooke's law

The force needed to stretch or compress a spring is proportional to its change in length: F = k × x. The constant k is the spring constant, in N/m. A stiff spring has a large k and a soft one a small k. The elastic potential energy stored in the spring is E = ½ × k × x².

Period of a spring mass system

A mass on a spring that is pulled from equilibrium and released performs simple harmonic motion. Its period depends only on the mass and the spring constant:

T = 2 × π × √(m / k)

Amplitude does not change the period: pulling the spring a little or a lot makes no difference to the timing. Four times the mass doubles the period. A spring four times as stiff halves it.

Example

A 0.5 kg mass is attached to a 200 N/m spring, pulled 10 cm and released. The angular frequency is √(200 / 0.5) = 20 rad/s, the period 2π / 20 = 0.314 s and the frequency 3.18 Hz. The speed through equilibrium is 0.1 × 20 = 2 m/s and the total energy is 0.5 × 200 × 0.1² = 1 J.

Hanging mass and measuring the spring constant

When a mass is hung on a vertical spring, the spring stretches by x = m × g / k. This is the easy way to measure a spring constant: hang a known mass, measure the stretch, and k = m × g / x. A second way is to time the oscillation: k = 4 × π² × m / T².

Limits

The relations hold while the spring stays within its elastic limit. The mass of the spring itself and friction are neglected. With several springs, find the equivalent constant first: for springs side by side (parallel) the constants add, and for springs end to end (series) the reciprocals add.