How Do You Calculate the Period of a Mass-Spring System?

For both SL and HL IB Physics, calculate the period using $T=2\pi\sqrt{\frac{m}{k}}$, where $m$ is the total oscillating mass in kilograms and $k$ is the spring constant in $\text{N m}^{-1}$. The resulting period $T$ is measured in seconds.

This equation belongs to simple harmonic motion in subtopic C.1. It applies when the restoring force follows Hooke's law, $F=-kx$, the oscillations are small, damping is negligible, and the spring's mass is negligible or included in an effective oscillating mass.

Why the Formula Works

Hooke's law gives a restoring force directed opposite to the displacement. Combining $F=ma$ with $F=-kx$ gives

$$a=-\frac{k}{m}x.$$

The defining equation of simple harmonic motion is $a=-\omega^2x$. Therefore,

$$\omega^2=\frac{k}{m}\quad\text{and}\quad\omega=\sqrt{\frac{k}{m}}.$$

Since $T=\frac{2\pi}{\omega}$, substitution gives

$$T=2\pi\sqrt{\frac{m}{k}}.$$

The square-root relationship means that period does not change in direct proportion to either mass or spring constant.

Worked Example

A $0.50\,\text{kg}$ mass is attached to a spring with $k=200\,\text{N m}^{-1}$:

$$T=2\pi\sqrt{\frac{0.50}{200}}=0.314\,\text{s}\approx0.31\,\text{s}.$$

The system therefore completes one oscillation every $0.31\,\text{s}$.

A common misconception is that doubling the mass doubles the period. Doubling $m$ actually multiplies $T$ by $\sqrt{2}$, while quadrupling $m$ doubles .

Experimentally, $k$ can be determined from the gradient of a force-extension graph. Repeated timing of several oscillations reduces percentage uncertainty: divide the total measured time by the number of oscillations.

For data analysis, square the relationship and plot period squared against oscillating mass. The model predicts a straight line through the origin. Its gradient equals four pi squared divided by the spring constant, so the spring constant can be determined from the gradient. A non-zero intercept may indicate unaccounted spring mass or systematic error.

Exam Technique

For a Calculate question, show the equation, substitution, and final value. Use SI units, select mass rather than weight, and never insert $g$ into the mass-spring period formula.