how-to · 5 min read
Simple harmonic motion: how spring and pendulum periods work
Simple harmonic motion is the simplest useful model of an oscillation: the restoring force is proportional to displacement and points back toward equilibrium. A mass on an ideal spring and a small-angle pendulum both follow it, which is why their periods can be calculated from just a couple of system properties.
This guide derives the two periods in practical terms, works through a spring and a pendulum example, connects period to frequency and angular frequency, and explains where the ideal model stops being accurate.
What makes an oscillation simple harmonic
A system oscillates when a displacement creates a force that pushes it back. In simple harmonic motion that force is proportional to the displacement, so the further the system is from equilibrium, the stronger the restoration. The resulting motion is sinusoidal, with a fixed period and frequency.
A stretched spring and a pendulum are the standard examples. For a spring the restoring force is −k·x, where k is the stiffness. For a pendulum the force along the arc is proportional to the displacement from vertical for small angles, which is what makes the small-angle version simple harmonic.
The period is the time for one complete cycle, and it is the property that matters for clocks, suspension systems, vibration analysis and musical instruments. In the ideal model the period does not depend on how large the oscillation is.
Spring oscillator: 0.5 kg on a 200 N/m spring
For a mass m on a spring with stiffness k, the period is T = 2π√(m ÷ k). With m = 0.5 kg and k = 200 N/m, the ratio is 0.5 ÷ 200 = 0.0025, whose square root is 0.05. Multiplying by 2π gives a period of about 0.314 s.
The angular frequency is ω = √(k ÷ m) = √400 = 20 rad/s, and the ordinary frequency is f = 1 ÷ T = 1 ÷ 0.314 ≈ 3.18 Hz. That means the mass completes about 3.18 cycles every second, which is a fast but easily measured oscillation.
The structure of the formula explains the behavior: adding mass increases the period, while a stiffer spring shortens it. Doubling the mass multiplies the period by √2, and quadrupling the stiffness halves it.
Pendulum: 1.00 m on Earth
For a simple pendulum of length L in gravity g, the period is T = 2π√(L ÷ g). With L = 1.00 m and g = 9.81 m/s², L ÷ g = 0.101936, the square root is 0.3193, and the period is about 2.006 s. The frequency is about 0.499 Hz, and the angular frequency is about 3.13 rad/s.
Notice that the mass of the bob does not appear in the formula. The restoring force grows in proportion to mass, and so does the inertia, so the mass cancels out. A heavy bob and a light bob on the same string behave the same way in the small-angle approximation.
The length and gravity are the two inputs worth measuring carefully. For a clock pendulum that is roughly one meter long, small changes in length or local gravity produce measurable changes in period, which is why pendulum clocks need temperature compensation and why a pendulum can be used to measure g.
Frequency, angular frequency and what changes the period
Frequency and period are reciprocals: f = 1 ÷ T, so a period of 0.314 s corresponds to 3.18 Hz. Angular frequency is ω = 2π ÷ T, measured in radians per second, and it appears naturally in the sine and cosine description of the motion. Since 2π radians is one full cycle, ω is simply the number of radians per second.
For a spring, the period rises with the square root of mass and falls with the square root of stiffness. For a pendulum, the period rises with the square root of length and falls with the square root of gravity. In both cases the amplitude does not appear, which is the defining simplification of simple harmonic motion.
Real oscillators depart from this ideal in three common ways: damping removes energy and reduces amplitude, a driving force can add energy at a resonant frequency, and large amplitudes make the small-angle approximation fail for a pendulum. Each effect adds terms the simple period formula does not contain.
Assumptions and when the simple model is enough
The spring model assumes an ideal spring: it obeys Hooke's law exactly, it has no mass of its own and there is no friction or damping. A real spring has mass, can bend nonlinearly when stretched far, and loses energy internally, so its measured period may differ slightly from the simple formula.
The pendulum model assumes a point mass on a massless string with no air resistance and a small angular amplitude. The small-angle approximation is usually reasonable up to roughly 15 degrees; beyond that the real period becomes longer than the formula predicts and a correction is needed.
For introductory calculations and first estimates, both formulas are excellent. For precision timing, vibration design or large-amplitude systems, use a more complete model and verify against measurement.