---
title: "Simple Harmonic Motion as a Second-Order Equation"
description: "Any restoring force proportional to displacement gives the same differential equation, whose solutions are sines and cosines at one natural frequency. Amplitude and phase come from the initial conditi"
canonical: https://lightmysky.com/learn/science/simple-harmonic-motion-as-a-second-order-equation-mt_HHmR-uCvwO
source: https://lightmysky.com/learn/science/simple-harmonic-motion-as-a-second-order-equation-mt_HHmR-uCvwO.md
retrieved: 2026-09-12
---

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# Simple Harmonic Motion as a Second-Order Equation

Any restoring force proportional to displacement gives the same differential equation, whose solutions are sines and cosines at one natural frequency. Amplitude and phase come from the initial conditions, never from the equation itself.

Subject: Science · Area: Forces & Motion · Ages 18 to 20
Page: https://lightmysky.com/learn/science/simple-harmonic-motion-as-a-second-order-equation-mt_HHmR-uCvwO

## Ready when they can

- Derives the harmonic equation for a mass on a spring and for a small-angle pendulum
- Writes the general solution with amplitude and phase and fits both to given initial conditions
- Shows that the natural frequency depends on the system, not on how hard it was started

## Lesson: The swing that keeps its tempo

Whenever the push back is proportional to the displacement, you get the same motion. A spring pulls with minus k times x, always toward rest. Acceleration proportional to minus displacement is the defining mark of simple harmonic motion.

The position follows a cosine: amplitude times cosine of angular frequency times time plus phase. Amplitude and phase never come from the equation itself. They come from where you release the mass and how fast it was moving then.

**Example.** A mass on a spring runs at square root of k over m, so stiffer springs sing higher and heavier masses sing lower. A pendulum at small swings obeys the same equation with square root of g over length. Pluck a guitar string gently or hard and the tone stays the same, since tempo ignores amplitude.

**Tip.** Use the two starting values to fit amplitude and phase: position at time zero gives one condition, velocity at time zero gives the other. For pendulums keep swings under about 15 degrees, where sine of the angle stays close to the angle itself.

**Recap.** A restoring push proportional to displacement gives cosine motion whose tempo the system sets.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Energy in Simple Harmonic Motion, Damping and Resonance](https://lightmysky.com/learn/science/energy-in-simple-harmonic-motion-damping-and-resonance-mt_3sh7E7k7j2)
- [Newton's Second Law as a Differential Equation](https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH)
- [Potential Energy Curves, Turning Points and Stability](https://lightmysky.com/learn/science/potential-energy-curves-turning-points-and-stability-mt_ZgRbrDaQzh)

## Opens up

- [The Quantum Harmonic Oscillator and Zero-Point Energy](https://lightmysky.com/learn/science/the-quantum-harmonic-oscillator-and-zero-point-energy-mt_HyDxmqyReX)
- [Damped Oscillations and the Three Damping Regimes](https://lightmysky.com/learn/science/damped-oscillations-and-the-three-damping-regimes-mt_JCpYpFVn6s)