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Simple Harmonic Motion and Pendulums, Explained

Here's how simple harmonic motion works, from a swinging pendulum to why bridges can shake themselves apart.

What is simple harmonic motion?

Simple harmonic motion is a back-and-forth motion around an equilibrium position at a constant period, which occurs whenever a restoring force acts β€” a force proportional to displacement but in the opposite direction. A mass bouncing on a spring or a swinging pendulum are classic examples.

What is a restoring force?

A restoring force is a force that pulls an object back toward its equilibrium position, and it acts more strongly the farther the object moves from equilibrium. Because of this restoring force, an object keeps being pulled back even after it passes the equilibrium point, which is why the vibration keeps repeating.

Hooke's law

Hooke's law states that a spring produces a restoring force proportional to how far it's stretched or compressed (its displacement), expressed as F = -kx. Here, k is the spring constant, representing the spring's stiffness β€” this relationship is what makes it possible to precisely use a spring's properties in designing scales or a car's suspension.

What determines a pendulum's period

A simple pendulum's period is determined only by its length and the acceleration due to gravity β€” it isn't affected by the pendulum's mass or its amplitude (as long as the swing is small). A longer pendulum has a longer period, a principle that was historically used to regulate the timekeeping of pendulum clocks.

Resonance

Every object has a natural frequency at which it tends to vibrate on its own, and when an externally applied force's frequency matches that natural frequency, resonance occurs and the amplitude grows sharply. Pushing a swing in time with its natural rhythm to make it go higher and higher, or a bridge shaking violently when it resonates with wind or marching footsteps, are both examples.

Everyday examples of simple harmonic motion

A clock pendulum, a swing, a speaker's vibrating diaphragm, a guitar string's vibration β€” simple harmonic motion explains many phenomena around us. This kind of periodic vibration is applied in many technologies, from timekeeping to generating sound.

Why it's worth understanding simple harmonic motion

From a swinging clock pendulum to a swing set to a speaker's vibration, our surroundings are full of motion that repeats back and forth at a steady period. Understanding the principle of simple harmonic motion explains why such repeating motions have a regular period, and why some vibrations grow larger while others gradually die out.

Sound is created by vibration, too

The vibration of an object β€” like a speaker or a musical instrument β€” is also the fundamental principle behind how sound is created. Understanding vibration is a natural stepping stone to understanding how sound waves work.

Frequently Asked Questions

Is a pendulum's period always the same no matter how big the swing is?

When the amplitude is small, a pendulum's period is nearly independent of the amplitude and stays constant, but once the amplitude becomes large, this approximation breaks down and the period gradually lengthens. Pendulum clocks and similar devices generally keep the amplitude small to minimize this error.

Is resonance always a good thing?

No. While it can be useful, like on a playground swing, if a bridge or building resonates with wind, an earthquake, or the rhythm of marching footsteps, the amplitude can grow so large that the structure is damaged or even collapses. That's why architects and bridge engineers carefully calculate a structure's natural frequency to avoid overlap with expected external vibrations.