AI Tutor
When you pull back a guitar string and let go, it doesn't just return to the center and stop β€” it vibrates back and forth hundreds of times per second.
The flat resting line is the equilibrium position, where the net force on the object is exactly zero. Any disturbance away from this center is called a displacement (xx).
Why doesn't the displaced object stay where you left it, and what pulls it back toward the center every single time?

Hooke's Law and the Restoring Force

In 1678, English scientist Robert Hooke published the Latin phrase ut tensio, sic vis ('as the extension, so the force') to describe how elastic materials push back against deformation.
We express this as Hooke's law: Fs=βˆ’kxF_s = -kx where kk is the spring constant (measured in N/m), and the negative sign indicates a restoring force that always opposes the direction of displacement.
What happens when this restoring force is the only unbalanced force acting on an object of mass mm?

The Defining Condition of SHM

Newton's second law states that net force causes acceleration (Fnet=maF_{\text{net}} = ma). Setting the restoring force equal to mass times acceleration gives: ma=βˆ’kxβ€…β€ŠβŸΉβ€…β€Ša=βˆ’(km)xma = -kx \implies a = -\left(\frac{k}{m}\right)x
This leads to the strict definition: simple harmonic motion (SHM) is any periodic oscillation where acceleration is directly proportional to displacement and directed toward the equilibrium position.
Let's test this with a concrete example: a 0.50Β kg0.50\text{ kg} block on a frictionless track attached to a spring with k=200Β N/mk = 200\text{ N/m} is pulled to x=+0.080Β mx = +0.080\text{ m} and released.