Stretched or compressed, a spring always tries to return to its natural length. The force that pulls it back is called the restoring force.
Only two things about it matter: its size is proportional to the displacement , and its direction is opposite to . Written out, . The minus sign means points the other way; it does not mean the value of the force is negative. When is negative, is positive.
Put the restoring force into the equation of motion: . Dividing both sides by gives .
Acceleration proportional to displacement and opposite in direction. Simple harmonic motion is the name for motion of this form. It is not simple harmonic because it is a spring. It is simple harmonic because it takes this form. So anything at all, spring or not, moves the same way once its force takes this form.
Here we gather and write it as . That is only naming it; nothing has been shown yet. The equation becomes .
We already know a motion that satisfies : uniform circular motion.
Shine a light straight down on a point going round a circle of radius at angular velocity . The shadow cast on the line below travels back and forth. Work out the acceleration of that shadow and it comes to exactly . The motion of the shadow satisfies the equation of motion just written.
So the position in simple harmonic motion can be written . Depending on where the start of time is taken it may be a or a , but it is the same oscillation. is called the amplitude and the angular frequency. The radius of the circle is the amplitude and the rate of turning is the angular frequency. is called an angular frequency because it began life as the angular velocity of a circle.
The shadow completes one round trip as the point completes one lap of the circle. So the period of the oscillation is the period of the circular motion itself, .
Since , this is . The heavier the weight the slower, the stiffer the spring the faster. Look at the numerator and denominator and you can read off which way each acts.
And does not appear in that equation. Large swing or small, one round trip takes the same time. This is called isochronism. Circles of different radii turning at the same angular velocity cast shadows that reach the ends together and come back together.