Simple harmonic motion occurs when motion in a circle with uniform acceleration is projected on a line through the centre of the circle. The motion is most naturally projected onto the x and y axes.

Suppose the motion is projected onto the– axis. Then (1)

We can differentiate to obtain the velocity: (2)

and again to get the acceleration: (3)

We can write (3) asso

This is often taken as the basic equation of simple harmonic motion, and means that the the acceleration is proportional to the acceleration, but directed towards the zero point on the– axis.

We can use the identityto obtain further equations.

From (1)and from (2)

Substituting these into the identity gives

Similarly for projection onto the– axis: (4)

We can differentiate to obtain the velocity: (5)

and again to get the acceleration: (5)

We can write (5) asso

We can use the identityto obtain a further equation.

From (4)and from (5)

Substituting these into the identity gives

If the frequency isthenso we can also write

andwith corresponding equations for