SHM as the shadow of uniform circular motion

If a particle P goes round a circle of radius A at a steady angular speed ω, its projection P′ on any diameter moves in simple harmonic motion.

When OP makes an angle φ with the +x axis at t = 0 and turns anticlockwise, the angle at time t is ωt + φ and the projection is x(t) = A cos(ωt + φ): radius = amplitude, angular speed = angular frequency, starting angle = phase constant.

P is called the reference particle and its circle the reference circle.

A ball whirled in a horizontal circle, watched edge-on or through its shadow on a wall perpendicular to the plane of the circle, appears to move to and fro along a straight line centred on the point of rotation.

The projection on the y-axis, y = A sin(ωt + φ), is also SHM of the same amplitude but differs in phase by π/2 from the x-projection.

A clockwise sense makes the angle decrease with time. A point that starts at 90° and turns clockwise with period T has x-projection B cos(π/2 − 2πt/T) = B sin(2πt/T). With a 30 s period this becomes B cos(πt/15 − π/2): phase constant −π/2.

Starting at 45° and turning anticlockwise with period 4 s, the x-projection is A cos(2πt/4 + π/4): amplitude A, period 4 s, phase constant π/4.

The link is geometric only. The force on a body in linear SHM acts along the line, towards the mean point, and grows with displacement; the centripetal force that keeps P on its circle has a constant magnitude.