While simple harmonic motion and uniform circular motion may be two separate concepts, they correlate and interlink with each other. Simple harmonic motion is an oscillatory motion in a system where the net force can be described by Hooke's law, while uniform circular motion is the motion of an object in a circular path at constant speed.
There is an easy way to produce simple harmonic motion by using uniform circular motion. For instance, consider a ball attached to a uniformly rotating vertical turntable, with its shadow projected on the floor. Here, the position of the shadow, also known as projection, performs simple harmonic motion. Hooke's law usually describes uniform circular motions (⍵ constant) rather than systems that have large visible displacements. Thus, observing the projection of uniform circular motion is often easier than observing a precise large-scale simple harmonic oscillator.
Another example is a record player undergoing uniform circular motion. Consider a dowel rod attached at one point on the outside edge of a turntable and a pen attached to the other end of the dowel. As the record player turns, the pen moves. If we drag a long piece of paper under the pen, its motion is captured as a wave. From this, it can be understood that the simple harmonic motion is a projection of uniform circular motion along the diameter of a circle in which the circular motion occurs.
This text is adapted from Openstax, College Physics, Section 16.6: Uniform Circular Motion and Simple Harmonic Motion and Openstax, University Physics Volume 1, Section 15.3: Comparing Simple Harmonic Motion and Circular Motion.
