15.6:
Simple Harmonic Motion and Uniform Circular Motion
Consider the Moon at a distance A from the center of the Earth, rotating in a circular motion with a constant angular velocity.
Let the Earth's center be the origin of the displacement-time coordinate system. When the Moon moves to a position P, its projection P' on the x-axis makes an angle Ф.
As the Moon moves further around the Earth at any time t, it makes an angle ωt+Ф. Based on the Moon's projection on the x or y-axis, the position of the projection can be denoted by either a cosine or sine function.
The period of the Moon can be given by the circumference of the Earth's orbit over its velocity. Recalling the velocity equation from energy conservation and modifying it, the period of the Moon's projection is determined.
The velocity of the Moon acts tangentially, while the acceleration of the Moon is directed radially inward.
The x-component of the velocity and acceleration of the Moon is equal to the velocity and acceleration of the Moon's projection. Their magnitudes are obtained by recalling the velocity and acceleration equations.
As observed, the time period, position, velocity, and acceleration equations of the Moon's projection are similar to that of a simple harmonic oscillator.
Therefore, the projection of a uniform circular motion along the diameter of a circle on which the circular motion occurs represents simple harmonic motion.
15.6:
Simple Harmonic Motion and Uniform Circular Motion
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.