Simple Harmonic motion is a physical phenomenon first observed by Galileo. The story goes that he was in mass one Sunday, and, bored with the service, noted that the chandeliers above the altar were swaying slightly. So bored he was in fact, that he took it upon himself to count out how long each swing of the chandelier took. To his surprise, he found that the time remained the same for each swing. Another Sunday, another boring mass and Galileo observed another chandelier and found that the time for each swing of this chandelier was also a constant. Galileo was later put under house arrest for life by the church, but this wasn’t just for being bored during mass.
Let’s break it down first, so that we have an initial idea about what simple harmonic motion might be. Simple – well that just means it’s easy (so far, so good). Motion – that’s just things moving (still pretty good). Harmonic – The idea of things being in harmony might sound familiar to musicians in terms of pure notes (a wave with constant amplitude), and the idea is the same here. So, simple harmonic motion is just an easy description of things moving with a constant amplitude (amplitude is the maximum distance from the starting point).
One of the best examples of simple harmonic motion is a mass hanging on a spring.
Initially the mass will be still, as the force of gravity acting on it will be cancelled out by the tension in the spring, the forces are said to be in equilibrium, and where this happens is known, unsurprisingly, as the equilibrium position. But, if you give the mass a pull downwards and then release it, it will move upwards, as the tension in the spring will have increased. However, the mass won’t stop moving when it returns to its original equilibrium position, even if there is no resultant force acting on it.
When the mass is moving above the equilibrium position, gravity creates a force in the downward direction. The resultant force acting downwards means that the mass will slow down, until it has stops moving for an instant, at the maximum displacement. Gravity will then continue to accelerate the mass downward back toward the equilibrium position.
Once the mass reaches the equilibrium position again, there will be no resultant force acting, but the mass will carry on moving down. As the mass passes the equilibrium position on its way down, the force of the spring acting up will overcome gravity, slowing the mass down, until it is at rest for an instant. The mass will then move upwards as the tension in the spring overcomes gravity. Then, well, skip up a couple of paragraphs.
There are a few key points in simple harmonic motion. The energy in the system is always the same, just constantly flipping from potential, locked up in the spring and the height of the mass, to kinetic, in the movement of the mass. The acceleration of the mass is always in the opposite direction to the displacement from the equilibrium. The amplitude of the system is fixed, as is the frequency of the mass.
There are a few equations that fully define simple harmonic motion, which are described here: https://isaacphysics.org/concepts/cp_shm