Simple harmonic motion is a fundamental type of periodic motion found throughout physics. It occurs when an object experiences a restoring force that is directly proportional to its displacement from equilibrium. The spring-mass system and pendulum are classic examples, where the motion is repetitive and predictable. Key characteristics include the equilibrium position where the net force is zero, the amplitude representing maximum displacement, and the restoring force that always acts toward equilibrium.
The mathematical description of simple harmonic motion begins with the restoring force equation F equals negative k x, where k is the spring constant and x is displacement. This leads to the differential equation m times the second derivative of x equals negative k x. The solution is x of t equals A cosine of omega t plus phi, where omega equals the square root of k over m. The graphs show the relationships between position, velocity, and acceleration, which are related through derivatives and have specific phase relationships.
The key parameters of simple harmonic motion include period T equals 2 pi times the square root of m over k, and frequency f equals 1 over 2 pi times the square root of k over m. The amplitude A represents the maximum displacement and affects the energy but not the frequency. The phase constant phi determines the initial conditions. Here we see three oscillators with different phase constants, demonstrating how they start at different positions but maintain the same frequency and period.