What defines simple harmonic motion (SHM)?

Simple harmonic motion (SHM) is defined as a type of oscillatory motion under a restoring force proportional to the displacement.

In more detail, simple harmonic motion is a special type of periodic motion where the restoring force is directly proportional to the displacement. The displacement is measured from the equilibrium position (the position where the object is at rest with no net force acting upon it). The motion is along a straight line and is oscillatory, meaning it moves back and forth over the same path.

The key characteristic of SHM is that the force responsible for the motion, known as the restoring force, is always directed towards the equilibrium position. This force is also proportional to the displacement from the equilibrium position. This relationship can be expressed mathematically as F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position. The negative sign indicates that the force is always directed towards the equilibrium position.

In SHM, the object's velocity and acceleration change as it oscillates. The object moves fastest as it passes through the equilibrium position and slows down as it reaches the extremes of its motion. The acceleration is highest at the extremes of the motion and zero at the equilibrium position. This is because the restoring force, and hence the acceleration, is highest when the object is furthest from the equilibrium position.

Examples of simple harmonic motion include the motion of a simple pendulum (for small angles of displacement), a mass on a spring, and the oscillations of a molecule about its equilibrium position. In all these cases, the system tends to return to its equilibrium position, and the force responsible for this is proportional to the displacement from equilibrium.

The mathematical description of SHM involves the use of sine and cosine functions. The displacement, velocity, and acceleration as functions of time can all be described using these functions. This is because these functions naturally describe a smooth, periodic oscillation, which is the hallmark of simple harmonic motion.