How is acceleration related to displacement in SHM?

In Simple Harmonic Motion (SHM), acceleration is directly proportional to displacement and is always directed towards the equilibrium position.

In more detail, Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement. This displacement is measured from the equilibrium position (the position where the object is at rest or not in motion), and is always directed towards it. The relationship between acceleration and displacement in SHM can be expressed by the equation a = -ω²x, where 'a' is acceleration, 'ω' is the angular frequency, and 'x' is displacement.

The negative sign in the equation indicates that the acceleration is always directed towards the equilibrium position, which means it is always trying to bring the object back to this position. This is a defining characteristic of SHM. The acceleration is maximum at the maximum displacement (either positive or negative) and is zero at the equilibrium position.

The angular frequency 'ω' is related to the period of the motion, which is the time it takes for one complete cycle of the motion. The larger the angular frequency, the faster the motion and the greater the acceleration for a given displacement.

This relationship between acceleration and displacement is a result of Hooke's Law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position. In SHM, this force is what causes the acceleration of the object. Therefore, the greater the displacement from the equilibrium position, the greater the acceleration towards it.

In summary, in Simple Harmonic Motion, the acceleration of an object is directly proportional to its displacement from the equilibrium position and is always directed towards it. This relationship is a fundamental characteristic of SHM and is a direct consequence of Hooke's Law.