In Simple Harmonic Motion (SHM), the acceleration of an object is directly proportional to its displacement from the equilibrium position and is always directed towards that equilibrium position.

To elaborate, Simple Harmonic Motion is a specific type of periodic motion characterized by a restoring force that is proportional to the displacement from the equilibrium position (the state where the object is at rest). This relationship can be mathematically expressed by the equation:

where:

• $a$ represents the acceleration,
• $\omega$ denotes the angular frequency, and
• $x$ is the displacement from the equilibrium position.

The negative sign in this equation signifies that the acceleration acts in the opposite direction to the displacement, meaning that it is always directed back towards the equilibrium position. This behavior is a hallmark of SHM. Notably, the acceleration reaches its maximum magnitude at the points of maximum displacement (either positive or negative) and is zero when the object is at the equilibrium position.

The angular frequency, $\omega$, is intrinsically linked to the period of the motion, which is the duration required for the object to complete one full cycle of movement. A larger angular frequency indicates a faster motion and a greater acceleration for a given displacement.

This fundamental relationship between acceleration and displacement stems from Hooke’s Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. In the context of SHM, this restoring force is what causes the object’s acceleration. Hence, the farther the object moves from the equilibrium position, the greater the acceleration experienced as it is pulled back toward that position.

In conclusion, 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 core characteristic of SHM and is a direct result of Hooke’s Law.