## Simple Harmonic Motion

Simple harmonic motion can be defined as the type of periodic and oscillatory motion, where the restoring force acts in the direction opposite to the displacement of the particle and is directly proportional to the displacement of the particle.

As we know, Newton’s second law of motion related the force acting on a system and the corresponding acceleration produced by the force. Thus, if the acceleration with which the particle is moving is known, the force acting on the particle can be determined.

For an object executing simple harmonic motion, the acceleration of the particle is given by,

a(t) = -ω^{2}x(t), as the acceleration of a particle is directly proportional to the displacement of the particle and is opposite to the displacement of the particle. Here, ω is the angular velocity of the particle.

From Newton’s second law of motion, the force acting on an object is proportional to the product of its mass and acceleration. Mathematically,

F = m × a

F × a (t) = – (mω^{2}x(t))

F = -k x(t)

Here we can see that the force acting on the particle executing simple harmonic motion is directly proportional to the displacement of the particle. This is known as the Hooke’s Law.

Let us understand this concept more clearly with the help of the following example.

The block-spring system forms a linear simple oscillator where linear indicates that F is proportional to the displacement of the block from its mean position.

For a simple spring, the spring constant is denoted as k = mω^{2}.

Angular frequency (ω)

As we know, angular frequency (ω)

Combining the two equations, we can write,

Time period (T)

Such a system is also referred to as a linear harmonic oscillator.

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