Define linear S.H.M. obtain a differential equation of linear S.H.M?

Linear Simple Harmonic Motion (S.H.M.) refers to a type of oscillatory motion in which a particle or object moves back and forth along a straight line. It occurs when the restoring force acting on the object is directly proportional to the displacement from its equilibrium position and acts in the opposite direction.

In linear S.H.M., the motion of the object is characterized by the following properties:

Restoring Force: The restoring force, denoted by F, acts in the opposite direction to the displacement of the object from its equilibrium position. It is proportional to the displacement and can be represented as F = -kx, where k is the force constant (also known as the spring constant) and x is the displacement.

Periodic Motion: The object undergoes repetitive motion with a constant period. The period, denoted by T, is the time taken for one complete oscillation (back and forth motion).

Amplitude: The amplitude, denoted by A, is the maximum displacement of the object from its equilibrium position.

To obtain the differential equation of linear S.H.M., we can apply Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.

Let's consider a particle of mass m undergoing linear S.H.M. Its displacement from the equilibrium position at any time t is given by x(t). The acceleration of the particle at any time t is the second derivative of the displacement with respect to time, which can be represented as d²x/dt².

Using Newton's second law, we can equate the force acting on the particle to its mass multiplied by its acceleration:

m * (d²x/dt²) = -k * x

Dividing both sides of the equation by m gives us:

(d²x/dt²) + (k/m) * x = 0

This is the differential equation that governs linear S.H.M. It is a second-order linear homogeneous differential equation, where (d²x/dt²) represents the second derivative of the displacement with respect to time, k/m is the angular frequency (ω), and x is the displacement. The negative sign indicates that the restoring force is opposite to the displacement.