Simple Harmonic Motion

Context

Consider a mass m attached to one end of an elastic spring can move freely on a frictional horizontal surface.

Hookes law

When the mass is displaced to right through a distance x, the force at that instant is given by Hooke’s law i.e. F =kx

Spring constant

k is a constant known as spring constant. its units are N/m or kg/s2

Restoring force F

Due to elasticity, spring opposes the applied force which produces the displacement. This opposing force is called restoring force which is equal and opposite to the applied force within elastic limit of the spring.Hence,
F = -kx

The negative sign indicates that F is directed opposite to x i.e. towards the equilibrium position. Thus in a system obeying Hooke’s law, the restoring force is directly proportional to the displacement of the system and is always directed towards it.

Acceleration due to restoring force

When the mass is released, it begins to oscillate to and fro about its mean position. The oscillatory motion taking place under the action of a restoring force is called simple harmonic motion. The acceleration due to restoring force can be calculated by second law of motion.

Derivation

F = ma

-kx = ma

a = -kx/m

a is directly proportional to -x

Definition of SHM

The oscillatory motion in which the acceleration of a body is directly proportional to the displacement and is always directed towards the mean position is called SHM.