What is meant by simple harmonic oscillation ? Give example

Explanation

Simple harmonic motion is a special type of oscillatory motion in which the acceleration or force on the particle is directly proportional to its displacement from a fixed point and is always directed towards that fixed point. In one dimensional case, let x be the displacement of the particle and `a_(x)` be the acceleration of the particle, then
`a_(x)propx" "...(1)`
`a_(x)=-bx" "...(2)`
where b is a constant which measures acceleration per unit displacement and dimensionally it is equal to `T^(-2)` By multiplying by mass of the particle on both sides of equation (1) and from Newton.s second law, the force is
`F_(x)=-kx" "...(3)`
where k is a force constant which is defined as force per Force verses displacement graph unit length. The negative sign indicates that displacement and force (or acceleration) are in opposite directions. This means that when the displacement of the particle is taken towards right of equilibrium position (x takes positive value), the force (or acceleration) will point towards equilibrium (towards left) and similarly, when the displacement of the particle is taken towards left of equilibrium position (x takes negative value), the force (or acceleration) will point towards equilibrium (towards right). This type of force is known as restoring force because it always directs the particle executing simple harmonic motion to restore to its original (equilibrium or mean) position. This force (restoring force) is central and attractive whose center of attraction is the equilibrium position.
In order to represent in two or three dimensions, we can write using vector notation
`vecF=-kvecr" "...(4)`
where `vecr` is the displacement of the particle from the chosn origin. Note that the force and displacement have a linear relationship. This means that the exponent of force `vecF` and the exponent of displacement `vecr` are unity. The sketch between cause (magnitude of force `|vecF|`) and effect (magnitude of displacement `|vecr|`) is a straight line passing through second and fourth quadrant.
By measuring slope `(1)/(k)`, one can find the numerical value of force constant k.