Show that the oscillation of the loaded sping is Simple Harmonic

Dear student

A particle is said to be simple harmonic  motion if its acceleration is directly proportional to the displacement from a fixed point and is always directed towards that point.

Horizontal and vertical oscillations of a loaded spring are some examples of simple harmonic motion.

The block-spring system is a linear simple harmonic oscillator .All oscillating systems like diving board,violin string have some element of springiness, k(spring constant) and some element of inertia,m.

Simple Harmonic Motion -horizontal Oscillations of Spring:

Consider a mass(m) attached to an end of a spiral spring(whcih obey's Hooke's law) whose other end is fixed to a support as shown in figure.

The body is placed on a smooth horizontal surface. Let the body be displaced through a distance x towards right and released. It oscillates about its mean position, The restoring force acts in the opposite direction and it is proportional to the displacement.

          Restoring force F = -kx

From Newton's second law, F= ma

       ma = -kx

      a = x

Comparing with the equation of simple harmonic motion a = - ²x

But time period T =

Time period T =

Frequency =