Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression of periodic time.

According to figure a block of mass $m$ fixed to a spring, which in turn is fixed to a rigid wall.

The block is placed on a friction less horizontal surface.

If the block is pulled on one side and is released it then executes a to and fro motion about a mean position.

Let $x=0$, indicate the position of the centre of the block when the spring is in equilibrium. The positions - A and + A indicate the maximum displacements to the left and the right of the mean position.

For spring Robert Hooke law, "Spring when deformed, is subject to a restoring force, the magnitude of which is proportional to the deformation or the displacement and acts is opposite direction."

Let any time $t$, if the displacement of the block from its mean position is $x$, the restoring force $\mathrm{F}$ acting on the block it

$\mathrm{F}(x)=-k x \quad \ldots$ $(1)$

Where $k$ is constant of proportionality and is called the spring constant or spring force constant.

Equation $(1)$ is same as the force law for $SHM$ and therefore the system executes a simple harmonic motion.