What is simple pendulum ? Deduce an expression for the time period of simple pendulum.

Simple pendulum : "A system of small massive body suspended by a light inexcusable and twice less string from a fixed (rigid) support is called a simple pendulum."

Whole mass of simple pendulum is concentred on the centre of gravity of suspended body.

The distance from the point of suspension to the centre of mass of the bob is called length of pendulum.

An ideal simple pendulum is not possible but a simple pendulum as shown in figure can be taken in practice.

Derivation of expression for periodic time of simple pendulum : Consider simple pendulum a small bob of mass $m$ tied to an inextensible massless string of length $\mathrm{L}$.

The other end of the string is fixed to a support in the ceiling.

The bob oscillates in a plane about the vertical line through the support.

Let $\theta$ be the angle made by the string with the vertical.

There are two forces acting on the body :

$(1)$ Tension $T$ along the string

$(2)$ Vertical force due to gravity $=m g$

The force $m g$ can be resolved into two components.

$(1)$ Parallel component $m g \cos \theta$ which is redial component along string.

$(2)$ Perpendicular component $m g \sin \theta$ which is a tangential component.

The motion of the bob is along a circle of the length $\mathrm{L}$ and centre at the support point, the bob has a radial acceleration $\left(\omega^{2} \mathrm{~L}\right)$ and also a tangential acceleration. The resultant radial force is $T$- $m g \cos \theta$ and tangential force is $m g \sin \theta$.