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

(N/A) Simple pendulum: $A$ system consisting of a small,heavy body (bob) suspended by a light,inextensible,and flexible string from a fixed (rigid) support is called a simple pendulum.
The entire mass of the simple pendulum is considered to be concentrated at the center of gravity of the suspended bob.
The distance from the point of suspension to the center of mass of the bob is called the length of the pendulum $(L)$.
Derivation of the expression for the time period $(T)$:
Consider a simple pendulum with a small bob of mass $m$ tied to an inextensible,massless string of length $L$.
The other end of the string is fixed to a support. Let $\theta$ be the angle made by the string with the vertical.
Two forces act on the bob:
$(1)$ Tension $T$ along the string.
$(2)$ Gravitational force $mg$ acting vertically downwards.
The force $mg$ can be resolved into two components:
$(1)$ Radial component: $mg \cos \theta$ (along the string).
$(2)$ Tangential component: $mg \sin \theta$ (perpendicular to the string).
The restoring force is $F = -mg \sin \theta$. For small oscillations,$\sin \theta \approx \theta$ (in radians).
So,$F = -mg \theta = -mg (x/L)$,where $x$ is the displacement.
Comparing with $F = -kx$,we get $k = mg/L$.
The time period is $T = 2\pi \sqrt{m/k} = 2\pi \sqrt{m / (mg/L)} = 2\pi \sqrt{L/g}$.