Simple Pendulum:
A simple pendulum consists of a small mass (known as the bob) attached to a string of length 'L'. When the bob is displaced from its equilibrium position and released, it oscillates back and forth under the influence of gravity.

Relation for Time Period:
The time period of a simple pendulum is the time taken for the pendulum to complete one full oscillation. The formula for the time period (T) of a simple pendulum is given by:
\[ T = 2 \pi \sqrt{\frac{L}{g}} \]
where 'g' is the acceleration due to gravity (approximately 9.81 m/s^2).

Derivation:
- The restoring force acting on the bob is given by F = -mg sinθ, where 'm' is the mass of the bob and 'θ' is the angle of displacement.
- Using Newton's second law, F = ma, we get -mg sinθ = ma.
- For small angles (sinθ ≈ θ), the equation becomes -mgθ = ma.
- Substituting a = -Lθ, we get -mgθ = -mLθ.
- Simplifying, we get θ = -(g/L)θ.
- The angular frequency ω of the pendulum is given by ω = √(g/L), and the time period T = 2π/ω.

Frequency:
The frequency (f) of a simple pendulum is the number of complete oscillations it makes in one second. The formula for frequency is given by:
\[ f = \frac{1}{T} = \frac{1}{2 \pi} \sqrt{\frac{g}{L}} \]