Derivation of Time Period of a Simple Pendulum
The time period of a simple pendulum is the time taken for the pendulum to complete one full oscillation. It is denoted by the symbol T. The time period of a simple pendulum can be derived using the formula:

T = 2π√(l/g)
where T is the time period, l is the length of the pendulum, and g is the acceleration due to gravity.

Explanation:
- Angle of Displacement: When a pendulum is displaced from its equilibrium position by an angle θ, it experiences a restoring force that is proportional to sinθ.
- Force Equation: The force equation for a simple pendulum is given by F = -mgl sinθ, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and l is the length of the pendulum.
- Angular Acceleration: Using Newton's second law, we can derive the equation of motion for the pendulum and find the angular acceleration as θ'' = - (g/l) sinθ.
- Small Angle Approximation: For small angles (sinθ ≈ θ), we can simplify the equation of motion to θ'' = - (g/l) θ.
- Solution: The general solution to this differential equation is θ(t) = A sin(√(g/l)t + φ), where A and φ are constants.
- Time Period: The time period of the simple pendulum is the time taken for one complete oscillation, which can be calculated as T = 2π√(l/g).
By deriving the time period formula for a simple pendulum, we can understand the factors affecting the oscillations of the pendulum and how the length of the pendulum and acceleration due to gravity influence its motion.