To derive an equation for the time period of a simple pendulum when the amplitude is not small, we can use the small-angle approximation to simplify the analysis.
The equation of motion for a simple pendulum is given by:
θ''(t) + (g/L) sin(θ(t)) = 0
Where: θ(t) is the angular displacement of the pendulum at time t, θ''(t) represents the second derivative of θ(t) with respect to time, g is the acceleration due to gravity, and L is the length of the pendulum.
When the amplitude of the pendulum's motion is small, we can make the approximation that sin(θ) ≈ θ. However, when the amplitude is not small, this approximation is not valid.
For larger amplitudes, the exact solution for the time period T of the pendulum is not readily obtainable in terms of elementary functions. However, an approximation can be made using an elliptic integral called the complete elliptic integral of the second kind (denoted as K(θ)), which is related to the period of the pendulum.
The equation for the time period T of a simple pendulum with arbitrary amplitudes is given by:
T = 4 * √(L/g) * K(θ)
Where: L is the length of the pendulum, g is the acceleration due to gravity, and θ is the amplitude of the pendulum's motion in radians.
In this equation, K(θ) represents the complete elliptic integral of the second kind, which involves more complex mathematical techniques for evaluation, such as numerical methods or special mathematical functions.
So, to obtain an equation for the time period of a simple pendulum with a non-small amplitude, we rely on the complete elliptic integral of the second kind to account for the effects of larger amplitudes.