To derive the equation for a large angle pendulum, we can start by considering the forces acting on the pendulum bob. Let's assume the pendulum bob has a mass 'm' and is initially displaced at an angle 'θ' from the vertical equilibrium position. The forces acting on the bob are the gravitational force and the tension force in the string.
The gravitational force can be broken down into two components: one along the string (tension force) and the other perpendicular to it. The tension force provides the restoring force that brings the pendulum back to its equilibrium position. At large angles, we need to take into account the fact that the restoring force is not strictly proportional to the displacement as it is in small angle approximations.
Using trigonometric relationships, we can express the gravitational force components as follows:
- The component along the string: mg * sin(θ)
- The component perpendicular to the string: mg * cos(θ)
The tension force in the string acts along the direction opposite to the displacement and can be represented as 'T'.
Now, applying Newton's second law in the radial (perpendicular) direction, we have:
ma_r = -mg * cos(θ) + T
where a_r is the radial acceleration of the pendulum bob.
Additionally, we can apply Newton's second law in the tangential (along the string) direction:
ma_t = T * sin(θ)
Since the length of the string is constant, we have a_t = L * d^2θ/dt^2, where 'L' is the length of the pendulum and 'd^2θ/dt^2' is the angular acceleration.
Combining the above equations, we get:
m * L * d^2θ/dt^2 = T * sin(θ) (1)
From the radial equation, we can rearrange it to solve for T:
T = mg * cos(θ) + ma_r (2)
Substituting equation (2) into equation (1), we have:
m * L * d^2θ/dt^2 = (mg * cos(θ) + ma_r) * sin(θ)
Expanding and simplifying, we get:
m * L * d^2θ/dt^2 = mg * cos(θ) * sin(θ) + ma_r * sin(θ)
Since ar = L * d^2θ/dt^2, the above equation becomes:
m * L * d^2θ/dt^2 = mg * cos(θ) * sin(θ) + L * d^2θ/dt^2 * sin(θ)
Dividing both sides by m * L, we get:
d^2θ/dt^2 = g * cos(θ) * sin(θ) + d^2θ/dt^2 * sin(θ)
Now, rearranging the terms, we obtain the equation for a large angle pendulum:
d^2θ/dt^2 + g * sin(θ) = 0
This is a nonlinear differential equation that describes the motion of a large angle pendulum. Solving this equation analytically is challenging, but numerical methods or approximations can be used to study the behavior of the system.