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The generalized coordinate for the pendulum is the angle of the arm off vertical, theta. Theta is 0 when the pendulum arm is down and pi when the arm is up.

M = mass of pendulum

L = length of pendulum arm

g = acceleration of gravity

\theta = angle of pendulum arm off vertical

\dot{\theta} = time derivative of \theta

What are the kinetic and potential energies?

Kinetic energy: T = (1/2)*M*(L*\dot{\theta})^2

Potential energy: V' = MLg(1-cos(\theta))

V = -MLg*cos(\theta)

--note: we can shift the potential by any constant, so lets choose to drop the MLg

The Lagrangian is L=T-V: L = (1/2)ML^2\dot{\theta}^2 + MLg*cos(\theta)

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Why a compound pendulum is called an equivalent simple pendulum?

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What is the Lagrangian equation for a double pendulum system and how does it describe the small oscillations of the system?

The Lagrangian equation for a double pendulum system is a mathematical formula that describes the system's motion based on its kinetic and potential energy. It helps analyze the small oscillations of the system by providing a way to calculate the system's behavior over time, taking into account the forces acting on the pendulums and their positions.


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Related Questions

What is the Lagrangian formulation for a rotating pendulum?

The Lagrangian formulation for a rotating pendulum involves using the Lagrangian function to describe the system's motion. This function takes into account the kinetic and potential energy of the pendulum as it rotates, allowing for the equations of motion to be derived using the principle of least action.


Is oscillation of compound pendulum or simple pendulum is perfectly simple harmonic motion?

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Why a compound pendulum is called equivalent simple pendulum?

Compound pendulum is a physical pendulum whereas a simple pendulum is ideal pendulum. The difference is that in simple pendulum centre of mass and centre of oscillation are at the same distance.


Why a compound pendulum is called an equivalent simple pendulum?

A compound pendulum is called an equivalent simple pendulum because its motion can be approximated as that of a simple pendulum with the same period. This simplification allows for easier analysis and calculation of its behavior.


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applications of simple pendulum


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