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)
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.
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.
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.
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.
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.
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.
A simple pendulum exhibits simple harmonic motion
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.
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.
A simple pendulum has one piece that swings. A complex pendulum has at least two swinging parts, attached end to end. A simple pendulum is extremely predictable, while a complex pendulum is virtually impossible to accurately predict.
applications of simple pendulum
The simple pendulum can be used to determine the acceleration due to gravity.
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.
A 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.
simple pendulum center of mass and center of oscillation are at the same distance.coupled pendulum is having two bobs attached with a spring.
The motion of the simple pendulum will be in simple harmonic if it is in oscillation.