Lagrangian of conic pendulum-rod

• Hanuda
In summary, the conversation discusses a pivoted stick with mass m and length l that swings around a vertical axis with angular frequency omega. The stick maintains an angle theta with the direction of gravity and has two degrees of freedom, theta and phi. The task is to determine the kinetic energy, which is shown to be a function of phi-dot, theta-dot, and theta, using spherical coordinates. The question also inquires about the meaning of the variable r in the solution.
Hanuda

Homework Statement

So, basically there is a stick, mass m and length l, that is pivoted at its top end, and swings around the vertical axis with angular frequency omega. The stick always makes an angle theta with the direction of gravity. I am told there are 2 degrees of freedom (theta, phi), with phi being the angle around the 'equator' (see attached picture). Phi-dot=omega.

a) Determine the kinetic energy and show that it is given by a function of the type T(phi-dot, theta-dot, theta).

The attempt at a solution

I assume I am supposed to use spherical coordinates for theta and phi, so it should look something like:

$$T=\frac{1}{2}m(\dot r^2 + r^2\dot \theta^2 + r^2sin^2(\theta)(\dot \phi)^2)$$

But we know that the kinetic energy of the pendulum only depends on theta, theta-dot, and phi-dot. So how should it look?

Attachments

• Screen shot 2014-03-15 at 20.38.51.png
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What is the meaning of your r ?

1. What is the Lagrangian of a conic pendulum-rod?

The Lagrangian of a conic pendulum-rod is a mathematical expression that describes the energy and motion of a pendulum attached to a rigid rod that is constrained to a conic section, such as an ellipse or hyperbola.

2. Why is the Lagrangian useful in analyzing the motion of a conic pendulum-rod?

The Lagrangian allows us to describe the motion of the pendulum-rod system using a single equation, rather than multiple equations for each individual component. This makes it easier to analyze and understand the motion and energy of the system.

3. How is the Lagrangian of a conic pendulum-rod derived?

The Lagrangian is derived using the principles of Lagrangian mechanics, which is a mathematical framework for analyzing the dynamics of physical systems. In the case of a conic pendulum-rod, the Lagrangian is derived by considering the kinetic and potential energy of the system.

4. Can the Lagrangian of a conic pendulum-rod be used to find the equations of motion?

Yes, the Lagrangian can be used to derive the equations of motion for the conic pendulum-rod system. By taking the derivative of the Lagrangian with respect to time, we can obtain the equations of motion for the pendulum and the rod.

5. Are there any limitations to using the Lagrangian for analyzing a conic pendulum-rod?

The Lagrangian approach assumes that the system is conservative, meaning that there is no external energy input or energy loss during the motion. This may not always be the case in real-world situations, so the Lagrangian may not provide an accurate representation of the system's behavior in all cases.

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