Lagrangian mechanics - rotating rod

In summary, the conversation discusses the concept of kinetic energy in a rotating rod and why it includes both translational and rotational components. The conversation also touches on the physical interpretation and intuition behind this concept. The conclusion is that while it may seem like a purely mathematical approach, understanding this concept can help in solving more complex problems.
  • #1
pj33
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Hello,

It might sound silly, but when I try to calculate the kinetic energy of a rotating rod to form the Langrangian (and in general), why it has both translational and rotational kinetic energy?

Is it because when I consider the moment of Inertia about the centre I need to include the translational since my "frame of reference" is the centre and it moves but when considering about end I only take into aacount the rotatioal since by "frame of reference" (end of rod) is stationary?

I am looking more for a physical interpretation/intuition.
I hope my explanation above is clear enough!

Thank you in advance!
 
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  • #2
You talk about a stationary end of rod, so you must have some specific configuration in mind. Can you divulge to us what it is ?

Note that ##\int {1\over 2} {\dot r}^2 \; dm\ ## is the kinetic energy in all cases. Often it is convenient/ useful/ practical to split it up in parts, e.g. as in your c.o.m plus rotation. A force like gravity works on all ##dm## but can comfortably be considered to work on the c.o.m. that way.

All in all nothing physical, just a mathematical approach :wink:

##\ ##
 
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  • #3
BvU said:
You talk about a stationary end of rod, so you must have some specific configuration in mind. Can you divulge to us what it is ?

Note that ##\int {1\over 2} {\dot r}^2 \; dm\ ## is the kinetic energy in all cases. Often it is convenient/ useful/ practical to split it up in parts, e.g. as in your c.o.m plus rotation. A force like gravity works on all ##dm## but can comfortably be considered to work on the c.o.m. that way.

All in all nothing physical, just a mathematical approach :wink:

##\ ##
I was just thinking of a simple beam attached at one end at a stationary point.
If I understand this, it helps to tackle harder problems. Thank you!
 
  • #4
pj33 said:
I was just thinking of a simple beam attached at one end at a stationary point.

That is a system with a constraint, so already not all that trivial, but surely instructive.

##\ ##
 
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1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of a system of particles, taking into account the forces acting on the particles and their positions and velocities. It was developed by Joseph-Louis Lagrange in the late 1700s.

2. How does Lagrangian mechanics differ from Newtonian mechanics?

In Newtonian mechanics, the motion of a system is described using Newton's laws of motion and the concept of forces. In Lagrangian mechanics, the motion is described using the principle of least action, which states that the actual path taken by a system is the one that minimizes the action (a mathematical quantity) of the system.

3. What is a rotating rod in Lagrangian mechanics?

A rotating rod is a common example used in Lagrangian mechanics to illustrate the principles of the framework. It consists of a rigid rod that is rotating about one end, with a mass attached at the other end. The rod is typically modeled as a simple pendulum, with the mass representing the bob of the pendulum.

4. How is the motion of a rotating rod described using Lagrangian mechanics?

The motion of a rotating rod can be described using the Lagrangian equation, which is derived from the principle of least action. This equation takes into account the kinetic and potential energies of the system, as well as any external forces acting on the system. By solving this equation, the equations of motion for the system can be determined.

5. What are some real-world applications of Lagrangian mechanics?

Lagrangian mechanics is used in a variety of fields, including physics, engineering, and robotics. It is commonly used to model the motion of complex systems, such as satellites and spacecraft, as well as the behavior of fluids and other materials. It is also used in the development of control systems for robots and other mechanical devices.

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