Mechanics of a torsion pendulum

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Discussion Overview

The discussion revolves around the mechanics of a torsion pendulum, specifically focusing on the formulation of its Lagrangian, the effects of damping, and the implications of Hamilton's Principle. Participants explore the interplay between rotational and translational kinetic energy, potential energy from gravity and spring forces, and the conditions for small oscillations.

Discussion Character

  • Technical explanation
  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about writing a Lagrangian for a torsion pendulum, considering both rotational and translational energies, and asks about potential forces of constraint.
  • Another participant suggests that the change in gravitational potential energy is negligible and provides expressions for kinetic and potential energy, leading to a proposed frequency for small vibrations.
  • A question is raised about how damping would affect the Lagrangian, prompting a response that indicates the Lagrangian remains unchanged but requires a nonconservative virtual work term.
  • Clarification is sought regarding the term "variational indicator," with a participant referencing Hamilton's Principle and its implications for the Lagrangian formulation.
  • A suggestion is made to consult a specific book for a detailed treatment of the topic, indicating the complexity of the formulation.
  • A participant expresses uncertainty about the previous explanations and seeks further clarification, indicating a lack of familiarity with the mechanics involved.

Areas of Agreement / Disagreement

There is no consensus on the impact of damping on the Lagrangian, as participants present differing views on how to incorporate it into the analysis. The discussion remains unresolved regarding the clarity of certain concepts and the implications of Hamilton's Principle.

Contextual Notes

Participants express varying levels of familiarity with the mechanics involved, indicating potential gaps in understanding certain terms and principles, such as "variational indicator" and Hamilton's Principle.

bojibridge
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I'm curious as to how you would write a Lagrangian for a torsion pendulum. Say you've got some disc that has mass and it's attached by a spring with spring constant k to a stationary "ceiling" of some kind and has gravity acting on it downwards. So, not only has it got rotational energy, but it's also moving up and down, so it's got some kind of translational kinetic energy as well. I guess the potential part of it would be from gravity and then from the spring potential. Can you think of any forces of constraints that might go along with it? Any other interesting points? Perhaps one could find the frequency of small oscillations using Taylor series? I guess I'm just looking to pick up all the pieces involved in such a problem.

(This is not a homework question per se, but if someone thinks it would go better in the homework discussions, I can move it.)
 
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Typically the change in gravitational potential energy is negligible in such a problem as this. Therefore, the kinetic energy is
T = (1/2)*J*w^2
ane the potential energy is
V = (1/2)*k*theta^2
where w = theta-dot
Then L = T-V as usual.

The frequency for small vibrations is f,

2*pi*f = sqrt(k/J)

It is unrealistic to expect significant axial (vertical) motion while talking about small angular motions only. Even for rather large angular motions, the axial motion is usually very slight, so this is just not much of a problem. It is hard to make axial motion a part of the problem short of mounting the rotor on a screw.
 
Would the Lagrangian change if the spring's motion was damped? If so, how would you take that into account?
 
The Lagrangian does not change. Damping has to be taken into account with a nonconservative virtual work term in the variational indicator.
 
I'm not familiar with the term "variational indicator." Do you mean when you set the Lagrangian equal to... um, where you set it to some generalized external force?
 
Are you acquainted with Hamilton's Principle? Hamilton's Principle is the statement that the time integral of the variation of the Lagrangian + the variation in the nonconservative virtual work is zero over the time between two states t1 and t2. This leads to the Lagrange eqns of motion. The potential energy portion of the Lagrangian gives rise to the potential driven generalized forces, and the nonservative virtual work accounts for the nonservative generalized forces (such as friction, time dependent forces, etc).
 
You might want to look fo the book Dynamics of Mechanical and Electromechanical Systems by Crandall, et. al. McGraw-Hill, 1968 (?). It givess an exceptionally fine treatment of this whole formulation.
 
Okay - can you expand on your last sentence? I'm not sure I totally understand what you're saying. (My mechanics is pretty rusty, sorry.)

Oh, I just read your second post. I'm going to see if I can find it.
 

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