Elliptic Pendulum: Homework on Kinetic & Potential Energies

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SUMMARY

The discussion focuses on deriving the kinetic and potential energies of a uniform rod AB of mass M and length 2l, which is attached to a slider constrained to move along a smooth horizontal wire. The participants aim to express these energies using the angle Ω and the x-coordinate of the center of mass, ultimately leading to the formulation of Lagrange's equations. Key equations mentioned include the kinetic energy expression KE = 1/2[Mx² + θ²Ml²sin²(θ)] and the velocity relation v = -2lcos(θ). The discussion emphasizes the necessity of treating the rod as a continuum to derive the Lagrangian correctly.

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  • Understanding of Lagrangian mechanics
  • Familiarity with kinetic and potential energy concepts
  • Knowledge of angular motion and trigonometric functions
  • Ability to work with calculus and differential equations
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  • Study the derivation of Lagrange's equations in classical mechanics
  • Learn about the treatment of continuous bodies in physics
  • Explore the concept of the center of mass and its implications in dynamics
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Homework Statement



A uniform rod AB of mass M and length 2l attatched to a slider at A which is constrained to move along a smooth horizontal wire. The rod, which is subject to gravity, is free to swing in the vertical plane containing the wire. Using Ω (the angle between rod and vertical) and x coordinate of the centre of mass, find an expression for the kinetic and potential energies, deduce Lagrange's equations. Show that the Lagrangian is regular.



Homework Equations





The Attempt at a Solution


I know I have to treat the rod as a continuum, but I can't get around it...

Sorry it doesn't seem like I've done much but I've tried every way I know and nothing's working, any advice would be greatly appreciated. Thank you

Could you please start me of...I know v=-2lmcosθ

But to get T could you please help me... I don't know what my x coordinate is
 
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KE = 1/2[Mx.^2 + θ.^2 Ml^2 sin^2(θ)]

I have seemed to get this...could any1 please check if I have done this right
 

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