Lagrangian for Conical Pendulum: T-V

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The Lagrangian for a conical pendulum is expressed as L = 1/2 I ω^2 - mgy, where I represents the moment of inertia and ω is the angular frequency. The pendulum has a fixed length l and a bob mass m, rotating in a horizontal circle at a constant angle θ with respect to the vertical z-axis. The distance r is defined as the distance from the fulcrum to the center of mass of the cone, and the relationship y = r sin(θ) is established. The discussion notes that this formulation does not account for spin motion and assumes two-dimensional motion, while acknowledging that three-dimensional motion could introduce precessional effects. Understanding these dynamics is crucial for accurately modeling the behavior of conical pendulums.
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What is the Lagrangian L=T-V for a conical pendulum? This is a pendulum with length l and bob mass m that rotates in a horizontal circle with theta (angle l makes with z axis) and phi(dot) (angular velocity-omega) are constant (cylindrical coordinate system).
 
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L =1/2 I w^2 - mgy

I is the moment of inertia
and w is the angular frequency w =d(theta)/dt and y =r sin(theta)

cos could be sin depending where you take your angle. and r is the distance from the fulcrum to the center of mass of the cone.
This doesn't include spin motion, and assumes the motion is two dimensional. If the motion were three dimensional and there was spin, you could get some precesional effects.
 
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