Rolling Object on Curved Surface: Lagrangian Mechanics + Constraint

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SUMMARY

The discussion focuses on plotting the trajectory of a ball rolling without slip on a curved surface using Lagrangian mechanics. The key variables include the radius, mass, and moment of inertia of the ball, along with the curvature formula and the relationship between path length and height. The participant identified a flaw in their assumption regarding the linear decrease of height with arc length, concluding that the potential term is a more complex function of arc length. This insight highlights the need for a deeper understanding of the relationship between height and path length in this context.

PREREQUISITES
  • Lagrangian mechanics fundamentals
  • Understanding of Euler-Lagrange equations
  • Knowledge of curvature in physics
  • Concept of rolling motion without slip
NEXT STEPS
  • Explore advanced applications of the Euler-Lagrange equations
  • Study the relationship between potential energy and path length in rolling objects
  • Investigate the effects of moment of inertia on rolling motion
  • Learn about non-linear dynamics in mechanical systems
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Students and professionals in physics, particularly those studying mechanics, as well as engineers and researchers working on dynamic systems involving rolling objects.

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Homework Statement



I want to be able to plot a trajectory wrt time of a ball that rolls without slip on a curved surface.

Known variables:
-radius/mass/moment of inertia of the ball.
-formula for the curvature of the path (quadratic)
-formula relating path length and corresponding height above the ground. (linearly decreasing, i.e. as you travel down the curve, you decrease in height).

Homework Equations


Euler-Lagrange Equations

The Attempt at a Solution


See attached for further details.

I've come up with an answer for the acceleration of the ball along the path length. However, it isn't a function of the starting height as I expected. I suspect I may have made an assumption that I'm not supposed to but I can't see what it is.
 

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After some though and comparison to classical mechanics, I've determined that the assumption that the height of the ball decreases linearly with arc length is not correct.

It seems my potential term is going to be some complicated function of arc length.

Anyone else have thoughts?
 

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