Normal and centrifugal force for arbitrary curves

AI Thread Summary
The discussion focuses on deriving equations for normal and centripetal forces acting on a ball rolling down a parametrized curve in a homogeneous gravitational field. The goal is to determine where the ball might leave the curve by calculating these forces at each point. The radius of curvature for the parametric curve is provided as a key formula, which is essential for the calculations. Participants are seeking a comprehensive equation that incorporates the curve's parametrization to find both forces. This approach aims to enhance understanding of the dynamics involved in the motion along arbitrary curves.
Gavroy
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Hi, i have a curve g:[0,t]->IR² with g(t)=(x(t),y(t)) in a homogenous gravitational field and i want to look at a ball rolling down this curve. therefore i want to derive some equations in order to calculate the normal force and the centripetal force at each point of this curve in order to see where the ball "leaves" the curve. therefore i am looking for an equation that gives me both forces just by using the parametrization of my curve. is there an equation for this?
 
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This might help, the radius of curvature for a parametric curve is:

radius(t) = ( (x'(t))2 + (y'(t))2 )3/2 / | (x'(t) y''(t)) - (y'(t) x''(t)) |
 

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