Kinematics, curvilinear motion (a couple of questions)

Thread starter srnixo
Start date Jan 27, 2024
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Position vector

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Homework Help Overview

The discussion revolves around kinematics, specifically curvilinear motion and the use of different coordinate systems such as tangential-normal (n-t) and Cartesian coordinates. Participants explore the existence of the position vector in these systems and the appropriateness of each coordinate system for analyzing motion along curved paths.

Discussion Character

Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

Participants question the existence of the position vector in the n-t coordinate system and discuss the implications of using different coordinate systems for curvilinear motion. There is also inquiry into the distinction between curvilinear and curved motion.

Discussion Status

The discussion is active, with various interpretations being explored regarding the position vector and the use of coordinate systems. Some participants express confusion and seek clarification, while others provide insights and references to support their points. No explicit consensus has been reached on the existence of the position vector in n-t coordinates.

Contextual Notes

Participants reference differing sources and interpretations regarding the position vector and coordinate systems, indicating a lack of uniformity in understanding. There is mention of constraints based on the professor's teaching and the nature of the problems being discussed.

Jan 28, 2024

haruspex

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srnixo said:

the tangential and normal coordinate systems do not have a fixed origin (the origin moves with the object). Therefore, the position vector of the object will constantly change

Ok, so in eqn 1, what does ##\vec r## mean? It is not the location of the object moving along the path s since that is by definition (0,0) in n-t. It must be the location of some other point expressed in n-t coordinates relative to the object, but what point?

##\rho \hat n## is the location of the centre of curvature, but what is ##R(t)##? I can find no online reference to "radius of curvature in tangent direction", and @Delta2's proposed interpretation in post #19 makes it the same as ##\rho##.

Here's another interpretation: ##R## and ##\rho## (or perhaps it is ##P##, not ##\rho##) have nothing to do with radii of curvature. All the eqn is saying is that given some arbitrary point ##\vec r## and a point on a curve as origin, ##\vec r## can be expressed in curvilinear coordinates as ##(R, P)##, meaning ##R\hat t+P\hat n##.