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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In curvilinear motion, the position vector does not exist in the tangential and normal coordinate system, as these coordinates move with the particle and do not have a fixed origin. The tangential and normal coordinate system is preferred for known curved paths, while Cartesian coordinates are more suitable for unknown paths. Curvilinear motion refers to movement along a curved path, independent of the coordinate system used, whereas rectilinear motion is along a straight line. The distinction between curvilinear and curved motion is subtle, with curvilinear specifically indicating one-dimensional motion along a curve. Understanding these differences is crucial for accurately analyzing motion in physics.

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##.