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Hi, can you explain why the position vector can never be described as a linear combination of r hat and theta hat whereas the velocity and acceleration vectors derived from the position vector are described in terms of a linear combination of r hat and theta hat. Indeed it seems velocity and acceleration vectors at each position are uniquely suited to this coordinate system since they are true vectors unlike the position vector (which starts at the origin and therefore only has an r hat component). This difference (between position vectors and their velocity/acceleration counterparts) seems to extend to the ability to take dot products in this coordinate system as well: dot products don't work for position vectors. Can you shed light on what all this means. Is there a deeper physical significance associated with this difference in the treatment of vectors which doesn't happen for the Cartesian coordinate system. I heard reference to velocity and acceleration being true vectors in the tangent spaces of each point etc and this fits well with a changing basis at each point (as with this coordinate system). I hope you can shed light on this.
If we use a polar coordinate system whose origin is either 1) moving with uniform velocity or 2) accelerating or 3) itself moving around another fixed point, can we use Newton's second law in the r hat and theta hat directions. I suspect we can still do so in case 1) but not if it's accelerating in 2) and 3). If not, how would we deal with such a system with an accelerating origin: I'm thinking something like a spinning ride which is itself on a spinning carousel. Hopefully you can comment on this. Thanks.
You need to derive the appropriate Newton's equations for an accelerating reference frame. You can find these equations in classical mechanics textbooks.
@@ProfJeffreyChasnov Thanks for your reply. Yes but we're not talking about simple LINEAR motion in an accelerating (non-inertial) reference frame, which is what you'll find in the typical mechanics text. That would be trivial enough. I'm talking about a ROTATING object in such an accelerating reference frame (so the frame itself can also be rotating or just accelerating linearly). Clearly, it's not as simple as the correction one would apply for an object in linear motion in such an accelerating (e.g. rotating) frame. Unless you're saying that all we do is add the components of acceleration ä of the origin of the accelerating frame to the r hat and theta hat components of the object's acceleration in that frame to get that object's acceleration in an inertial frame. If you are, surely this isn't trivial, is it? For example take a spinning object B (with its own r hat and theta hat acceleration components) in the frame of another spinning object A (whose centre, for the sake of argument is fixed). So the r hat and theta hat components of acceleration of the CENTRE of spinning object B would first have be obtained (using a polar coordinate system whose origin is the centre of object A). Then we would have to get the r hat and theta hat components of object B using a polar coordinate system whose origin this time is the centre of spinning object B. Are you saying the respective r hat and theta hat components of objects A and B can just be added together then. Surely that can't be the case. From what I understand, we can't add components "across" two polar coordinate systems (with different origins) like this, can we? Maybe you're not saying this at all, sorry.🙂 If not, can you say how one would proceed in this case. Cheers.
