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In this question if magnetic field $B$ is directed into the paper and charge $q$ is attached to the ring of negligible mass. And surface is rough.

Here if I assume bottom most point as 'ICOR' and try to balance torque with respect to 'ICOR'.

Then according to me force on charge $q$ is perpendicular to tangent at surface at this force $F=qV×B$ is passing through centre of ring.

But my book stated that we assumed very bottom most point as 'ICOR' so we will take direction of this force with respect to very bottom most point it means $F$ will pass through its bottommost point.

Please clear my concepts I don't know this concept.

Thank you.

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  • $\begingroup$ Consider to spell out acronyms. $\endgroup$
    – Qmechanic
    Jun 9 at 6:03
  • $\begingroup$ Hi, please don't make trivial/meaningless edits just for the sake of bumping your post to the top. Thanks! $\endgroup$ Jun 9 at 12:58
  • $\begingroup$ @Vincent Thacker sorry brother really very sorry $\endgroup$
    – Florine
    Jun 9 at 17:40
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Well, you've missed a little thing. And here's what:

The equation τ=Iα is valid as long as the point of reference you are using is itself the frame of reference, and lies on the body (even if the point is actually outside the body, we extend the body so that the whole new extended body rotates about the same ICOR) . Thus, the point about which you wish to write the torque need not be an ICOR itself in ground frame. It can be any point, provided you apply the necessary modification (which is the addition of pseudo-force on centre of mass).

The problem you are facing is because you have not written the complete Lorentz Force quantity in the equation. F=q(E + v x B). The given magnetic field is the entire electromagnetic field in ground frame. When you change the frame from ground to the COM, which happens to be moving with an velocity v=w x r with respect to ground, an additional electric field E appears which is necessary for keeping the total Lorentz force constant (as you have only chosen a different inertial frame, so the individual forces does not change.)

So, yes, the electromagnetic force vector, in this question, definitely passes through the ICOR in any frame with respect to the ground. Thus, in order to write the equations with respect to the COM, you have to add an additional force F=qE where E + (w x r) x B= (w x r')xB, where r is the position vector of charge with respect to centre of mass, and r' is the position vector of charge with respect to ICOR. Add this term and you will get the same answer as that given in the book.

Have a nice day!

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