Q. A ring of radius `R` rolls without sliding with a constant velocity .The radius of curvature o f the path followed by any particle of the ring at the highest point of its path will be:- A) `R` B) `2R` C) `4R` D) None.

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When a ring moves by rotation without slipping on a surface the highest points P of the ring will have a speed `2v` and the lowest points will have the speed 0. Thus the highest points of the ring will rotate about the lowest points of the ring with an angular frequency `omega` and a radius of gyration `2R` .

`2v =omega*2R`

It is like when the motion of the upper points is composed of a linear translation with speed `v` of the center of ring and a rotation about the center of the ring with the same angular speed `omega` (see the figure below)

`v_("tot") = v +omega*R = v+v =2v`

Therefore the radius of the curvature of a particle at the highest point of the ring is `2R` . The correct answer is B.

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