TITLE: Bead on a rotating hoop, direction of normal force
QUESTION [1 upvotes]: Not having been introduced to Lagrangians, we were discussing this problem in lecture, trying to solve just by using the acceleration of a particle in spherical coordinates and solving the diff eqs that follow from F=ma.
My question is just this-
Why should there be a normal force in the $\phi$ direction?
Given the bead is in contact with the hoop at all times, shouldn't the force just be along the radial direction? I thought normal force always acted perpendicular to the surface the object was in contact with.

REPLY [1 votes]: Your intuition seems correct. If $\phi$ points in the direction of angular velocity, there will be no normal reaction in this direction. As you rightly suggest, it instead points towards the centre of the 'sphere' swept out by the hoop along the radial direction.
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