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Question:

A smooth wire of length 2π is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed ω about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of ω² is equal to:

3g/2r

2g/(3r)

2g/r

(g/3)r

Solution:

Correct option is D.
2g/(3r)
Nsinθ = mrω².. (1)
Ncosθ = mg.. (2)
tanθ = rω²/2g
23r/2r = rω²/2g
ω² = 2g/(3r)