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Question

A bead 'B' is threaded on a smooth circular wire frame of radius r. The radius at a certain position of the bead makes an angle θ with the vertical axis. The entire frame is rotated with an angular speed ω about the vertical axis. The distance of the bead from the vertical axis is R. If the bead is in equilibrium (with respect to the rotating frame), then identify the correct statements:
  1. ω=gtanθR
  2. R=rsinθ
  3. Nsinθ=mω2R
  4. ω=rgcosθ


Solution

The correct options are
A ω=gtanθR
B R=rsinθ
C Nsinθ=mω2R

The bead is in equilibrium with respect to the rotating wire frame, which is rotating about a vertical axis passing through its centre.
Hence a centrifugal force Fcentrifugal=mω2r will appear to act in the outward direction, when observed from the rotating frame.

Applying the condition of equilibrium for bead as per FBD:
Nsinθ=mω2R   .....(1)
Ncosθ=mg      .....(2)

Distance of bead from vertical axis
R=rsinθ............(3)

Dividing (1) by (2):
sinθcosθ=ω2Rg
ω=gtanθR

Option a,b,c are correct.

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