Two beads of mass m and 2m are connected by a massless rod of length R and threaded onto a smooth circular wire of radius R. When the wire rotates uniformly with angular velocity ω about a vertical diameters, the beads do not slide on it and rod remains vertical as shown. What must be the angular velocity ω (in rad/s) of the wire ? (Take R = 15 m)

Engineering

Physics

Variable Mass System

Question

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F – N₁ cos 60° = mg  .... (1)
N₁ sin 60° = mω² $(\frac{\sqrt{3} R}{2})$ ... (2)
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F – $\frac{{mω}^{2} R}{2}$ = mg .....(A)
N₂ cos 60° = F + 2mg ..... (3)
N₂ sin 60° = 2mω² $(\frac{\sqrt{3} R}{2})$ .... (4)
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mω²R = F + 2mg ..... (B)
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$\frac{1}{2}$ mω²R = 3mg   $∴$ $ω = \sqrt{\frac{6 g}{R}}$

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