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A thin smooth straight tube $OA$ is constrained to rotate with constant angular velocity $ω$ about a fixed vertical axis through $O$, and a particle is free to move in the tube. The angle between $OA$ and the upward vertical is a fixed acute angle $α$ and describe a fixed horizontal circle of radius $a$, While the particle is in a state of steady motion the angular velocity of the tube is suddenly reduced to $\dfrac{\omega}{2}$ and is then maintained constant at the new value. Find the time the particle takes to reach O,

My problem is

Is this particle always equilibrium vertically that is $Rsin(\alpha)=mg$ where $R$ is reaction on particle

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If you resolve the forces, then $R cos(\alpha) + mg = ma sin(\alpha)$ as the motion of the particle is constrained within the tube, and it will try to move towards the center. Hence vertical motion will be not be in equillibrium always, as there will be a vertical component of the net acceleration.

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