A tube of length 1 m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is $F$ then angular velocity of the tube is $\sqrt{\frac F{\alpha M}}$ in SI unit. The value of $\alpha$ is __________.

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] The liquid element at distance $r$ needs a centripetal force from the pressure gradient. The force at the far end equals the centripetal force on the whole liquid: $F=\int_0^L\frac{2M}L\,\omega^2r\,dr=M\omega^2L$. With $L=1$ m, $\omega=\sqrt{\frac FM}$, so $\alpha=1$. Answer: 1.

Explanation

Reviewed by Dr. Abubacker

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