A tube of length 1 m is filled completely with an ideal liquid of… | JEE Main 2025 (22 Jan, Shift 2)

Q8JEE Main 2025 · 22 JAN · SHIFT 2Difficult

Centripetal Acceleration & Force

A tube of length 1 m is filled completely with an ideal liquid of mass 2 M, 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 _________.

Numerical answer type. Work it out, then open the answer.

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Answer: 1

The end wall must supply the centripetal force for the whole liquid column.
Element $dm = \frac{2M}{L}dx$ at distance $x$ needs $dm\,\omega^2 x$ .
$F = \int_0^L \frac{2M}{L}\omega^2 x\,dx = M\omega^2 L = M\omega^2$ (with $L = 1$ m).
$\omega = \sqrt{\frac{F}{M}} \Rightarrow \alpha = 1$
Answer: $\alpha = 1$ .

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