Force in a Rotating Liquid-Filled Tube – Rankers Physics
Topic: Circular Motion
Subtopic: Dynamics of Circular Motion

Force in a Rotating Liquid-Filled Tube

A tube of length \(L\) is filled completely with an incompressible liquid of mass \(M\) and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity \(omega\). The force exerted by the liquid at the other end is:

(2006)

\(\frac{ML\omega^2}{2}\)
\(\frac{ML\omega^2}{2}\)
\(2ML\omega^2\)
\(\frac{ML^2\omega^2}{2}\)

Solution:

Concept: Centrifugal force in a rotating system.
Formula: The force \(F\) is the integral of centrifugal force elements \(dF = dm \cdot r \cdot \omega^2\) from \(0\) to \(L\). Here, \(dm = (M/L)dr\).
Solution: \(F = \int_0^L \frac{M}{L} \omega^2 r dr = \frac{M\omega^2}{L} \left[\frac{r^2}{2}\right]_0^L = \frac{ML\omega^2}{2}\).

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