Rotation 5 Question 9
19. A tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both the 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
(a) $\frac{M \omega^{2} L}{2}$
(b) $M \omega^{2} L$
(c) $\frac{M \omega^{2} L}{4}$
(d) $\frac{M \omega^{2} L^{2}}{2}$
(1992, 2M)
Objective Question II (One or more correct option)
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Answer:
Correct Answer: 19. (a)
Solution:
- Mass of the element $d x$ is $m=\frac{M}{L} d x$.
This element needs centripetal force for rotation.
$$ \begin{array}{ll} \therefore & d F=m x \omega^{2}=\frac{M}{L} x \omega^{2} d x \\ \therefore & F=\int _0^{L} d F=\frac{M}{L} \cdot \omega^{2} \int _0^{L} x d x=\frac{M \omega^{2} L}{2} \end{array} $$
This is the force exerted by the liquid at the other end.
