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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 $ω$ . The force exerted by the liquid at the other end is

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${Mω}^{2} L / 2$

${Mω}^{2} L$

${Mω}^{2} L / 4$

${Mω}^{2} L^{2} / 2$

answer is A.

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Detailed Solution

Consider a small element dl at a distance l from the axis. Mass of the element is

$\frac{M}{L} dl$ Centrifugal force on this element $= (\frac{Mdl}{L}) (ω^{2} l)$

Centrifugal force on the whole liquid $= \frac{M}{L} ω^{2} \int_{0}^{L} ldl = \frac{{MLω}^{2}}{2}$

This is the force exerted by the liquid at the other end.

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