Question

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 ends is:
(A) $\dfrac{{ML{\omega ^2}}}{2}$
(B) $\dfrac{{M{L^2}\omega }}{2}$
(C) $ML{\omega ^2}$
(D) $\dfrac{{M{L^2}{\omega ^2}}}{2}$

Answer 1

Centripetal force is the force that acts along the rotational motion that acts always towards the centre of the rotation. Use the formula of the centripetal force to find the force that acts at the other end of the tube by substituting the known values in it.

Formula used:
The formula of the centripetal force is given by
$F = m{v^2}l$
Where $F$ is the centripetal force, $m$ is the mass of the fluid in the pipe and $l$ is the distance from the center.

Complete step by step solution:
It is given that the tube is filled with the liquid which is of the mass $M$ and its one of the ends has the angular velocity of $\omega$ .
Since the tube is rotated in a horizontal plane, the force exerted by it is the centripetal force. So, both the ends also exerted by the centripetal force. Hence let us apply the formula of the centripetal force to find the angular velocity at the other end.
$F = m{v^2}l$
The ends are at the distance of half the length of the tube. So, substituting the known factors in the above relation, we get,
For the given condition, the mass is not same in all points, so we have to integrate, then
$F = \int\limits_0^L {\dfrac{M}{L}} {\omega ^2} \times dx$
By integrating the terms in the above equation, then
$F = \dfrac{M}{{2L}}{\omega ^2}{L^2}$
By cancelling the terms in the above equation, then
$F = \dfrac{{ML{\omega ^2}}}{2}$
Hence the force that acts at the end of the tube is obtained as $\dfrac{{ML{\omega ^2}}}{2}$ .
Thus, the option (A) is correct.

Note: The incompressible fluid has the constant flow rate and density throughout the flow and their pressure variation is very small when compared to that of the total absolute pressure. These fluids take up the shape of the container and are generally liquids or gases.