Question

A horizontally oriented tube AB of length $l$ rotates with a constant angular velocity $\omega$ about a stationary vertical axis OO' passing through the end A (fig). The tube is filled with an ideal fluid. The end A of the tube is open, the closed end B has a very small orifice. Find the velocity of the fluid relative to the tube as a function of the column 'height' $h$.

• A. $v=\omega \sqrt{h(l-h)}$
• B. $v=\omega \sqrt{h(2l-h)}$
• C. $v=\omega h$
• D. $v=\omega \sqrt{2h(l-h)}$

Answer 1

Answer: (B)

$v=\omega \sqrt{h(2l-h)}$

As the tube rotates, the centrifugal force (as seen in the rotating frame of the tube) creates a pressure head causing the water to be jetted out of the orifice at the end of tube

Consider, an infinitesimally thin slice of water of width $dx$ at a distance $x$ from the one end of the tube. Let the area of cross-section of the tube be $s$. The mass of water contained in this infinitesimally thin slice of water is given by $dm=\rho sdx$, here $\rho$ is the density of water. the centrifugal force by this slice of water on the next is given by,

$df=\left(dm\right)w^{2}x=\rho s{\omega }^{2}xdx$...............(1)

the net force acting at the end of the tube just before the orifice can be obtained by integrating over all such thin slice as,

${\int }_0^{f}df={\int }_{l-h}^l\rho s{\omega }^{2}xdx$ or,

$f=\frac{1}{2}\rho s{\omega }^{2}h(2l-h)$ or,

$P=f/s=\frac{1}{2}\rho {\omega }^{2}h(2l-h)$.................(2)

In (2), $P$ is the pressure head generated by the centrifugal force acting on the water. the net pressure at the end of the tube just before the orifice is given by $P+P_o$, when $P_o$ is the atmosphereic pressure. We assume that at this point the water is still i.e its velocity is zero. Right outside the tube, the water jets into an area of pressure $P_i$ (atmospheric pressure), say at a velocity $v$. Using Bernoulli's equation at point just before and just outside the orifice we have,

$P+P_o+\frac{1}{2}\rho (0{)}^2=P_0+\frac{1}{2}\rho v^2$ or,

$P=\frac{1}{2}\rho v^2$

or, $v=\omega \sqrt{h(2l-h)}$