A horizontal oriented tube AB of length l=2.5 m rotates with a constant angular velocity $ω = 5 rad / s$ about a stationary vertical axis ${OO}^{1}$ passing through the end A. Initially 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 (in m/s) relative to the tube when the liquid column length in tube reduce to h=1m.

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answer is 25.

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Detailed Solution

Using Bernoulli’s theorem, we have

$P_{B} = P_{atm} + \frac{1}{2} ρ v^{2} . . . . . . (i)$

and $P_{B} - P_{atm} = \int_{l - h}^{l} dr \cdot ρ \cdot ω^{2} x = \frac{{ρω}^{2}}{2} [l^{2} - {(l - h)}^{2}]$

$P_{B} - P_{atm} = \frac{1}{2} ρ ω^{2} (2 l - h) h . . . . . . (ii)$

$\frac{1}{2} ρ ω^{2} (2 l - h) h = \frac{1}{2} ρ v^{2} \Rightarrow v = ω h \sqrt{(\frac{2 l}{h} - 1)}$

$= 5 \times 2.5 \sqrt{(\frac{2 \times 2.5}{1} - 1)} = 25 m / s$

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