A tank of cross-sectional area $A_{0}$ is filled with a liquid. A small orifice of area $A$ ( $A < < A_{0}$ ) is present at the bottom of tank. At a moment, the height of liquid in the tank is $H$ , what is magnitude is acceleration of top layer of liquid at this moment as a multiple of ${(\frac{A}{A_{0}})}^{2}$ g?

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Detailed Solution

Velocity of efflux, $V = \sqrt{2 g h}$
Velocity of top layer = $\frac{A V}{A_{0}} = \frac{A}{A_{0}} \sqrt{2 g h}$

$∴$ Acceleration of top layer = $\frac{d}{d t} (\frac{A}{A_{0}} \sqrt{2 g h})$
$\frac{A}{A_{0}} \sqrt{2 g} \times \frac{1}{2 \sqrt{h}} \times \frac{d h}{d t}$

= $\frac{A}{A_{0}} \sqrt{2 g} \times \frac{1}{2 \sqrt{h}} \times \frac{A}{A_{0}} \sqrt{2 g h}$

${(\frac{A}{A_{0}})}^{2} g = 1$

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