Q64 / 90 · JEE Main 2012 (07 May Online) · Physics

A square hole of side length $\ell$ is made at a depth of $\mathrm{h}$ and a circular hole of radius $\mathrm{r}$ is made at a depth of $4 \mathrm{~h}$ from the surface of water in a water tank kept on a horizontal surface. If $\ell< < h, r< < h$ and the rate of water flow from the holes is the same, then $r$ is equal toFigure

  • Correct answer
    A
    $\frac{\ell}{\sqrt{2 \pi}}$
  • B
    $\frac{\ell}{\sqrt{3 \pi}}$
  • C
    $\frac{\ell}{3 \pi}$
  • D
    $\frac{\ell}{2 \pi}$

Solution

As $A_1 v_1=A_2 v_2$ (Principle of continuity) or, $\ell^2 \sqrt{2 g h}=\pi r^2 \sqrt{2 g \times 4 h}$ (Efflux velocity $=\sqrt{2 \mathrm{gh}})$ $\therefore \quad r^2=\frac{\ell^2}{2 \pi} \quad$ or $\quad r=\sqrt{\frac{\ell^2}{2 \pi}}=\frac{\ell}{\sqrt{2 \pi}}$

Hence, the correct answer is option: 'A'.

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