Fluid Mechanics

JEE Main level, single correct, medium

76 of 192

Question 76 of 192

A square hole of side length ℓ\ell is made at a depth of h\mathrm{h} and a circular hole of radius r\mathrm{r} is made at a depth of 4 h4 \mathrm{~h} from the surface of water in a water tank kept on a horizontal surface. If ℓ<<h,r<<h\ell\lt \lt h, r\lt \lt h and the rate of water flow from the holes is the same, then rr is equal to

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Answer: (A) ℓ2π\frac{\ell}{\sqrt{2 \pi}}

As A1v1=A2v2A_1 v_1=A_2 v_2 (Principle of continuity) or, ℓ22gh=πr22g×4h\ell^2 \sqrt{2 g h}=\pi r^2 \sqrt{2 g \times 4 h}
(Efflux velocity =2gh)=\sqrt{2 \mathrm{gh}})
∴r2=ℓ22π\therefore \quad r^2=\frac{\ell^2}{2 \pi} \quad or r=ℓ22π=ℓ2π\quad r=\sqrt{\frac{\ell^2}{2 \pi}}=\frac{\ell}{\sqrt{2 \pi}}

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