Question

A vessel completely filled with water has holes 'A' and 'B' at depths 'h' and '3h' from the top respectively. Hole 'A' is a square of side 'L' and 'B' is a circle of radius 'r'. The water flows out per second from both the holes in the same. Then 'L' is equal to ______. 

विकल्प

• `"r"^(1/2)(π)^(1/2)(3)^(1/2)`

• `"r"(π)^(1/2) (3)^(1/2)`

• `"r"(π)^(1/2) (3)^(1/4)`

• `"r"^(1/2)(π)^(1/3)(3)^(1/2)`

Answer 1

A vessel completely filled with water has holes 'A' and 'B' at depths 'h' and '3h' from the top respectively. Hole 'A' is a square of side 'L' and 'B' is a circle of radius 'r'. The water flows out per second from both the holes in the same. Then 'L' is equal to `underlinebb("r"(π)^(1/2) (3)^(1/4))`

Explanation:

Velocity of efflux for A: v₁ = `sqrt(1"gh")`

Velocity of efflux for B: v₂ = `sqrt(2"g" xx 3"h") = sqrt(6"gh")`

Water flowing out from A = Water flowing out from B .....(Given)

∴ v₁A₁ = v₂A₂

Since, Area of square (A₁) = L²

Area of circle (A₂) = πr²

∴ `sqrt(2"gh") xx "L"^2 = sqrt(6"gh") xx π"r"^2`

∴ `"L"^2 = sqrt(6"gh")/sqrt(2"gh") xx π"r"^2`

= `sqrt3 xx π"r"^2`

L = `3^(1/4) xx π^(1/2)`"r"

`"r"(π)^(1/2) (3)^(1/4)`