A vessel completely filled with water has holes 'A' and 'B' at depths 'h' and '2h' from the top, respectivel. Hole 'A' is a square of side 'L' and 'B' is circle of radish 'r' The water flowing out per second from both the holes is same Then, 'L'is equal to

Options

A. `r^((1)/(2))(pi)^((1)/(2))(3)^((1)/(2))`
B. `r*(pi)^((1)/(4))(3)^((1)/(4))`
C. `r*(pi)^((1)/(2))(3)^((1)/(4))`
D. `r^((1)/(2))(pi)^((1)/(3))(3)^((1)/(2))`

Answer

C

Explanation

Given, depth of hole A from the top = h Depth of hole B from the top =3h According to question, we can draw the following diagram

According the continuity equation,
`A_(a)v_(a)=A_(2)v_(2)`
Velocity of efflux,
`v_(a)=sqrt(2ght)` (for square hole
`v_(2)=sqrt(2g(3h))=sqrt(6gh)` (for circular hole)
`L^(2)sqrt(2gh)=pir^(2)sqrt(6gh)`
`(because"area of square "-L^(2) "area of circle"=pir^(2))`
On squaring both sides =, we get
`2L^(4)gh=6pi^(2)r^(4)gh" "2L^(4)=6pi^(2)r^(4)`
`L^(4)=3pi^(2)r^(4)" "L=(3pi^(2))^(1//4)`
`L=rpi^(1//2)(3)^(1//4)`