Question

A large open tank has two holes in the wall. One is a square hole of side L at a depth y from the top and the other is a circular hole of radius R at a depth 4y from the top. When the tank is completely filled with water the quantities of water flowing out per second from both the holes are the same. Then R is equal to

• A. $2 \pi L$
• B. $\frac { L } { \sqrt { 2 \pi } }$
• C. L
• D. $\frac { L } { 2 \pi }$

Answer 1

Answer: (B)

$\frac { L } { \sqrt { 2 \pi } }$

Answer 2

Velocity of efflux when the hole is at depth h, $v = \sqrt { 2 g h }$

Rate of flow of water from square hole $L ^ { 2 } \sqrt { 2 g y }$

Rate of flow of water from circular hole

$Q _ { 2 } = a _ { 2 } v _ { 2 }$= $\pi R ^ { 2 } \sqrt { 2 g ( 4 y ) }$

and according to problem $Q _ { 1 } = Q _ { 2 }$

⇒ $L ^ { 2 } \sqrt { 2 g y } = \pi R ^ { 2 } \sqrt { 2 g ( 4 y ) }$ $R = \frac { L } { \sqrt { 2 \pi } }$