MHT CET · Physics · Mechanical Properties of Fluids

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

A $r^{\frac{1}{2}} {(π)}^{\frac{1}{2}} {(3)}^{\frac{1}{2}}$
B $r {(π)}^{\frac{1}{4}} {(3)}^{\frac{1}{4}}$
C $r {(π)}^{\frac{1}{2}} {(3)}^{\frac{1}{4}}$
D $r^{\frac{1}{2}} {(π)}^{\frac{1}{3}} {(3)}^{\frac{1}{2}}$

Verified Solution

Answer & Solution

Correct Answer

(C) $r {(π)}^{\frac{1}{2}} {(3)}^{\frac{1}{4}}$

Step-by-step Solution

Detailed explanation

A_{1} V_{1} = A_{1} V_{2}

L^{2} \sqrt{2 gh} = π r^{2} \sqrt{6 gh}

L^{4} gh = π^{2} r^{4} 3 gh

L = (π)^{\frac{1}{2}} (r) {(3)}^{\frac{1}{4}}

L = (r) {(π)}^{\frac{1}{2}} {(3)}^{\frac{1}{4}}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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

Step-by-step Solution

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