\(A\) આડછેદનું ક્ષેત્રફળ ધરાવતા નળાકાર પાત્રમાં \(h\) ઊંચાઈ સુધી પાણી ભરેલ છે.તેના તળિયે \(a\) ત્રિજ્યાનું કાણું છે તો પાત્રને ખાલી થતાં કેટલો સમય લાગે?

A \(\frac{{2A}}{{\pi {a^2}}}\sqrt {\frac{h}{g}} \)
B \(\frac{{\sqrt 2A }}{{\pi {a^2}}}\sqrt {\frac{h}{g}} \)
C \(\frac{{2\sqrt 2A }}{{\pi {a^2}}}\sqrt {\frac{h}{g}} \)
D \(\frac{A}{{\sqrt 2 \pi {a^2}}}\sqrt {\frac{h}{g}} \)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{{\sqrt 2A }}{{\pi {a^2}}}\sqrt {\frac{h}{g}} \)

Step-by-step Solution

Detailed explanation

Let the rate of falling water level be \( - \frac{{dh}}{{dt}}\) Initially at \(t = 0\,;\,h = h\) \(t = t\,;h = 0\) \(Then,\,A\left( { - \frac{{dh}}{{dt}}} \right) = \pi {a^2}.v\) \(\left[ {Velocity\,of\,efflux\,of\,liquid\,v = \sqrt {2gh} } \right]\) Integrating both sides…

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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