Two identical cylindrical vessels with their bases at the same level each contain a liquid of density ρ (rho). The height of the liquid in one vessel is h1 and in other is h2. The area of either base is A. What is the work done by gravity in equalizing the levels when the tow vessels are connected?

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Grade 11Mechanics

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density ρ (rho). The height of the liquid in one vessel is h₁ and in other is h₂. The area of either base is A. What is the work done by gravity in equalizing the levels when the tow vessels are connected?

Radhika Batra

12 Years agoGrade 11

2 Answers

Kevin Nash

12 Years ago

Hello Student,

Please find the answer to your question

P.E. of liquid in cylinder 1

U₁ = (m) g h₁ / 2 = ( ρ x A x h₁ ) g h₁ / 2 = ρAgh²₁ / 2

Note :

[The total mass can be supposed to be concentrated at the center of the filled part which will be at height h₁ / 2]

Similarly P.E. of liquid in cylinder 2 U₂ = ρAgh²₂ / 2

∴ Total P.E. initially U = U₁ + U₂ (h²₁ + h²₂)

After the equalising of levels.

P.E. of liquid in cylinder 1 U₁ = mg h / 2 = ρAg / 2 h²

P.E. of liquid in cylinder 2 U₂’ = mg h / 2 = ρAg / 2 h²

∴ Total P.E. finally U’ = U₁’ + U₂’ = ρAgh²

The change in P.E.

∆U = U – U’ = ρAg [h²₁ / 2 + h²₂ / 2 – h²]

Total volume remains the same.

Ah₁ + Ah₂ = Ah + Ah

⇒ h = h₁ + h₂ / 2

Therefore,

∆U = ρAg [h²₁ / 2 + h²₂ / 2 – (h₁ + h₂ / 2)²]

= ρAg / 4 (h₁ – h₂)²

This change in P.E. is the work done by gravity

Thanks
Kevin Nash
askIITians Faculty

ankit singh

5 Years ago

wo identical cylindrical vessels with their bases at the same levelTwo identical cylindrical vessel with their bases at the same level each contain a liquid of density ρ. The height of the liquid in one vessel is h1 and in the other is h2 the area of either base is A