A tank of cross-sectional area A₀ contains a liquid; a small hole of area A (A ≪ A₀) is at the bottom. At liquid height H, express the downward acceleration of the top surface as x(A/A₀)²g. Find x.
Physics · Mixed Chapter
JEE Main Practice

Question

A tank of cross-sectional area A₀ contains a liquid; a small hole of area A (A ≪ A₀) is at the bottom. At liquid height H, express the downward acceleration of the top surface as x(A/A₀)²g. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Torricelli's law for efflux gives v = √(2gH). Step 2: Continuity gives downward speed of upper surface u = (A/A₀)√(2gH). Step 3: Because dH/dt = −u, differentiate u² = 2gH(A/A₀)²: 2u(du/dt) = 2g(A/A₀)²(−u). Step 4: The downward acceleration has magnitude g(A/A₀)², hence x = 1.