A cylindrical tank has constant cross-sectional area A and a small bottom orifice of area a. The ratio A/a is 100. Initially the water depth above the orifice is 5 m. Neglecting viscosity and taking g = 10 m/s², the total time required to empty the tank is N s. Find N.
Physics · Fluid Mechanics · equation of continuity and bornullie theorem
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Question

A cylindrical tank has constant cross-sectional area A and a small bottom orifice of area a. The ratio A/a is 100. Initially the water depth above the orifice is 5 m. Neglecting viscosity and taking g = 10 m/s², the total time required to empty the tank is N s. Find N.

Physics · Fluid Mechanics · equation of continuity and bornullie theorem

G1 Physics · Accepted Answer

At water depth h, efflux speed is v = √(2gh). Continuity of volume gives −A dh/dt = a√(2gh). Therefore, dt = −(A/a) dh/√(2gh). Integrate from h = H to h = 0: T = (A/a)√(2H/g). Given, A/a = 100, H = 5 m, g = 10 m/s². Thus, T = 100√(2×5/10) = 100√1 = 100 s. Hence, N = 100.