5. Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From Torricelli's principle in hydrodynamics (see Problem 6 in Section 2.3), it follows that the rate at which water flows through the hole is aa √//2gh, where h is the current depth of water in the tank, g is the acceleration due to gravity, and a is a contraction coefficient that satisfies 0.5 ≤ x ≤ 1.0. (a) Show that the depth of water in the tank at any time satisfies the equation dh/dt = (kaa√√/2gh)/A. (b) Determine the equilibrium depth he of water, and show that it is asymptotically stable. Observe that he does not depend on A.
5. Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From Torricelli's principle in hydrodynamics (see Problem 6 in Section 2.3), it follows that the rate at which water flows through the hole is aa √//2gh, where h is the current depth of water in the tank, g is the acceleration due to gravity, and a is a contraction coefficient that satisfies 0.5 ≤ x ≤ 1.0. (a) Show that the depth of water in the tank at any time satisfies the equation dh/dt = (kaa√√/2gh)/A. (b) Determine the equilibrium depth he of water, and show that it is asymptotically stable. Observe that he does not depend on A.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5. Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a
constant rate k and leaks out through a small hole of area a in the bottom of the tank. From
Torricelli's principle in hydrodynamics (see Problem 6 in Section 2.3), it follows that the rate at
which water flows through the hole is aa √//2gh, where h is the current depth of water in the tank, g
is the acceleration due to gravity, and a is a contraction coefficient that satisfies 0.5 ≤ x ≤ 1.0.
(a) Show that the depth of water in the tank at any time satisfies the equation
dh/dt = (kaa√√/2gh)/A.
(b) Determine the equilibrium depth he of water, and show that it is asymptotically stable. Observe
that he does not depend on A.
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