A woman is draining her fish tank by siphoning the water...

A woman is draining her fish tank by siphoning the water into an outdoor drain, as shown in Figure P14.74. The rectangular tank has footprint area $A$ and depth $h$. The drain is located a distance $d$ below the surface of the water in the tank, where $d>>h$. The cross-sectional area of the siphon tube is $A^{\prime}$. Model the water as flowing without friction. (a) Show that the time interval required to empty the tank is given by $$\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}$$ (b) Evaluate the time interval required to empty the tank if it is a cube $0.500 \mathrm{m}$ on each edge, if $A^{\prime}=2.00 \mathrm{cm}^{2},$ and $d=10.0 \mathrm{m}$ (FIGURE CAN'T COPY)

A woman is draining her fish tank by siphoning the water into an outdoor drain, as shown in Figure P14.74. The rectangular tank has footprint area $A$ and depth $h$. The drain is located a distance $d$ below the surface of the water in the tank, where $d>>h$. The cross-sectional area of the siphon tube is $A^{\prime}$. Model the water as flowing without friction.
(a) Show that the time interval required to empty the tank is given by
$$\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}$$
(b) Evaluate the time interval required to empty the tank if it is a cube $0.500 \mathrm{m}$ on each edge, if $A^{\prime}=2.00 \mathrm{cm}^{2},$ and $d=10.0 \mathrm{m}$
(FIGURE CAN'T COPY)

Key Concepts

Integration of Differential Equations

Integrating differential equations is a common method in fluid mechanics to determine the time evolution of systems where variables change continuously. In this problem, it is used to relate the rate at which water leaves the tank (linked to the cross-sectional area and velocity) to the total volume of water in the tank, ultimately leading to the expression for the emptying time.

Bernoulli's Principle

Bernoulli's Principle is a fundamental concept in fluid dynamics that states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or potential energy. In the context of siphoning water from a tank, this principle helps relate the pressure difference between the water surface and the drain to the fluid’s velocity, assuming frictionless flow.

Torricelli's Law

Torricelli's Law is derived from Bernoulli's Principle and states that the speed of efflux of a fluid under the force of gravity from an orifice is equal to the square root of twice the product of the gravitational acceleration and the vertical distance between the fluid surface and the opening (v = ?(2gd)). This law is crucial for determining the outflow velocity in the siphoning process.

Continuity Equation

The Continuity Equation in fluid dynamics asserts that the mass flow rate of a fluid must remain constant along a streamline in the absence of any additions or losses. This means that the product of the cross-sectional area of the flow and the fluid velocity is conserved. In the siphon problem, it relates the outlet area of the tube to the volumetric flow rate out of the tank.

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