A legendary Dutch boy saved Holland by plugging a 1.20 -cm...

A legendary Dutch boy saved Holland by plugging a 1.20 -cm diameter hole in a dike with his finger. If the hole was 2.00 $\mathrm{m}$ below the surface of the North Sea (density $1030 \mathrm{kg} / \mathrm{m}^{3} ),$ (a) what was the force on his finger? (b) If he pulled his finger out of the hole, during what time interval would the released water fill 1 acre of land to a depth of 1 $\mathrm{ft}$ ? Assume the hole remained constant in nsize. (A typical U.S. family of four uses 1 acre-foot of water, $1234 \mathrm{m}^{3},$ in 1 year.)

A legendary Dutch boy saved Holland by plugging a 1.20 -cm diameter hole in a dike with his finger. If the
hole was 2.00 $\mathrm{m}$ below the surface of the North Sea (density $1030 \mathrm{kg} / \mathrm{m}^{3} ),$ (a) what was the force on his finger? (b) If he pulled his finger out of the hole, during what time interval would the released water fill 1 acre of land to a depth of 1 $\mathrm{ft}$ ? Assume the hole remained constant in nsize. (A typical U.S. family of four uses 1 acre-foot of water, $1234 \mathrm{m}^{3},$ in 1 year.)

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure describes how pressure in a fluid increases with depth due to the weight of the overlying fluid. It is calculated using the relation P = ?gh, where ? is the density of the fluid, g is the acceleration due to gravity, and h is the depth below the surface. This concept is fundamental when determining the pressure acting on any submerged surface or object.

Force on a Submerged Surface

The force exerted by a fluid on a submerged surface is determined by the product of the pressure acting on the surface and the area over which it acts. In problems like these, the pressure is generally assumed to be uniform across a small area, allowing the force to be calculated simply as F = P × A. This concept links the hydrostatic pressure to practical applications such as calculating the force on a barrier or plug in a fluid system.

Torricelli’s Law

Torricelli's Law provides a way to determine the speed of a fluid discharging from an opening under the influence of gravity. It states that the speed v is given by sqrt(2gh), where h is the vertical distance from the free surface of the fluid to the orifice. This law is crucial in fluid dynamics for estimating the flow rate from small openings and is applied when analyzing discharge scenarios in various systems.

Flow Rate and Discharge Calculations

Flow rate is the volume of fluid that passes through an opening per unit time and is typically calculated as Q = A × v, where A is the cross-sectional area of the opening and v is the speed of the fluid. In discharge problems, this concept is used to determine how quickly a specific volume of fluid is delivered or removed, linking the dynamics of fluid flow to time-dependent volume changes.

Unit Conversions and Acre-Foot Concept

Volume measurements in water resource management often use units like the acre-foot, which is defined as the volume of water required to cover one acre of area to a depth of one foot. Converting between units such as cubic meters and acre-feet is important in practical applications, allowing engineers and scientists to contextualize large volumes of water in more accessible terms. This concept bridges theoretical calculations with real-world water usage and resource planning.

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