A legendary Dutch boy saved Holland by plugging a hole of...

A legendary Dutch boy saved Holland by plugging a hole of diameter $1.20 \mathrm{~cm}$ in a dike with his finger. If the hole was $2.00 \mathrm{~m}$ below the surface of the North Sea (density $\left.1030 \mathrm{~kg} / \mathrm{m}^{3}\right),$ (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 size.

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Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational force acting on it. It increases linearly with depth, and is given by 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 free surface. This concept is fundamental in determining the force acting on submerged surfaces.

Force Due to Fluid Pressure

The force on a surface due to fluid pressure is obtained by multiplying the pressure acting on the surface by the area of that surface. In the context of a circular hole, the force is calculated as F = P·A, where A is the area of the hole (?(d/2)²). This concept is crucial for understanding how pressures translate into forces on physical objects in fluids.

Orifice Flow and Torricelli's Law

When a fluid exits through an orifice, the speed of the fluid can be determined by Torricelli's law, which states that the exit velocity is v = ?(2gh), assuming the fluid is ideal and the flow is streamlined. The volumetric flow rate Q is then given by Q = A·v, where A is the area of the opening. This principle is essential for analyzing problems involving discharge of fluid from an opening, and for determining the time required to fill a given volume.

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