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

A legendary Dutch boy saved Holland by plugging a hole in a dike with his finger, which is 1.20 $\mathrm{cm}$ in diameter. 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, how long would it take the released water to fill 1 acre of land to a depth of 1 $\mathrm{ft}$ , assuming the hole remained constant in size? (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 is the pressure exerted by a fluid at rest due to the force of gravity. It increases linearly with depth and is calculated using the formula 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 key concept helps in determining the force exerted on submerged surfaces.

Force Calculation from Pressure

Force on a surface in a fluid can be determined by multiplying the pressure by the area over which the pressure is acting (F = pA). When the pressure is uniform across the surface, this relationship directly links the hydrostatic pressure to the force experienced by objects immersed in the fluid.

Fluid Flow Rate Through an Orifice

When fluid exits through an orifice under gravitational force, the speed of exit can be estimated using Torricelli's law, which gives v = ?(2gh). The flow rate, defined as the volume of fluid passing through an area per unit time, is then calculated by multiplying the cross-sectional area of the orifice by the exit velocity (Q = A?(2gh)).

Time to Fill a Volume Based on Flow Rate

Determining the time required to fill a given volume involves dividing the total volume by the volumetric flow rate. This calculation necessitates consistent unit conversion and application of the connection between flow rate and time, highlighting the relationship between fluid dynamics and practical volume measurements.

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