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 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 $\mathrm{ft}$ ? Assume the hole remained constant in size.

Key Concepts

Hydrostatic Pressure

This concept involves the pressure exerted by a fluid at a given depth due to the weight of the overlying fluid. It is calculated using the relation P = ?gh, where ? is the fluid density, g is the acceleration due to gravity, and h is the vertical depth below the free surface. Hydrostatic pressure increases linearly with depth and is fundamental in understanding how forces act on submerged surfaces.

Force Calculation by Pressure

The force exerted by a fluid on a submerged surface is determined by multiplying the pressure acting on the surface by the area over which it is applied (F = PA). This principle is critical in many fluid mechanics applications, where knowing the distributed pressure and the area involved helps compute the resultant force acting on structures or objects in a fluid.

Discharge Flow Rate from an Orifice

When a hole or orifice is submerged and a fluid flows out due to hydrostatic pressure, the volumetric flow rate can be estimated using principles from fluid dynamics. Typically, Torricelli's law is applied, which relates the exit velocity of the fluid as v = ?(2gh), and the flow rate is then determined by multiplying this velocity by the area of the orifice. This concept is essential for understanding how fast fluids discharge under gravity from openings.

Unit Conversion and Volume Calculations

Problems in fluid mechanics often require conversions between different units of volume, area, and time. This involves converting between systems (e.g., metric to imperial) or between different scales (e.g., acres to square meters, feet to meters). Proper unit conversion is critical to ensure consistency in calculations, particularly when determining volumes and the time required to fill or empty containers or areas.

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