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Chapter 9: Problem 16

In the Netherlands, a dike holds back the sea from a town below sea level. The dike springs a leak \(3.0 \mathrm{m}\) below the water surface. If the area of the hole in the dike is \(1.0 \mathrm{cm}^{2},\) what force must the Dutch boy exert to save the town?

Short Answer

Expert verified
Answer: The force required is approximately 2.943 N.

Step by step solution

01

Identify the given values

The given values are: - Depth of the leak below the water surface: \(h = 3.0\,\mathrm{m}\) - Area of the hole in the dike: \(A = 1.0\,\mathrm{cm}^2\) The density of water is approximately \(\rho = 1000\,\mathrm{kg/m^3}\) and the acceleration due to gravity is \(g = 9.81\,\mathrm{m/s^2}\).
02

Convert the area to SI units

Since we are dealing with SI units (meters and kilograms), we need to convert the area of the hole from square centimeters to square meters. \(1\,\mathrm{cm}^2\) = \(0.0001\,\mathrm{m}^2\) Therefore, the area of the hole in the dike is: \(A = 0.0001\,\mathrm{m}^2\)
03

Calculate the pressure exerted by the water

We are now ready to calculate the pressure exerted by the water at a depth of 3 meters. We'll use the formula \(P = h \rho g\), where \(h = 3.0\,\mathrm{m}, \rho = 1000\,\mathrm{kg/m^3}\), and \(g = 9.81\,\mathrm{m/s^2}\) \(P = (3.0\,\mathrm{m})(1000\,\mathrm{kg/m^3})(9.81\,\mathrm{m/s^2}) = 29,430\,\mathrm{Pa}\)
04

Calculate the force required to stop the leak

Finally, we can now calculate the force required to stop the leak using the formula \(F = PA\), where \(P = 29,430\,\mathrm{Pa}\) and \(A = 0.0001\,\mathrm{m}^2\) \(F = (29,430\,\mathrm{Pa})(0.0001\,\mathrm{m}^2) = 2.943\,\mathrm{N}\) The force that the Dutch boy must exert to save the town is approximately \(2.943\,\mathrm{N}\).

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