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Chapter 1: Problem 1

The force, \(F\), of the wind blowing against a building is given by \(F=C_{D} \rho V^{2} A / 2,\) where \(V\) is the wind speed, \(\rho\) the density of the air, \(A\) the cross-sectional area cf the building, and \(C_{D}\) is a constant termed the drag coefficient. Determine the dimensions of the drag coefficient.

Short Answer

Expert verified
The dimensions of the drag coefficient, \(C_D\), is dimensionless, often denoted by \(M^{0}L^{0}T^{0}\).

Step by step solution

01

Analyze the given equation

The given equation is \(F=C_{D} \rho V^{2} A / 2\), where each variable represents a physical quantity. Here, \(F\) is force, \(C_{D}\) is the drag coefficient, \(\rho\) is the air density, \(V\) is the wind speed, and \(A\) is the cross-sectional area of the building.
02

Write down the dimensions of known quantities

We have the dimensions of the following quantities: \n - Force, \(F\) is measured in \(ML/T^{2}\) where \(M\) is mass, \(L\) is length and \(T\) is time. \n - Air density, \(\rho\) is mass per unit volume, so it's measured in \(M/L^{3}\). \n - Speed, \(V\) is distance per unit time, so it's measured in \(L/T\). \n - Cross-sectional Area, \(A\) is measured in \(L^{2}\).
03

Determine the dimensions of the drag coefficient

Setting up the equation with respect to the dimensions gives: \n \([F] = [C_D][\rho][V]^2[A]\), where \([x]\) denotes the dimension of \(x\). \nSubstituting the dimensions of the known quantities gives: \n \(ML/T^{2} = [C_D](M/L^{3})(L/T)^{2}(L^{2})\). \nSolving this for \([C_D]\) gives the dimensions of the drag coefficient, \(C_D\), as \(M^{0}L^{0}T^{0}\) or dimensionless.

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