Chapter 1.2, Problem 1P

To determine

The dimension of coefficient of drag.

Answer to Problem 1P

The drag coefficient $C_D$ is dimensionless.

Explanation of Solution

Write the expression for the wind force $\left(F\right)$.

  $F=C_D\rho V^2\frac{A}{2}$                                                (I)

Here, drag coefficient is $C_D$, density of air is $\rho$, velocity of air is $V$, cross-sectional area of the building is $A$.

Conclusion:

The equation (I) can be rearranged to get $C_D$ as follows:

  $C_D=\frac{2F}{\rho V^{2}A}$                                                 (II)

The dimensions of the various terms in the equation (II) are given by:

  $\begin{array}{l}F=ML{T}^{-2}\\ \rho =M{L}^{-3}\\ V=L{T}^{-1}\\ A=L^2\end{array}$

Substitute the above in equation (II) to get the dimensions of $C_D$.

  $\begin{array}{l}{C}_D=\frac{ML{T}^{-2}}{\left(M{L}^{-3}\right){\left(L{T}^{-1}\right)}^2\left(L^2\right)}\\ C_D=\frac{ML{T}^{-2}}{\left(M{L}^{-3}\right)\left(L^2{T}^{-2}\right)\left(L^2\right)}\\ C_D=M^0{L}^0{T}^0\end{array}$

Thus, the drag coefficient $C_D$ is $\text{dimensionless}$.