Chapter 1, Problem 60P

To determine

The relation for the drag force based on unit considerations alone.

Answer to Problem 60P

The drag force relation is $F_D=C_{drag}{A}_{frontal}\rho V^2$.

Explanation of Solution

Given information:

The drag force is a function of drag coefficient, frontal area, air density and velocity.

Write the expression for the drag force.

    $\begin{array}{c}{F}_D\propto \left(C_{drag},A_{frontal},\rho ,V\right)\\ F_D=C_{drag}{\left(A_{frontal}\right)}^a{\rho }^b{V}^c\end{array}$   ..... (I)

Here, the drag coefficient is $C_{drag}$, the frontal area is $A_{frontal}$, the density of air is $\rho$, the velocity is $V$ and the constant values for unit dimensions are $a,b,c$.

Write the expression for the dimensions of all variables in $MLT$ system.

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

    $C_{drag}=\left[M^0{L}^0{T}^0\right]$

Calculation:

Substitute $\left[ML{T}^{-2}\right]$ for $F_D$, $\left[L^2\right]$ for $A_{frontal}$, $\left[M{L}^{-3}\right]$ for $\rho$, $\left[M^0{L}^0{T}^0\right]$ for $C_{drag}$, $\left[L{T}^{-1}\right]$ for $V$ in Equation (I).

    $\begin{array}{l}\left[ML{T}^{-2}\right]=\left[M^0{L}^0{T}^0\right]{\left[L^2\right]}^a{\left[M{L}^{-3}\right]}^b{\left[L{T}^{-1}\right]}^c\\ \left[ML{T}^{-2}\right]={\left[M\right]}^{0+b}{\left[L\right]}^{0+2a+3b+c}{\left[T\right]}^{0-1}\\ \left[ML{T}^{-2}\right]={\left[M\right]}^b{\left[L\right]}^{2a+3b+c}{\left[T\right]}^{-c}\end{array}$   ..... (II)

Compare powers of $M$ in Equation (II).

    $b=1$

Compare powers of $L$ in Equation (II).

    $2a-3b+c=1$   ..... (III)

Compare powers of $T$ in Equation (II).

    $\begin{array}{l}-c=-2\\ c=2\end{array}$

Substitute $1$ for $b$ and $2$ for $c$ in Equation (III).

    $\begin{array}{l}2a-3\left(1\right)+2=1\\ 2a=2\\ a=1\end{array}$

Substitute $1$ for $a$, $1$ for $b$ and $2$ for $c$ in Equation (I).

    $\begin{array}{l}{F}_D=C_{drag}{\left(A_{frontal}\right)}^1{\rho }^1{V}^2\\ F_D=C_{drag}{A}_{frontal}\rho V^2\end{array}$

Conclusion:

The drag force relation is $F_D=C_{drag}{A}_{frontal}\rho V^2$.