Question

Question: 49. Interpret the graph in Figure 6.16(b), which describes the results for falling coffee filters discussed in Example 6.10. Proceed as follows. (a) Find the slope of the straight line, including its units. (b) From Equation 6.6$\mathit{R}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{D}\mathit{\rho }\mathit{A}{\mathit{v}}^{\mathbf{2}}$, identify the theoretical slope of a graph of resistive force versus squared speed. (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters. Model the cross-sectional area of the filters as that of a circle of radius 10.5 cmand take the density of air to be$\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$. (d) Arbitrarily choose the eighth data point on the graph and find its vertical separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph, state what the graph demonstrates and compare it with the theoretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d).

Short answer

(a) The slope of the straight line is 0.016Kg/m.

(b) The theoretical slope of the graph is $\frac{1}{2}D\rho A$.

(c) The value of the drag coefficient is 0.778.

(d) The scatter percentage is 1.6%.

(e) The resistive force is proportional to the square of the terminal speed as suggested by the relation$\left(\frac{1}{2}D\rho A{v}^2\right)$ to about 1.6% uncertainty.

Step-by-step solution

Expression for resistive force

The resistive force acting on an object moving through a liquid or gas as proportional to the object’s velocity, the resistive force can be expressed as

$\overrightarrow{R}=-b\overrightarrow{v}$

Here$\overrightarrow{R}$ is the resistive force,b is a constant whose value depends upon the medium and$\overrightarrow{v}$ is the velocity of the object relative to the medium.

Finding the slope of a straight line

(a)

The graph of the figure 6.16 (b) is shown below.

In order to find the slope of a straight line, choose any two points on the line. Let us choose the origin and ( 10, 0.16 ).

$\begin{array}{c}\text{Slope}=\frac{\Delta R}{\Delta v^2}\\ =\frac{(0.16-0)}{10-0}\\ =0.016\text{\hspace{0.17em}Kg}/\text{m}\end{array}$

Therefore, the slope of the straight line is 0.016Kg/m.

Finding the theoretical slope

(b)

The expression for the resistive force is given by

$R=\frac{1}{2}D\rho A{v}^2$

The expression for the theoretical slope is given by

$\text{Theoretical\hspace{0.17em}slope}=\frac{R}{v^2}$

Substitute$\frac{1}{2}D\rho A{v}^2$ forR in the above equation.

$\begin{array}{c}\text{Theoretical\hspace{0.17em}slope}=\frac{\frac{1}{2}D\rho A{v}^2}{v^2}\\ =\frac{1}{2}D\rho A\end{array}$

Therefore, the theoretical slope of the graph is $\frac{1}{2}D\rho A$.

Finding the experimental slope

(c)

The value of the experimental slope is calculated subpart (a).

Therefore, the experimental slope of the straight line is 0.016 Kg/m.

Now the drag coefficient can be calculated from the expression of theoretical slope.

The theoretical slope of the graph is $\frac{1}{2}D\rho A$.

Here D is the drag coefficient, $\rho$ is the density and A is the area.

The expression for the area is given by

$A=\pi r^2$

Substitute 0.105m for r in the above equation.

$\begin{array}{c}A=\pi {\left(0.105\right)}^2\\ =0.03461{\text{\hspace{0.17em}m}}^{\text{2}}\end{array}$

Now substitute 0.03461 m² for A , 1.20 Kg/m³ for $\rho$and 0.016 Kg/m for theoretical slope in the expression of theoretical slope.

$\begin{array}{c}\text{Theoretical\hspace{0.17em}slope}=\frac{1}{2}D\rho A\\ 0.016=\frac{1}{2}D\left(1.20\right)\left(0.03461\right)\\ D=0.778\end{array}$

Therefore, the value of the drag coefficient is 0.778.

Finding the scatter percentage

(d)

In order to find the scatter percentage, we need to find the resistive force at this point.

$R=mg$

Here m is the mass of eight nested coffee filters.

$\begin{array}{c}R=8\left(1.64\times {10}^{-3}\right)\left(9.8\right)\\ =0.129\text{N}\end{array}$

Find the corresponding vertical coordinate to the point, by multiplying the value of the point by the slope.

$\begin{array}{c}{R}^{\text{'}}=\left(0.0162\right)\left(2.8\right)\\ =0.127\text{N}\end{array}$

$\begin{array}{c}\text{Scatterpercentage}=\frac{R-R^{\text{'}}}{R^{\text{'}}}\times 100\\ =\frac{0.129-0.127}{0.127}\times 100\\ =1.6\%\end{array}$

Therefore, the scatter percentage is 1.6%.

Finding the terminal speed

(e)

The resistive force is plotted as a function of the square of the terminal speed. It can be clearly observed that the resistive force is proportional to the square of the terminal speed as suggested by the relation$\left(\frac{1}{2}D\rho A{v}^2\right)$ to about 1.6% uncertainty.