Chapter 15, Problem 36P

To determine

The plot for car speed against power required to overcome (a) rolling resistance, (b) aerodynamic resistance, and (c) combined effect.

Explanation of Solution

Given:

Mass of the car $\left(m\right)$ is 950 kg.

Drag coefficient $\left(C_D\right)$ is 0.32.

Frontal area of the car $\left(A\right)$ is $1.8{\text{ m}}^2$.

Coefficient of rolling resistance $\left(C_{RR}\right)$ is 0.04.

The speed of car $\left(V\right)$ varies from $0\text{km/h}$ to $150\text{km/h}$.

Density of air $\left({\rho }_a\right)$ is $1.20\text{kg}/{\text{m}}^3$.

The maximum power output of the engine is 80 kW.

Calculation:

Calculate the rolling resistance of the car.

  $\begin{array}{c}{F}_{RR}=C_{RR}W\\ =C_{RR}\left(mg\right)\\ =\left(0.04\right)\left(950\text{kg}\times \text{9}\text{.81}{\text{m/s}}^2\right)\\ =327.78\text{N}\end{array}$

When the velocity of the air over the car is $30\text{km/h}$.

  $V=30\text{km/h}$

Calculate the drag resistance of the car.

  $\begin{array}{c}{F}_D=\left(0.32\right)\left(1.8{\text{m}}^2\right)\frac{\left(1.20{\text{kg/m}}^3\right){\left(30\text{km/h}\times \frac{1000\text{m}}{1\text{km}}\times \frac{1\text{h}}{3600\text{s}}\right)}^2}{2}\\ =24\text{N}\end{array}$

Calculate the power required to overcome rolling resistance.

  $\begin{array}{c}{W}_{\text{rolling}}=F_{RR}V\\ =\left(327.78\text{N}\right)\left(30\text{km/h}\times \frac{1000\text{m}}{1\text{km}}\times \frac{1\text{h}}{3600\text{s}}\right)\\ =3106.5\text{W}\\ =3.1065\text{kW}\end{array}$

Calculate the power required to overcome drag resistance.

  $\begin{array}{c}{W}_{\text{drag}}=F_{D}V\\ =\left(24\text{N}\right)\left(30\text{km/h}\times \frac{1000\text{m}}{1\text{km}}\times \frac{1\text{h}}{3600\text{s}}\right)\\ =200\text{W}\\ =0.20\text{kW}\end{array}$

Calculate the total power required to overcome the combined drag and rolling.

  $\begin{array}{c}{W}_{\text{total}}=W_{\text{rolling}}+W_{\text{drag}}\\ =3.1065\text{kW}+0.20\text{kW}\\ =3.3065\text{kW}\end{array}$

Similarly, use excel spread sheet to calculate the $W_{\text{rolling}}$, $W_{\text{drag}}$, and $W_{\text{total}}$ for various speed of car ranging from the $0\text{km/h}$ to $150\text{km/h}$. The calculated values are tabulated and shown below.

$V$	$F_{RR}$	$F_{DD}$	$W_{RR}$	$W_{FD}$	$W_{\text{total}}$
0	372.78	0	0	0	0
15	372.78	6	1.55	0.03	1.58
30	372.78	24	3.11	0.20	3.31
45	372.78	54	4.66	0.68	5.33
60	372.78	96	6.21	1.60	7.81
75	372.78	150	7.77	3.13	10.89
90	372.78	216	9.32	5.40	14.72
105	372.78	294	10.87	8.58	19.45
120	372.78	384	12.43	12.80	25.23
135	372.78	486	13.98	18.23	32.20
150	372.78	600	15.53	25.00	40.53

Refer the above table, plot the graph of speed of car versus $W_{\text{rolling}}$, $W_{\text{drag}}$, and $W_{\text{total}}$. The plot is shown in below Figure.