Question

A major force opposing the motion of a vehicle is the rolling resistance of the tires, \(F_{r}\), given by $$ F_{\mathrm{r}}=f^{\circ} W $$ where \(f\) is a constant called the rolling resistance coefficient and \(W\) is the vehicle weight. Determine the power, in \(\mathrm{kW}\), required to overcome rolling resistance for a truck weighing \(322.5 \mathrm{kN}\) that is moving at \(110 \mathrm{~km} / \mathrm{h}\). Let \(f=0.0069\).

Short answer

The power required to overcome rolling resistance is 68.009 kW.

Step-by-step solution

Write the Given Values

Identify the values given in the problem:o \(W = 322.5 \text{ kN}\)o \(v = 110 \text{ km/h}\)o \(f = 0.0069\)

Calculate the Rolling Resistance Force

Using the formula for rolling resistance force:o \(F_{\text{r}} = f W\)Substitute the given values:o \(F_{\text{r}} = 0.0069 \times 322.5 \text{ kN}\)Calculate the result:o \(F_{\text{r}} = 2.22525 \text{ kN}\)

Convert the Speed Unit

Convert the speed from km/h to m/s:o \(v = 110 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}\)Calculate the result:o \(v = 30.56 \text{ m/s}\)

Calculate the Power in Watts

Use the formula for power to overcome rolling resistance:o \(P = F_{\text{r}} \times v\)Substitute the calculated force and speed:o \(P = 2.22525 \text{ kN} \times 30.56 \text{ m/s}\)Since 1 kN = 1000 N:o \(P = 2225.25 \text{ N} \times 30.56 \text{ m/s}\)Calculate the result:o \(P = 68009.04 \text{ W}\)

Convert Power to Kilowatts

Convert the power from watts to kilowatts:o \(P_{\text{kW}} = \frac{P}{1000}\)Substitute the value of power in watts:o \(P_{\text{kW}} = \frac{68009.04 \text{ W}}{1000}\)Calculate the result:o \(P_{\text{kW}} = 68.009 \text{ kW}\)

Key concepts

Rolling Resistance Coefficient

Rolling resistance is a key concept in vehicle dynamics. The rolling resistance coefficient (\f\text{\textsuperscript{\text{o}}}\)) represents the frictional force opposing the motion of a vehicle due to the deformation of tires and the surface they travel on.
The formula for calculating the rolling resistance force is:

\( F_{\text{r}} = f^o W \times W \times W \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} \times \frac{1 \text{ MN}}{1000 \text{ kN}} \times \times \times P \times F_{\text{r}} \times v, \times \times P_{\text{kW}} \times P_{\text{kW}} \times 1000 ,$$ \times \frac{1000 \text{ W in}}{1000}$ #00225 is a key concept in vehicle dynamics.