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Chapter 1: Problem 54

The rolling resistance of the tires, \(F_{r}\), opposing the motion of a vehicle is given by $$ F_{r}=f W $$ where \(f\) is a constant called the rolling resistance coefficient and \(W\) is the vehicle weight. For a vehicle weighing, \(W=300 \mathrm{kN}\) that is traveling at \(100 \mathrm{~km} / \mathrm{h}\), calculate the power in \(\mathrm{kW}\) required to overcome the vehicle rolling resistance for \(f=0.0070\).

Short Answer

Expert verified
The power required to overcome the vehicle rolling resistance is \(58.338\) kW.

Step by step solution

01

Calculate rolling resistance

First, find the rolling resistance \(F_{r}\) using the provided formula \(F_{r}=f W\). Subsitute the given values: \(f=0.0070\) and \(W=300\) kN into the formula to get \(F_{r} = 0.0070*300 = 2.1\) kN.
02

Convert speed to m/s

The speed is given in km/h but our calculations require m/s. To convert, multiply the given \(100\) km/h by \(\frac{1000}{3600}\) to get \(27.78\) m/s.
03

Calculate power

Power (\(P\)) is calculated by multiplying force (\(F_{r}\)) by velocity (\(v\)) (i.e. \(P=F_{r}*v\)). Subsitute the given values: \(F_{r} = 2.1\) kN converted to N by multiplying by \(1000\) and \(v= 27.78\) m/s into the formula to get \(P = 2100*27.78 = 58338\) W.
04

Convert power to kW

Lastly, convert power from Watts to kilowatts by dividing by \(1000\). Therefore, \(P = \frac{58338}{1000} = 58.338\) kW.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vehicle Dynamics
Understanding vehicle dynamics is crucial for analyzing how a vehicle behaves under the influence of different forces. When a vehicle is in motion, it is subject to numerous forces, including rolling resistance, gravitational force, and aerodynamic drag.
  • Rolling Resistance: This is the force that opposes the motion of the vehicle's tires on the surface. It is caused by the deformation of the tire and the road surface.
  • Gravitational Force: This force always acts downwards towards the center of the Earth, affecting the vehicle's motion, especially on slopes.
  • Aerodynamic Drag: This is the air friction a vehicle faces while moving, impacting its speed and fuel efficiency.
In our scenario, we deal predominantly with rolling resistance which can be calculated using the formula: \[ F_{r} = f \times W \]Where \( f \) is the rolling resistance coefficient and \( W \) is the vehicle weight. These forces are important as they directly influence the performance and fuel consumption of the vehicle.
Power Calculation
Power calculation in vehicle dynamics involves determining the amount of energy required to keep the vehicle moving against various opposing forces. Here, we specifically focus on the power needed to overcome rolling resistance.
  • Formula: The power \( P \) can be determined using the equation:\[ P = F_r \times v \]where \( F_r \) is the rolling resistance force and \( v \) is the vehicle velocity.
  • Calculation Example: In our exercise, the rolling resistance force \( F_r \) is calculated to be \( 2.1 \text{ kN} \). The vehicle's speed is converted to \( 27.78 \text{ m/s} \) (from \( 100 \text{ km/h} \)). Therefore, the power can be calculated as:\[ P = 2100 \times 27.78 = 58338 \text{ W} \]
  • From Watts to Kilowatts: Since power is often expressed in kilowatts, we convert from watts by dividing by \( 1000 \), resulting in:\[ P = 58.338 \text{ kW} \]
Understanding power calculation helps in assessing the engine's capability and efficiency in overcoming resistance.
Unit Conversion
Unit conversion is a fundamental aspect of solving physics problems, especially in vehicle dynamics where measurements often need to be in the same units to perform calculations.
  • Speed Conversion: The vehicle's speed is initially given in kilometers per hour. However, most calculations require meters per second (m/s) for consistency with force calculations. By multiplying by \( \frac{1000}{3600} \), we convert \( 100 \text{ km/h} \) to \( 27.78 \text{ m/s} \).
  • Force Conversion: The rolling resistance is initially calculated in kilonewtons (kN) which needs to be converted into newtons (N) for correct power calculations. This is done by multiplying by \( 1000 \): \[ 2.1 \text{ kN} \rightarrow 2100 \text{ N} \]
  • Power Conversion: For the final presentation, power is often expressed in kilowatts. We convert watts to kilowatts by dividing by \( 1000 \), which simplifies comparisons and interpretations:\[ 58338 \text{ W} \rightarrow 58.338 \text{ kW} \]
Unit conversions ensure precision and accuracy in calculating and interpreting the results.

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