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Chapter 10: Problem 41

A car manufacturer makes a luxury sedan that weighs \(20 \mathrm{kN}\) and has a frontal area of \(3.1 \mathrm{~m}^{2}\). When placed on an inclined test track, the car rolls to a terminal speed of \(27 \mathrm{~m} / \mathrm{s}\). The slope of the incline is \(6 \%,\) and it is estimated that \(50 \%\) of the resistance to motion is provided by the aerodynamic drag on the car. Estimate the drag coefficient of the car. Assume standard air.

Short Answer

Expert verified
The drag coefficient of the car is approximately 0.37.

Step by step solution

01

Determine the weight force along the slope

The weight of the car can be resolved into a component that's parallel to the slope. Since the slope is 6%, it makes a small angle \(\theta\) with the horizontal. The parallel component is given by \(F_{\text{parallel}} = \text{weight} \times \sin(\theta)\). Here, the weight \(W = 20 \text{kN} = 20000 \text{N}\) and \(\sin(\theta) = 0.06\). Thus, \(F_{\text{parallel}} = 20000 \times 0.06 = 1200 \text{N}\).
02

Calculate the aerodynamic drag force

We know that 50% of the resistance to motion is due to aerodynamic drag. Since the car is in equilibrium (terminal speed), the total resistance force equals the parallel component of the weight. Therefore, the drag force \(F_{d} = 0.5 \times 1200 = 600 \text{N}\).
03

Use the drag equation to find the drag coefficient

The aerodynamic drag force is calculated using the formula \( F_d = \frac{1}{2} \cdot \rho \cdot C_d \cdot A \cdot v^2 \), where \( \rho \) is the air density (approximately \(1.225 \text{kg/m}^3\) at sea level), \( C_d \) is the drag coefficient, \( A = 3.1 \text{m}^2 \) is the frontal area of the car, and \( v = 27 \text{m/s} \) is the velocity. Substituting the values, we get \(600 = \frac{1}{2} \times 1.225 \times C_d \times 3.1 \times 27^2\).
04

Solve for the drag coefficient \(C_d\)

Rearrange the equation to solve for \(C_d\): \(C_d = \frac{2 \times F_d}{\rho \times A \times v^2}\). Substituting \(F_d = 600\), \(\rho = 1.225\), \(A = 3.1\), and \(v = 27\), we find \(C_d = \frac{2 \times 600}{1.225 \times 3.1 \times 27^2}\). Calculate to find \(C_d \approx 0.37\).

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Key Concepts

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

Aerodynamic Drag
Aerodynamic drag is a resistive force that acts against the motion of objects moving through the air. It is an important consideration for vehicles like cars and airplanes, as it impacts their efficiency and fuel consumption. When a car moves, it displaces the air around it, creating pressure differences that result in drag.
  • Drag increases with speed: As the speed of a vehicle increases, the aerodynamic drag increases exponentially, making high-speed travel particularly challenging.
  • Impact factors: The shape, size, and speed of the vehicle, along with air density, significantly influence the drag force experienced.
Understanding aerodynamic drag is crucial for designing energy-efficient vehicles, reducing fuel consumption, and improving performance.
Drag Coefficient
The drag coefficient, denoted as \( C_d \), is a dimensionless number that quantifies the resistance an object faces as it moves through the air. It is a critical aspect in fluid mechanics, representing the aerodynamic efficiency of a body.
  • Calculating \( C_d \): It is determined using the drag equation \( F_d = \frac{1}{2} \cdot \rho \cdot C_d \cdot A \cdot v^2 \), where \( F_d \) is the drag force, \( \rho \) is the air density, \( A \) is the frontal area, and \( v \) is the velocity.
  • Significance: A lower drag coefficient indicates a more aerodynamically efficient design, allowing the vehicle to move smoothly with less resistance.
  • Optimization: Engineers strive to minimize the drag coefficient through improved design and material selection to enhance vehicle performance.
This coefficient is essential for vehicle designers aiming to create models that conserve energy and reduce emissions.
Inclined Plane
An inclined plane is a flat surface tilted at an angle, simplifying the analysis of forces acting parallel and perpendicular to the surface. It is a classic problem setup in physics that helps in understanding force components and motion.
  • Components of force: On an incline, gravity can be decomposed into two components - one parallel and one perpendicular to the plane.
  • Applications: This concept is crucial in analyzing vehicles on slopes, machinery where tilt is involved, and for understanding resistance to motion.
  • Benefits: Solving inclined plane problems helps in grasping the basics of forces and mechanics.
Learning about inclined planes is fundamental for students dealing with mechanics and physics problems, aiding their understanding of real-world applications.
Weight Force Calculation
The weight force is the gravitational force acting on an object, calculated as the product of its mass and the acceleration due to gravity (\( F = mg \)). When dealing with inclined planes, it's important to resolve this force into components.
  • Parallel and perpendicular components: The weight vector can be split into a component acting down the slope and another normal to the surface.
  • Calculation on inclines: For a slope of angle \( \theta \), the parallel force component is \( F_{\text{parallel}} = W \sin(\theta) \).
  • Practical application: Knowing how to calculate and resolve weight helps analyze various engineering and physics problems.
Correct calculation of weight force components is vital in applications such as vehicle dynamics, construction, and even sports science.

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Most popular questions from this chapter

Show that the terminal velocity of a rising spherical body in a surrounding fluid of different density can be expressed in nondimensional terms using the so-called densimetric Froude number, \(\mathrm{Fr}_{\mathrm{d}},\) as $$ \mathrm{Fr}_{\mathrm{d}}^{2}=\frac{4}{3 C_{\mathrm{D}}} $$ where \(C_{\mathrm{D}}\) is the drag coefficient and the densimetric Froude number, \(\mathrm{Fr}_{\mathrm{d}},\) is defined by the relation $$ \mathrm{Fr}_{\mathrm{d}}=\frac{V}{\sqrt{\frac{\Delta \rho}{\rho} g D}} $$ where \(V\) is the terminal velocity, \(\Delta \rho\) is the difference in density between the surrounding fluid and the sphere, \(\rho\) is the density of the surrounding fluid, \(g\) is gravity, and \(D\) is the diameter of the sphere. Comment on whether Equation 10.57 is also applicable to a falling spherical body. Note that \(\Delta \rho / \rho \cdot g\) is the effective gravity on a submerged body, commonly represented by \(g^{\prime}\).

An average person on a recreational bicycle can maintain a speed of \(8.05 \mathrm{~m} / \mathrm{s}\). The frontal area of a typical bicycle with a typical person aboard in riding position is \(0.5 \mathrm{~m}^{2},\) and the drag coefficient of this configuration is approximately \(0.5 .\) Under the stated typical conditions, estimate the power that the rider must exert to overcome the aerodynamic drag. Assume standard air.

A balloon is filled with helium and is supported by a light string. The diameter of the balloon is \(650 \mathrm{~mm}\), and the tension in the string is measured as \(2.5 \mathrm{~N}\) when the balloon is in standard air at sea level. If a wind speed of \(5 \mathrm{~m} / \mathrm{s}\) is imposed on the balloon and the string deflects \(45^{\circ}\) from the vertical, estimate the drag coefficient of the balloon.

An aircraft has a wing planform area of \(180 \mathrm{~m}^{2}\), an aspect ratio of \(7.5,\) a zero-lift drag coefficient of \(0.0185,\) and a weight of \(800 \mathrm{kN}\) when fully loaded. Estimate the speed of the aircraft that will minimize the required engine thrust when the aircraft is flying under standard sea-level atmospheric conditions.

Firing tests on a \(10-\mathrm{mm}\) -diameter bullet with a mass of \(20 \mathrm{~g}\) show that in standard air, the bullet speed decreases from \(300 \mathrm{~m} / \mathrm{s}\) to \(180 \mathrm{~m} / \mathrm{s}\) over a distance of \(200 \mathrm{~m}\). Estimate the average drag coefficient of the bullet. Neglect compressibility effects.
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