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Chapter 9: Problem 24

A kite may be regarded as equivalent to a rectangular aerofoil of \(900 \mathrm{~mm}\) chord and \(1.8 \mathrm{~m}\) span. When it faces a horizontal wind of \(13.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) at \(12^{\circ}\) to the horizontal the tension in the guide rope is \(102 \mathrm{~N}\) and the rope is at \(7^{\circ}\) to the vertical. Calculate the lift and drag coefficients, assuming an air density of \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\)

Short Answer

Expert verified
The lift coefficient is approximately 0.55, and the drag coefficient is approximately 0.068.

Step by step solution

01

- Convert measurements to consistent units

Convert the chord and span from millimeters and meters to meters. The chord is given as \(900 \text{ mm}\) which is equivalent to \(0.9 \text{ m}\). The span is already in meters: \(1.8 \text{ m}\).
02

- Calculate the area of the kite

The area \(A\) of the kite can be obtained by multiplying the chord and the span.\[ A = \text{chord} \times \text{span} = 0.9 \text{ m} \times 1.8 \text{ m} = 1.62 \text{ m}^2 \]
03

- Resolve tension into components

Resolve the tension force \(T\) into vertical and horizontal components. The tension in the guide rope \(T = 102 \text{ N}\), and the rope angle is \(7^\text{∘}\) to the vertical.\[ T_{vertical} = T \times \text{cos}(7^\text{∘}) = 102 \times \text{cos}(7^\text{∘}) \]\[ T_{horizontal} = T \times \text{sin}(7^\text{∘}) = 102 \times \text{sin}(7^\text{∘}) \]
04

- Calculate dynamic pressure

Dynamic pressure \(q\) is calculated using the wind speed \(V\) and air density \(\rho\).\[ q = \frac{1}{2} \rho V^2 \]Given: \(V = 13.5 \text{ m/s}\), \(\rho = 1.23 \text{ kg/m}^3\).\[ q = \frac{1}{2} \times 1.23 \times (13.5)^2 = 112.48 \text{ N/m}^2 \]
05

- Calculate lift force

Lift force \(L\) is equivalent to the vertical component of the tension.\[ L = T_{vertical} = 102 \times \text{cos}(7^\text{∘}) \approx 101.0 \text{ N} \]
06

- Calculate drag force

Drag force \(D\) is equivalent to the horizontal component of the tension.\[ D = T_{horizontal} = 102 \times \text{sin}(7^\text{∘}) \approx 12.4 \text{ N} \]
07

- Calculate lift and drag coefficients

Lift coefficient \(C_L\) and drag coefficient \(C_D\) are calculated using lift and drag forces respectively.\[ C_L = \frac{L}{q \times A} = \frac{101.0}{112.48 \times 1.62} \approx 0.55 \]\[ C_D = \frac{D}{q \times A} = \frac{12.4}{112.48 \times 1.62} \approx 0.068 \]

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Key Concepts

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

aerofoil
An aerofoil, or airfoil, is a shape designed to produce lift when air flows over it. In this exercise, the kite is considered as an aerofoil. The key to understanding an aerofoil is its profile, which allows the air to travel faster over the top surface than the bottom. This difference in speed creates a pressure difference, leading to lift. This principle is crucial in everything from aircraft wings to the kite in this problem.
dynamic pressure
Dynamic pressure is the pressure of the moving fluid, denoted by the symbol q. It is given by the formula \[q = \frac{1}{2} \rho V^2\] where \( \rho \) is air density and \( V \) is the wind speed. In the exercise, this pressure is a result of the wind blowing at 13.5 meters per second. Calculating dynamic pressure is essential as it helps determine the forces acting on the kite.
wind speed
Wind speed is the velocity of the air moving past the kite, given here as 13.5 meters per second. Wind speed directly influences the dynamic pressure, lift, and drag forces on the kite. Higher wind speeds generally increase the lift and drag, making it an important variable in aerodynamics.
air density
Air density \( \rho \) represents the mass of air per unit volume. In this exercise, it is given as 1.23 kg/m². The air density affects the dynamic pressure; denser air exerts more force on objects moving through it. Understanding air density is key in calculating aerodynamic forces, as shown in the problem.
vertical and horizontal components
Forces acting on the kite are split into vertical and horizontal components. Given the tension in the guide rope as 102 N at an angle of 7 degrees to the vertical, the vertical component \( T_{vertical} \) is calculated using \[ T_{vertical} = T \times \cos(7^{\circ}) \] and the horizontal component \( T_{horizontal} \) using \[ T_{horizontal} = T \times \sin(7^{\circ}) \] These components help determine the lift and drag forces.
tension force
The tension force in the guide rope is essential to analyze the forces on the kite. It is given as 102 N. By resolving this tension into vertical and horizontal components, we can determine the respective lift and drag forces. The tension force ensures that the kite remains stable and is an indication of how much force the wind applies to keep the kite in the air.
lift force
Lift force \( L \) is the upward force acting on the kite, keeping it in the air. It is equivalent to the vertical component of the tension force: \[ L = T_{vertical} \approx 101.0 \text{ N} \] Lift is calculated using \[ C_L = \frac{L}{q \times A} \approx 0.55 \] where \( C_L \) is the lift coefficient, \( q \) is dynamic pressure, and \( A \) is the area of the kite.
drag force
Drag force \( D \) is the horizontal force acting against the kite's motion through the air. It is equivalent to the horizontal component of the tension force: \[ D = T_{horizontal} \approx 12.4 \text{ N} \] The drag can be calculated using \[ C_D = \frac{D}{q \times A} \approx 0.068 \] where \( C_D \) is the drag coefficient, \( q \) is dynamic pressure, and \( A \) is the area of the kite. Understanding drag is crucial as it affects the stability and efficiency of the kite.

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Most popular questions from this chapter

A tall cylindrical body having an oval cross-section with major and minor dimensions \(2 X\) and \(2 Y\) respectively is to be placed in an otherwise uniform, infinite, two-dimensional air stream of velocity \(U\) parallel to the major axis. Assuming irrotational flow and a constant density, show that an appropriate flow pattern round the body may be deduced by postulating a source and sink each of strength \(|m|\) given by the simultaneous solution of the equations $$ m / \pi U=\left(X^{2}-b^{2}\right) / b \quad \text { and } \quad b / Y=\tan (\pi U Y / m) $$ Determine the maximum difference of pressure between points on the surface.

At a speed of \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) the resistance to motion of a rotor-ship is \(80 \mathrm{kN}\). It is propelled by two vertical cylindrical rotors, each \(3 \mathrm{~m}\) diameter and \(9 \mathrm{~m}\) high. If the actual circulation generated by the rotors is \(50 \%\) of that calculated when viscosity and end effects are ignored, determine the magnitude and direction of the rotational speed of the rotor necessary when the ship travels steadily south-east at \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) in a \(14 \mathrm{~m} \cdot \mathrm{s}^{-1}\) north-east wind. For these conditions use inviscid flow theory to determine the positions of the stagnation points and the difference between the maximum and minimum pressures. (Assume an air density of \(1.225 \mathrm{~kg} \cdot \mathrm{m}^{-3}\).)

A set of paddles of radius \(R\) is rotated with angular velocity \(\omega\) about a vertical axis in a liquid having an unlimited free surface. Assuming that the paddles are close to the free surface and that the fluid at radii greater than \(R\) moves as a free vortex, determine the difference in elevation between the surface at infinity and that at the axis of rotation.

To the two-dimensional infinite flow given by \(\psi=-U y\) are added two sources, each of strength \(m\), placed at \((0, a)\) and \((0,-a)\) respectively. If \(m>|2 \pi U a|\), determine the stream function of the combined flow and the position of any stagnation points. Sketch the resulting body contour and determine the velocity at the point where the contour cuts the \(y\)-axis.

Which of the following functions could represent the velocity potential for the two-dimensional flow of an inviscid fluid? (a) \(x+5 y ;\) (b) \(3 x^{2}-4 y^{2} ;\) (c) \(\cos (x-y)\); (d) \(\ln (x+y)\) (e) \(\arctan (x / y) ;\) (f) \(\operatorname{arccosec}(x / y)\)
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