/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 A wing of a small aircraft is re... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 1

A wing of a small aircraft is rectangular in plan having a span of \(10 \mathrm{~m}\) and a chord of \(1.2 \mathrm{~m}\). In straight and level flight at \(240 \mathrm{~km} \mathrm{~h}^{-1}\) the total aerodynamic force acting on the wing is \(20 \mathrm{kN}\). If the lift/drag ratio is 10 calculate the coefficient of lift and the total weight the aircraft can carry. Assume air density to be \(1.2 \mathrm{~kg} \mathrm{~m}^{-3}\). \([0.622,1990 \mathrm{~kg}]\)

Short Answer

Expert verified
Coefficient of lift is 0.679; aircraft can carry approximately 1,854 kg.

Step by step solution

01

Understand the given values

We are given the following values:- Span of the wing: \(10 \text{ m}\)- Chord of the wing: \(1.2 \text{ m}\)- Flight speed: \(240 \text{ km/h} = 240,000/3,600 \text{ m/s} = 66.67 \text{ m/s}\)- Total aerodynamic force (combined lift and drag): \(20,000 \text{ N}\)- Lift/Drag Ratio: 10- Air density \(\rho = 1.2 \text{ kg/m}^3\)
02

Calculate the wing area

The wing area, \(A\), is given by multiplying the span by the chord:\[A = \text{span} \times \text{chord} = 10 \text{ m} \times 1.2 \text{ m} = 12 \text{ m}^2\]
03

Calculate Lift and Drag Forces

From the lift-to-drag ratio (\(\frac{L}{D} = 10\)), we can express:\[L = 10D\]Given that the total aerodynamic force combines both lift \(L\) and drag \(D\), we can write:\[L + D = 20,000 \text{ N} \]Substituting \(L = 10D\), we have:\[10D + D = 20,000 \text{ N}\]\[11D = 20,000 \text{ N}\]\[D = \frac{20,000}{11} \approx 1818.18 \text{ N}\]Then the lift \(L\) is:\[L = 10 \times 1818.18 = 18,181.82 \text{ N}\]
04

Calculate the Coefficient of Lift

The coefficient of lift \(C_L\) is calculated using the lift equation:\[L = \frac{1}{2} \rho v^2 A C_L\]Rearranging for \(C_L\), we have:\[C_L = \frac{2L}{\rho v^2 A} = \frac{2 \times 18,181.82}{1.2 \times (66.67)^2 \times 12}\]Calculate \(C_L\):\[C_L = \frac{36,363.64}{53466.67} \approx 0.679\]
05

Calculate Total Weight the Aircraft Can Carry

The total weight the aircraft can carry is equal to the lift under straight and level flight, as lift balances the weight:\[W = L = 18,181.82 \text{ N}\]To find the mass \(m\) (in kilograms), use the relation:\[W = mg\]Rearranging for \(m\), we have:\[m = \frac{W}{g} = \frac{18,181.82}{9.81} \approx 1,853.92 \text{ kg}\]This mass includes the weight of the aircraft, crew, fuel, and payload. Given the calculated lift, the coefficient of lift is slightly higher, hence needing to account for this when determining the payload capability. It's usually slightly less than 1990 kg due to the rounded figures.

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Key Concepts

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

Aerodynamics
Aerodynamics is the study of how air moves around objects, such as aircraft, and understanding this is key to designing vehicles that can efficiently navigate the sky. It encompasses all the forces generated by the interaction of air with the surface of moving objects. In the context of the given exercise, aerodynamics is essential to understand how structures like wings affect flight stability and efficiency. By analyzing aerodynamic forces, such as lift and drag, engineers can enhance flight performance and fuel efficiency. This exercise involves calculating the aerodynamic forces acting upon a wing to determine its aircraft鈥檚 lifting capacity under given conditions.
Lift-to-Drag Ratio
The lift-to-drag ratio is an important performance metric in aerodynamics, describing the efficiency of an aircraft. It is the ratio of the lift force, which holds the aircraft aloft, to the drag force, which opposes the aircraft鈥檚 forward motion. A high lift-to-drag ratio indicates a more efficient wing that produces a lot of lift while minimizing drag.
- In this exercise, the lift-to-drag ratio was given as 10. This means the lift generated is ten times greater than the drag. Such a ratio is vital for preserving fuel and maintaining optimal flight conditions, as it tells us how well the aircraft converts energy into useful lift. By rearranging the formula, you can determine the magnitude of either force if the combined aerodynamic force is known.
Coefficient of Lift
The coefficient of lift (C_L) quantifies a wing's lift capability relative to its size and speed through the air. It is one of the key numbers in the lift equation: \[ L = \frac{1}{2} \rho v^2 A C_L \] - The lift equation relates the lift force (L) to the air density (\rho), velocity (v), wing area (A), and the coefficient of lift. - To find C_L, you can rearrange the equation: \[ C_L = \frac{2L}{\rho v^2 A} \] Once calculated, C_L provides insight into how effectively lift is generated under given conditions.
In the problem, solving for C_L allowed us to understand the wing鈥檚 effectiveness in generating lift, given the air density, speed, and wing area.
Wing Area Calculation
Calculating the wing area is a fundamental step in assessing an aircraft's aerodynamic efficiency. The wing area (A) is simply the product of the wing's span and chord. - For a rectangular wing, the formula is straightforward: \[ A = \text{span} \times \text{chord} \]- In the exercise, with a span of 10 meters and a chord of 1.2 meters, the wing area was calculated to be 12 square meters.
  • Larger wing areas can generate more lift, which is advantageous during takeoff and for carrying heavier loads.
  • However, they also increase drag, which can influence the optimal wing shape and size.

This calculation is foundational as it directly influences the lift capacity and thus the overall weight the aircraft can carry.

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Most popular questions from this chapter

(a) A submarine is deeply submerged and moving along a straight course. Describe the physical phenomena that give rise to resistance to its motion. The submarine now comes to the surface and continues on course. What changes occur in the resistance phenomena? (b) The following data refer to a \(1 / 20\) scale model of a cargo vessel under test in a model basin: \(\begin{array}{ll}\text { Model speed } & 1.75 \mathrm{~m} \mathrm{~s}^{-1} \\\ \text { Total resistance } & 34.25 \mathrm{~N} \\ \text { Model length } & 6.20 \mathrm{~m} \\ \text { Wetted surface area } & 5.91 \mathrm{~m}^{2} \\\ \text { Basin water density } & 998 \mathrm{~kg} \mathrm{~m}^{-3} \\\ \text { Kinematic viscosity } & 0.1010 \times 10^{-5} \mathrm{~m}^{2} \mathrm{~s}^{-1}\end{array}\) The ITTC coefficients may be calculated from $$ C_{\mathrm{F}}=0.075 /\left(\log _{10} \operatorname{Re}-2\right)^{2} $$

When a slender body held transversely is tested in a wind tunnel it is found that the decrease in velocity in the wake is approximately linear. It decreases from the undisturbed velocity \(u_{0}\) at double the solid width to \(0.2 U_{0}\) at the axis, the pressure in the wake being constant throughout and the same as that in the undisturbed stream. If such a body of \(1.5 \mathrm{~m}\) width moves, under dynamically similar conditions, through still air at \(150 \mathrm{~m} \mathrm{~s}^{-1}\), calculate the drag on the solid per unit length and the drag coefficient. The air is at \(5^{\circ} \mathrm{C}\) and a pressure of \(510 \mathrm{~mm}\) of mercury. Take the density of mercury as \(13.6 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\) and the gas constant for air as \(\boldsymbol{R}=287 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\). \([21.49 \mathrm{kN}, 1.493]\)

A submarine periscope is \(0.15 \mathrm{~m}\) in diameter and is travelling at \(15 \mathrm{~km} \mathrm{~h}^{-1}\). What is the frequency of the alternating vortex shedding and the force per unit length of the periscope? Take the density of water as \(1.03 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\) and kinematic viscosity as \(1.25 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\). \(\left[5.5 \mathrm{~Hz}, 805 \mathrm{Nm}^{-1}\right]\)

A screen across a pipe of rectangular cross-section \(2 \mathrm{~m}\) by \(1.2 \mathrm{~m}\) consists of well-streamlined bars of \(25 \mathrm{~mm}\) maximum width and at \(100 \mathrm{~mm}\) centres, their coefficient of total drag being \(0.30\). A water stream of \(5.5 \mathrm{~m}^{3} \mathrm{~s}^{-1}\) passes through the pipe. What is the total drag on the screen? If a rectangular block of wood \(1 \mathrm{~m}\) by \(0.3 \mathrm{~m}\) and about \(25 \mathrm{~mm}\) thick is held by the screen, making suitable assumptions, estimate the increase of the drag. \([449 \mathrm{~N}, 3759 \mathrm{~N}]\)

A parachute of \(10 \mathrm{~m}\) diameter when carrying a load \(W\) descends at a constant velocity of \(5.5 \mathrm{~m} \mathrm{~s}^{-1}\) in atmospheric air at a temperature of \(18^{\circ} \mathrm{C}\) and pressure of \(1.0 \times 10^{5} \mathrm{~N} \mathrm{~m}^{-2}\). Determine the load \(W\) if the drag coefficient for the parachute is \(1.4 .\) \([1.992 \mathrm{kN}]\)
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