To solve the problems, we need to calculate the required lift coefficient for the Boeing 747 flying at a given equivalent airspeed, and then we can approach the next part regarding the tail surface area in relation to stability. Let's break it down step by step.
Part 1: Required Lift Coefficient
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Calculate the necessary parameters:
- Weight (W) = Mass (m) × g (acceleration due to gravity)
- Conversion: \( g \approx 9.81 , \text{m/s}^2 \)
\[ W = 250,000 , \text{kg} \times 9.81 , \text{m/s}^2 = 2,452,500 , \text{N} \]
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Convert True Airspeed to m/s:
- Given equivalent airspeed = 267 knots.
- Conversion factor = 1 knot = 0.51444 m/s
\[ V = 267 , \text{knots} \times 0.51444 , \text{m/s per knot} \approx 137.76 , \text{m/s} \]
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Calculate the dynamic pressure (q): \[ q = \frac{1}{2} \rho V^2 \]
- At Flight Level 350, the air density (\(\rho\)) is approximately 0.000 ~ (lower density due to altitude, let’s assume roughly 0.4 kg/m³ for quick calculations)
\[ q \approx \frac{1}{2} \times 0.4 , \text{kg/m}^3 \times (137.76 , \text{m/s})^2 \approx 0.5 \times 0.4 \times 18956.45 \approx 3774.37 , \text{N/m}^2 \]
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Use lift equation to find the required lift coefficient (Cl): The lift equation is given by:
\[ L = C_L \times q \times S \] Where \(L\) is lift, \(C_L\) is lift coefficient, \(q\) is dynamic pressure, and \(S\) is wing area.
Rearranging for \(C_L\):
\[ C_L = \frac{L}{q \times S} = \frac{2,452,500 , \text{N}}{3774.37 , \text{N/m}^2 \times 511 , \text{m}^2} \]
\[ C_L \approx \frac{2,452,500}{1926460.57} \approx 1.27 \]
Part 2: Minimum Required Tail Surface Area
To ensure stability, the static margin needs to be defined in the aircraft design.
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Calculate the static margin (SM): The static margin should be adequate for the CG position:
The formula for static margin often relates the tail surface area to the overall stability of the aircraft. A common approach is using the following equation to determine tail effectiveness:
\[ S_T = \frac{W}{q C_L_{tail}} \times \frac{l_H}{x_{cg}} \]
Where \(C_L\) from the tail given is: \[ C_L_{tail} = dC_L/d\alpha_{tail} \times ( \alpha - \text{downwash}) \]
But, without more detail on the CG (shown as less than 1.8m), we can assume:
\[ S_T = \text{(use some properties of } \text{stability, aerodynamic moment)} \]
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Calculate Tail Area: Rearranging to find tail area:
\[ S_T \text{ will derive similar using known values of lift coefficients for tail components and stability necessary metrics.} \]
Depending on aircraft requirements, empirical considerations could place requirements between certain coefficients for configurations.
So, for complete validity, simulators or more depth data inquiry would likely substantiate these quick assumptions given typical values for stability.
Final Output
- Required Lift Coefficient (C_L) = 1.27
- Calculating tail area typically would require specific angle stats or approximations based on stability defined flight metrics that consider the tail effect, which was numerical but lacked explicit values from referenced documentation.
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