To determine the required lift coefficient and the minimum required tail surface area for the Boeing 747, we can follow these steps:
1. Calculate Required Lift Coefficient
Lift Equation: \[ L = \frac{1}{2} \rho V^2 S C_L \] Where:
- \( L \) = Lift (equal to weight for steady horizontal flight, \( L = W = mg \))
- \( \rho \) = Air density (at FL350 = ~0.5 kg/m³)
- \( V \) = Velocity in m/s
- \( S \) = Wing area (511 m²)
- \( C_L \) = Lift coefficient
From the data given, we need the weight of the aircraft:
- \( m = 250,000 , \text{kg} \)
- \( g \approx 9.81 , \text{m/s}^2 \)
- Therefore, \( W = mg = 250,000 , \text{kg} \times 9.81 , \text{m/s}^2 = 2,452,500 , \text{N} \)
Assuming the aircraft is flying at cruising altitude at a speed of, say, 240 knots (about 123.5 m/s) (an example airspeed; exact speed should be provided if known):
Air Density at FL350 (about 35,000 feet):
- Generally, the density at this altitude is roughly 0.5 kg/m³.
Now, substituting into the lift equation:
\[ W = \frac{1}{2} \rho V^2 S C_L \]
Setting \( W = 2,452,500 , \text{N} \):
\[ 2,452,500 = \frac{1}{2} \cdot 0.5 \cdot (123.5)^2 \cdot 511 \cdot C_L \]
Now compute the right-hand side:
\[ W = 0.25 \cdot 15270.25 \cdot 511 \cdot C_L \] \[ W = 0.25 \cdot 7805086.35 \cdot C_L \] \[ W = 1951271.588 \cdot C_L \] Now, solving for \( C_L \):
\[ 2,452,500 = 1951271.588 \cdot C_L \] \[ C_L = \frac{2,452,500}{1951271.588} \] \[ C_L \approx 1.256 \]
2. Minimum Required Tail Surface Area
To ensure stability, we will employ the static stability criteria, involving the tail and the wing. The requirement that the center of gravity (c.g.) should be less than 1.8 meters behind the aerodynamic center (a.c.) of the central wing implies we'd need sufficient lift from the tail to balance change in center of gravity.
The condition for stability can be expressed using the following equation for the sum of moments:
Static Margin: \[ \tan(\alpha) \cdot (h_{tail} - h_{wing}) - \frac{Cl_{w}}{Cl_{t}} \cdot \frac{S_{t}}{S} < constant \] Where \(S_{t}\) is the tail area we need to solve for.
Taking into account:
- \(dCL/d\alpha\) for the wing = \(0.098\) per degree
- And for the tail = \(0.047\) per degree.
To proceed further we need more data, particularly \( h_{tail} \), but assuming a general stability analysis, and conditions usually provide:
\[ S_{t} = \frac{W \cdot l_H}{C_L} \]
Where:
- (Assume \(l_H = 31.5 m \), and substitute other parameters as needed).
In a simplified version, we would be able to directly substitute and compute once all parameters are provided. However, without direct values for height relationships, further calculations could vary.
This provides a schematic assessment for stability under flight conditions which may approximate or provide ranges based on flight control design specifics. Further specifics can enhance accuracy which should be elucidated based on direct functional contexts provided for the aircraft.
If additional data or the specific wing-based height differences are available we can derive more accurate conditions.
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