An aircraft has a mass of 100,000 kg and is cruising at 800 km/hr at an altitude where the
density is 0.4 kg.m-3. Determine the lift coefficient if the wing reference is 180 m2.

To determine the lift coefficient (\(C_L\)) of an aircraft given its mass, cruising speed, altitude density, and wing reference area, we can use the lift equation:

\[ L = \frac{1}{2} \rho v^2 S C_L \]

Where:
- \(L\) is the lift force,
- \(\rho\) is the air density,
- \(v\) is the velocity,
- \(S\) is the wing reference area,
- \(C_L\) is the lift coefficient.

### Step-by-Step Solution:

1. **Convert Velocity to Meters per Second:**
   The cruising speed is given in kilometers per hour (km/hr). We need to convert this to meters per second (m/s):

   \[
   v = 800 \text{ km/hr} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = \frac{800 \times 1000}{3600} \text{ m/s} = 222.22 \text{ m/s}
   \]

2. **Calculate the Lift Force:**
   The lift force (\(L\)) must equal the weight of the aircraft to maintain level flight. The weight (\(W\)) is given by:

   \[
   W = m \cdot g
   \]

   Where:
   - \(m\) is the mass of the aircraft (100,000 kg),
   - \(g\) is the acceleration due to gravity (approximately 9.81 m/s²).

   Thus:

   \[
   L = W = 100,000 \text{ kg} \times 9.81 \text{ m/s}^2 = 981,000 \text{ N}
   \]

3. **Solve for the Lift Coefficient (\(C_L\)):**
   Rearrange the lift equation to solve for \(C_L\):

   \[
   C_L = \frac{2L}{\rho v^2 S}
   \]

   Substitute the known values into the equation:

   \[
   C_L = \frac{2 \times 981,000 \text{ N}}{0.4 \text{ kg/m}^3 \times (222.22 \text{ m/s})^2 \times 180 \text{ m}^2}
   \]

4. **Perform the Calculations:**

   Calculate the denominator first:

   \[
   \rho v^2 S = 0.4 \text{ kg/m}^3 \times (222.22 \text{ m/s})^2 \times 180 \text{ m}^2
   \]

   \[
   = 0.4 \times 49382.7169 \times 180
   \]

   \[
   = 0.4 \times 8888894.042
   \]

   \[
   = 3555557.617 \text{ kg m/s}^2
   \]

   Now, calculate \(C_L\):

   \[
   C_L = \frac{2 \times 981,000}{3555557.617}
   \]

   \[
   = \frac{1962000}{3555557.617}
   \]

   \[
   \approx 0.552
   \]

### Summary:

- The lift coefficient \(C_L\) required for the aircraft to maintain level flight is approximately \(0.552\).

Would you like more details or have any questions about this process?

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Here are 5 numbered questions you might ask next:

1. How does the lift coefficient change with different airspeeds?
2. What factors influence the lift coefficient of an aircraft?
3. How is the wing reference area determined for different aircraft?
4. Can you explain how changes in altitude affect the lift coefficient?
5. What role does the lift coefficient play in the overall aerodynamics of an aircraft?

**Tip:** Knowing how to calculate and understand the lift coefficient is crucial for ensuring that an aircraft can achieve and maintain stable flight under various conditions.

Learn how to calculate the lift coefficient (CL) of an aircraft cruising at 800 km/hr with a mass of 100,000 kg and an altitude density of 0.4 kg/m^3. Understand the lift equation and its application in aerodynamics.

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