91Ó°ÊÓ

To understand how a light aircraft propels itself forward, we first need to discuss thrust. Thrust is the force that moves an aircraft through the air. In this case, we know the aircraft requires a thrust of 10.3 kN, which translates to 10,300 N (Newtons). This force is produced by pushing air backward through the propeller, propelling the aircraft in the opposite direction.

The key equation to remember for thrust force is:
\[ F = \frac{1}{2} \rho A v_{\text{relative}}^2 (v - v_{\text{relative}}) \] where:

• \(F\) is the thrust force
• \(\rho\) is the air density
• \(A\) is the propeller area
• \(v_{\text{relative}}\) is the relative velocity
• \(v\) is the absolute velocity

Remembering this equation helps you understand how changing any of these variables will impact the overall thrust of the aircraft.

Relative velocity is crucial in propeller design. It represents the effective speed of the aircraft against the wind. In our exercise, the aircraft's absolute speed is 240 km/h, with a headwind of 48 km/h. To calculate the relative velocity:

\[ v_{\text{relative}} = 240 \frac{\text{km}}{\text{h}} + 48 \frac{\text{km}}{\text{h}} = 288 \frac{\text{km}}{\text{h}} \]
Converting this to meters per second (1 km/h = 0.27778 m/s):

\[ v_{\text{relative}} = 288 \frac{\text{km}}{\text{h}} \times 0.27778 = 80 \frac{\text{m}}{\text{s}} \]
This relative velocity is what the thrust has to act against to propel the aircraft forward

Calculating the power requirement is vital for determining the engine's capability to propel the aircraft. To find the power needed to drive the propeller, we use the equation:
\[ P = \frac{T v_a}{\text{efficiency}} \]
where:

• \(T\) is the thrust force
• \(v_a\) is the absolute velocity of the aircraft
• \(\text{efficiency}\) is the efficiency of the propeller

Substituting the known values into the formula:

\( T = 10,300 \text{N} \), \( v_a = 66.67 \frac{\text{m}}{\text{s}} \), \(\text{efficiency} = 0.90\), we get

\[ P = \frac{10,300 \text{N} \times 66.67 \frac{\text{m}}{\text{s}}}{0.90} \]
\[ P = 763 \text{kW} \]
This is the power output required to keep the aircraft moving at the given speed.

Determining the propeller area is necessary for ensuring the thrust can be generated efficiently. Starting with the thrust force equation:
\[ 10,300 \text{N} = \frac{1}{2} \times 1.2 \times A \times 80^2 \times (240 - 80) \]
Solving for \(A\), the propeller area, yields:
\[ A = \frac{10,300}{(0.6 \times 6,400 \times 160)} = 0.0168 \text{m}^2 \]
Lastly, converting this area to the diameter of the propeller using the area of a circle formula:
\[ A = \frac{\text{Ï€} d^2}{4} \]
Solving for \(d\) (diameter):
\[ d^2 = \frac{0.0168 \times 4}{π} \text{d} ≈ \text{sqrt}(0.0214) \text{d} ≈ 0.146 \text{m} \]
This result shows the diameter needed for the propeller to achieve the calculated thrust.