Question

An airplane of mass 1000 kg is in a steady level flight with a speed of 50 m/s. The wing has an elliptic planform with a span of 20 m and planform area 31.4 m$^2$. Assuming the density of air at that altitude to be 1 kg/m$^3$ and acceleration due to gravity to be 10 m/s$^2$, the induced drag on the wing is ............ N (rounded off to 1 decimal place).

Hint:

For induced drag, always use $C_{D_i} = \dfrac{C_L^2}{\pi AR e}$. For elliptic wings, $e=1$.

Answer 1

Step 1: Compute lift required in steady level flight.
\[ L = W = mg = 1000 \times 10 = 10000 \, \text{N} \] Step 2: Aspect ratio of wing.
Span = $b = 20$ m, Area = $S = 31.4$ m$^2$. \[ AR = \frac{b^2}{S} = \frac{20^2}{31.4} = \frac{400}{31.4} \approx 12.74 \] Step 3: Dynamic pressure.
\[ q = \tfrac{1}{2} \rho V^2 = 0.5 \times 1 \times 50^2 = 1250 \, \text{N/m}^2 \] Step 4: Lift coefficient.
\[ C_L = \frac{L}{q S} = \frac{10000}{1250 \times 31.4} = \frac{10000}{39250} \approx 0.255 \] Step 5: Induced drag coefficient (elliptical wing, efficiency $e=1$).
\[ C_{D_i} = \frac{C_L^2}{\pi AR e} = \frac{0.255^2}{\pi \times 12.74} \] \[ C_{D_i} = \frac{0.0650}{39.98} \approx 0.00163 \] Step 6: Induced drag.
\[ D_i = q S C_{D_i} = 1250 \times 31.4 \times 0.00163 \] \[ D_i \approx 12.7 \, \text{N} \] \[ \boxed{12.7 \, \text{N}} \]