An airplane is cruising at 50\,m/s at standard sea level in straight and level flight. The airplane has a wing area of 15\,m^2 and a wing span of 10\,m, and has the maximum gross weight of 1000\,kg. The airfoil of the wing has lift slope of 0.102 per degree and angle of attack at zero lift of -2.5^{\circ}. Considering, \tau = 0.1, find the angle of attack for the wing. Also, find the induced drag for the airplane, if span efficiency factor is 0.6.
An airplane is cruising at 50\,m/s at standard sea level in straight and level flight. The airplane has a wing area of 15\,m^2 and a wing span of 10\,m, and has the maximum gross weight of 1000\,kg. The airfoil of the wing has lift slope of 0.102 per degree and angle of attack at zero lift of -2.5^{\circ}. Considering, \tau = 0.1, find the angle of attack for the wing. Also, find the induced drag for the airplane, if span efficiency factor is 0.6.
Induced drag is created due to downwash. Downwash is produced by the wing-tip vortices during flight. Aspect ratio of the wing is, AR=\frac{b^{2}}{S}=\frac{\left ( 10 \right )^{2}}{15}=6.67
At, standard sea level density of air is 1.225\,kg/m^{3}
Velocity of airplane is 50\,m/s
Dynamic pressure, q_{\infty} = \frac{1}{2}\rho _{\infty}V_{\infty}^{2}=\frac{1}{2}\times1.225\times\left ( 50 \right )^{2}=1531.25\,N/m^{2}
Lift slop of the airfoil of the wing is 0.102 \,per\,degree = 5.8442\,per\,radian
Lift produced by the wing is, L=\frac{1}{2}\rho _{\infty}V_{\infty}^{2}SC_{L}=\rho _{\infty}SC_{L}
L = Weight of the airplane = 1000\times9.8 = 9800\,N
Therefore, C_{L} = \frac{9800}{1531.25\times15}=0.427
Lift slope of the wing will be, a = \frac{a_{0}}{1 + \frac{a_{0}}{\pi AR}\left ( 1+\tau \right ) }=\frac{5.8442}{1+\frac{5.8442}{\pi\left ( 6.67 \right )}\left ( 1+0.1 \right )}=4.472\,per\,radian = 0.0781\,per\,degree
Also, Coefficient of lift is given as C_{L}=a\left ( \alpha -\alpha _{L=0} \right )\Rightarrow 0.427=0.0781\left ( \alpha -\left ( -2.5^{\circ} \right ) \right )\Rightarrow \alpha = 2.97\,degree
Induced drag is given as D_{i}=q_{\infty}SC_{D,i}
Therefore, D_{i} = \left ( 1531.25 \right )\left ( 15 \right )\left ( 0.0145 \right ) = 333.05\,N