GATE 2021 AE – Question 65
An aircraft has wing area 350 m², chord 9 m, wing lift slope 0.1 per degree, CG at 30% chord and wing aerodynamic center at 25% chord. Tail area is 80 m², lift slope 0.068 per degree, aerodynamic-center separation 28 m and downwash gradient 0.4. The pitching-moment slope per degree is (three decimals)
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Correct answer: -0.034 to -0.014
Explanation
**Pitching-moment slope** of the complete aircraft (wing plus tail), per degree:
$$C_{m_\alpha}=C_{L_{\alpha,w}}\left(\bar x_{cg}-\bar x_{ac}\right)-C_{L_{\alpha,t}}\,\frac{S_t}{S}\,\frac{\ell_t}{\bar c}\left(1-\frac{d\varepsilon}{d\alpha}\right).$$
**Wing term** (CG at 30% chord, wing aerodynamic centre at 25%; the CG is behind the AC, which is destabilising):
$$0.1\times(0.30-0.25)=+0.005 .$$
**Tail term.** The tail lever arm is measured from the CG: the aerodynamic centres are 28 m apart, the wing AC is 5% of the chord (0.45 m) ahead of the CG, so $\ell_t=28-0.05\times9=27.55$ m.
$$0.068\times\frac{80}{350}\times\frac{27.55}{9}\times(1-0.4)=0.068\times0.22857\times3.0611\times0.6=0.02855 .$$
**Total:**
$$C_{m_\alpha}=0.005-0.02855=\mathbf{-0.0235\ per\ degree}.$$
(The negative sign means the aircraft is statically stable in pitch.)