Discussion
At a fixed true airspeed an aeroplane climbs until the air density has halved. If its short-period frequency was \( 4.0 \ \mathrm{rad/s} \) at the lower altitude, it becomes approximately
Answer: B
The stiffness term \( M_\alpha \) is proportional to dynamic pressure and therefore to density at fixed true airspeed, and frequency goes as its square root, so
\[ \omega_{n2} = 4.0 \times \sqrt{0.5} = 2.83 \ \mathrm{rad/s} \]
giving about 2.8 rad/s. Halving the frequency to 2.0 would require the density to fall to a quarter. 5.7 has the trend backwards, and would follow from the density doubling. The unchanged 4.0 rad/s is what would be seen at constant equivalent airspeed rather than constant true airspeed, since then dynamic pressure is held fixed by definition, and that distinction is the point of the question.
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