Map each axis to its primary surface: roll \(\Rightarrow\) ailerons, pitch \(\Rightarrow\) elevator (longitudinal), yaw \(\Rightarrow\) rudder (directional). When in doubt about a control surface's effect on stability, match axis-to-axis and you'll avoid confusions like (D).
Step 1: Recall the primary aerodynamic roles.
The rudder is the primary control surface for yaw about the vertical axis; it provides directional control and contributes to directional (weathercock) stability. The surfaces linked to longitudinal (pitch) stability are the horizontal tail and elevator, not the rudder.
Step 2: Evaluate each option.
(A) Adverse yaw in turns. During aileron-induced roll, the downgoing aileron increases drag on the rising wing, yawing the nose opposite to the turn (adverse yaw). Coordinated rudder input counters this yaw to keep the turn coordinated (ball centered). \(\Rightarrow\) True.
(B) Asymmetric thrust / engine failure. In multi-engine aircraft, failure of one engine creates a yawing moment toward the dead engine. Applying rudder toward the live engine (and sometimes banking a few degrees) provides the required yawing moment to maintain directional control. \(\Rightarrow\) True.
(C) Crosswind landing. Two common techniques are crab-and-decrab and sideslip. In both, rudder is used to align the longitudinal axis with the runway centerline just before or during touchdown, countering the crosswind-induced drift. \(\Rightarrow\) True.
(D) Longitudinal stability. Longitudinal stability concerns restoring moments in pitch, governed by the horizontal tail/elevator and static margin. The rudder affects directional (yaw) stability; it does not "enhance longitudinal stability." \(\Rightarrow\) False.
\[ \boxed{\text{Correct uses of the rudder: (A), (B), (C).}} \]
A jet-powered airplane is steadily climbing at a rate of 10 m/s. The air density is 0.8 kg/m³, and the thrust force is aligned with the flight path. Using the information provided in the table below, the airplane’s thrust to weight ratio is ___________ (rounded off to one decimal place). 
While taking off, the net external force acting on an airplane during the ground roll segment can be assumed to be constant. The airplane starts from rest. \( S_{LO} \) and \( V_{LO} \) are the ground roll distance and the lift-off speed, respectively. \( \alpha V_{LO} \) (\( \alpha>0 \)) denotes the airplane speed at 0.5 \( S_{LO} \). Neglecting changes in the airplane mass during the ground roll segment, the value of \( \alpha \) is _________ (rounded off to two decimal places).
A jet-powered airplane is steadily climbing at a rate of 10 m/s. The air density is 0.8 kg/m³, and the thrust force is aligned with the flight path. Using the information provided in the table below, the airplane’s thrust to weight ratio is __________ (rounded off to one decimal place).

Two designs A and B, shown in the figure, are proposed for a thin-walled closed section that is expected to carry only torque. Both A and B have a semi-circular nose, and are made of the same material with a wall thickness of 1 mm. With strength as the only criterion for failure, the ratio of maximum torque that B can support to the maximum torque that A can support is _________ (rounded off to two decimal places).
A thin flat plate is subjected to the following stresses: \[ \sigma_{xx} = 160 \, {MPa}; \, \sigma_{yy} = 40 \, {MPa}; \, \tau_{xy} = 80 \, {MPa}. \] Factor of safety is defined as the ratio of the yield stress to the applied stress. The yield stress of the material under uniaxial tensile load is 250 MPa. The factor of safety for the plate assuming that material failure is governed by the von Mises criterion is _________ (rounded off to two decimal places).
A prismatic vertical column of cross-section \( a \times 0.5a \) and length \( l \) is rigidly fixed at the bottom and free at the top. A compressive force \( P \) is applied along the centroidal axis at the top surface. The Young’s modulus of the material is 200 GPa and the uniaxial yield stress is 400 MPa. If the critical value of \( P \) for yielding and for buckling of the column are equal, the value of \( \frac{l}{a} \) is __________ (rounded off to one decimal place).
A uniform rigid bar of mass 3 kg is hinged at point F, and supported by a spring of stiffness \( k = 100 \, {N/m} \), as shown in the figure. The natural frequency of free vibration of the system is ___________ rad/s (answer in integer).
F and G denote two points on a spacecraft’s orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If \( OF = 8000 \, {km} \), \( OG = 10000 \, {km} \), \( \theta_F = 0^\circ \), and \( \theta_G = 60^\circ \), the eccentricity of the spacecraft's orbit is __________ (rounded off to two decimal places).