A rocket engine gimbaling mechanism tilts the thrust vector by an angle $\delta = 5.0^\circ$ away from the vehicle longitudinal centerline to generate pitch control. If the engine produces total thrust $F = 500\text{ kN}$ and is located a distance $L = 8.0\text{ m}$ aft of the rocket center of gravity, the control pitching moment $M_{\text{pitch}}$ (in $\text{kNm}$) generated about the center of gravity is closest to:

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### 1. Strategy & Theoretical Principles When a gimbaled rocket engine is pivoted by an angle $\delta$: - Transverse side-force: $F_{\text{trans}} = F \sin \delta$ - Moment arm to the center of gravity: $L$ - Control moment: $M = F \sin \delta \cdot L$ ### 2. Analytical Formulation & Calculation Given: - $F = 500\text{ kN} = 500{,}000\text{ N}$ - $\delta = 5.0^\circ$ - $L = 8.0\text{ m}$ $$\sin(5.0^\circ) \approx 0.0871557$$ $$M_{\text{pitch}} = 500{,}000 \times 0.0871557 \times 8.0$$ $$M_{\text{pitch}} = 4{,}000{,}000 \times 0.0871557 = 348{,}623\text{ Nm} \approx 348.6\text{ kNm}$$ ### 3. Distractor Forensics - Candidate using $\tan(5^\circ) = 0.087488$ gets $349.9\text{ kNm}$, within the safe tolerance range. - Candidate using radians directly ($5 \times 4000 = 20,000$) forgets degree conversion. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always ensure calculator is in DEG mode when evaluating $\sin(5^\circ)$. - **Standard Reference:** Sutton, G.P. & Biblarz, O., *Rocket Propulsion Elements*, Chapter 18 (Thrust Vector Control).

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