An airplane of mass 5000 kg is flying at a constant speed of 360 km/h at the bottom of a vertical circular loop of radius 400 m. Taking $g = 9.8$ m/s$^2$, the lift force (in kN) generated by the wings at the bottom of the loop is _________ (round off to the nearest integer).
Options:
(A) The governing parameter increases monotonically with Space Dynamics & Orbital Mechanics
(B) The governing parameter remains invariant across the boundary
(C) The parameter satisfies the conservation equation for Space Dynamics & Orbital Mechanics
(D) The parameter depends exclusively on viscous dissipation
Official Answer Key:
174.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2020 Q.25)**
**1. Intuitive Physical Setup & Given Parameters**:
- Airplane mass: $m = 5000\text{ kg}$
- Speed: $V = 360\text{ km/h} = \frac{360 \times 1000}{3600} = 100\text{ m/s}$
- Loop radius: $R = 400\text{ m}$
- Gravitational acceleration: $g = 9.8\text{ m/s}^2$
**2. Step-by-Step Flight Dynamics Derivation**:
1. **Equations of Motion at Bottom of Vertical Loop**:
At the lowest point of the circular vertical loop, lift $L$ acts vertically upwards, and weight $W = mg$ acts vertically downwards.
Centripetal acceleration $\frac{V^2}{R}$ is directed upwards toward the center of the loop:
$L - mg = m \frac{V^2}{R}$
$L = mg + m \frac{V^2}{R} = m \left( g + \frac{V^2}{R} \right)$
2. **Substituting Numerical Values**:
$\frac{V^2}{R} = \frac{100^2}{400} = \frac{10000}{400} = 25\text{ m/s}^2$
$g + \frac{V^2}{R} = 9.8 + 25 = 34.8\text{ m/s}^2$
$L = 5000\text{ kg} \times 34.8\text{ m/s}^2 = 174000\text{ N} = 174\text{ kN}$
The lift generated at the bottom of the loop is exactly **174 kN** (load factor $n = 34.8/9.8 \approx 3.55$).
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Evaluation**: **174.0 kN** (Official Range: [173, 175]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **174.0**
- **Key Formula**: $L = m\left(g + \frac{V^2}{R}\right)$ at the bottom of a vertical loop.
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