Subtopic: Rigid Body Dynamics & Moment of Inertia Tensor
Question Statement:
A symmetric rigid body has moment of inertia I1, I2, I3 about its principal axes 1, 2, and
3, respectively, with I1 = I3 = I⊥ and I2 ≠ I⊥. It is rotating in space with no torque on it
so that its angular momentum L⃗ is constant. Let \omega1, \omega2, \omega3 be the components of its
angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following
quantities is/are constant during the motion of this rigid body?
Options:
(A) \omega1 + \omega3
(B) \omega1
2 + \omega3
2
(C) Angle between axis 2 and L⃗
(D) \omega2
Official Answer Key:
B;C;D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.55)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 2: Classical Mechanics* | **Topic**: *Small oscillations* (`Normal Modes of Coupled Oscillators`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Small oscillations** (`Normal Modes of Coupled Oscillators`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `𝜔1
2 + 𝜔3
2`**.
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