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🔬 KnowJEE Advanced Rotational Motion: Compound Physical Pendulum Optimization

In this advanced tutorial, we solve a complex JEE-level problem that combines rotational dynamics, calculus optimization, and physical intuition - the compound physical pendulum with variable mass position!

🎯 PROBLEM:
A uniform rod with attached point mass oscillates as physical pendulum. Find the time period and optimal mass position for minimum period.

✨ KEY FEATURES:
• Real physical pendulum (not simple pendulum)
• Variable mass position optimization
• Calculus application in physics
• Moment of inertia calculations
• Center of mass determination
• Equivalent simple pendulum concept

📚 STEP-BY-STEP SOLUTION:
1. Physical pendulum fundamentals
2. Moment of inertia calculation
3. Center of mass determination
4. Time period derivation
5. Calculus optimization setup
6. Quadratic equation solution
7. Physical interpretation
8. Special cases analysis

💡 UNIQUE ASPECTS:
• Combines mechanics + calculus
• Practical optimization problem
• Physical intuition development
• Multiple solution verification
• JEE Advanced pattern matching

🎓 PERFECT FOR:
• JEE Advanced 2024/2025 aspirants
• Students wanting to master rotational motion
• Those struggling with physical pendulums