🔄 Rotational Dynamics 

📌 1. Basic Concepts

🔹 Circular Motion

• Motion where an object moves along a circular path.

• Rotation: Motion about an axis through the object.

• Revolution: Motion around an axis outside the object.

• Velocity changes direction constantly in circular motion, so there is acceleration. 

🔹 Uniform vs Non-uniform Circular Motion

• Uniform: Speed is constant; acceleration (centripetal) towards the centre.

• Non-uniform: Speed changes with time. 

📌 2. Forces in Circular Motion

🌀 Centripetal Force

• Keeps object moving in a circle.

• Formula:
  Fc = m v² / r = m r ω²
  where v = linear speed, ω = angular velocity, r = radius. 

🌀 Centrifugal Force (pseudo force)

• Apparent outward force seen in a rotating frame. 

📌 3. Moment of Inertia (I)

• Rotational analogue of mass (resistance to change in angular motion). 

• Discrete mass system:
  I = Σ mᵢrᵢ²

• Continuous body:
  I = ∫ r² dm

• SI unit: kg·m². 

🔹 Theorems

• Parallel Axis Theorem 

• Perpendicular Axis Theorem
  Useful for calculating I for various shapes. 

📌 4. Angular Quantities

• Angular displacement θ, Angular velocity ω, Angular acceleration α. 

• Relation to linear variables:
  v = ω r, aₜ = α r. 

📌 5. Torque (τ)

• Rotational equivalent of force.

• Formula:
  τ = I α

• Causes change in rotational motion. 

📌 6. Angular Momentum (L)

• Rotational analogue of linear momentum.

• L = I ω 

🔹 Conservation of Angular Momentum

In absence of external torque:
Initial L = Final L. 

📌 7. Kinetic Energy of Rotation

• A body rotating with angular speed ω has:
  K = ½ I ω². 

📌 8. Rolling Motion

• Combination of rotation and translation without slipping. 

• Total kinetic energy:
  E = ½ Mv² + ½ Iω². 

• Where v = speed of centre of mass. 

📌 9. Applications in Problems

🚗 Banking of Roads

• Helps vehicles take turns at higher speeds safely using component forces. 

🪢 Conical Pendulum

• Bob moves in a horizontal circle; tension components give centripetal force. 

🎡 Vertical Circular Motion

• Motion in a vertical plane needs certain minimum speed at the top to maintain string tension. 

Formula List (Quick Revision) 

Law of Conservation of Angular Momentum

• State and prove the principle of conservation of aular momentum.

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum. N

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum. G

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum. 

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum. 

• Numericals = guaranteed marks (almost every year) 

• 1 derivation OR 1 long theory question is fixed 

• Angular momentum + MOI together contribute 8–10 marks 

• Memorise derivations + final formulae 

• Practice board-level numericals (simple but conceptual) 

• Write answers with proper steps & units (marks depend on it!) 

Law of Conservation of Angular Momentum 

• State and prove the principle of conservation of angular momentum. 

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum. 

• Explain the principle with two suitable examples.

• State the law of conservation of angular momentum.