🔄 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) 

• 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!) 

1. Moment of Inertia 

1. Define moment of inertia. Write its SI unit.

2. Derive an expression for the moment of inertia of a solid circular disc about an axis through its centre and perpendicular to its plane.

2. Parallel Axis Theorem 

3. A uniform rod of length L and mass M rotates about an axis perpendicular to it and passing through one end. Using the parallel axis theorem, find its moment of inertia.

3. Torque and Angular Acceleration 

4. A wheel of moment of inertia I is acted upon by a constant torque τ. Derive the expression for its angular velocity as a function of time, assuming initial angular velocity is zero.

4. Rolling Motion 

5. A solid sphere and a hollow cylinder both roll without slipping down the same incline. Which reaches the bottom first? Explain using energy considerations.

5. Angular Momentum 

6. State the law of conservation of angular momentum. A rotating disc (initially at rest) receives a second disc and is pressed on top of it so they rotate together. Find their final angular velocity if their moments of inertia are I₁ and I₂.

6. Kinetic Energy of Rotation 

7. Derive the expression for the rotational kinetic energy of a rigid body in terms of its angular speed and moment of inertia.

7. Combined Translational + Rotational Motion 

8. A cylinder of mass m and radius R is rolling without slipping with velocity v. Find its total kinetic energy.

8. Dynamic Equilibrium 

9. A wheel of radius 50 cm is subjected to three torques: 20 N·m clockwise, 15 N·m counterclockwise, and 10 N·m clockwise. Find the net torque and the direction of rotation.

9. Physical Pendulum 

10. A metre-stick pivoted at 20 cm from one end is used as a physical pendulum. Find its period of small oscillations. (Take moment of inertia about centre and use parallel axis theorem.)

10. Numerical Mixed Question 

11. A solid disc (radius 0.4 m, mass 5 kg) starts from rest and reaches angular speed 10 rad/s in 4 s under constant torque. Calculate:

(a) Angular acceleration        (b) Torque       (c) Rotational kinetic energy at 4 s