Chapter 1: Rotational Dynamics Class 12 Physics

1.1 Angular Motion & Relation with Linear Motion

• Angular displacement (θ): Angle rotated by a body about an axis (in radian).

Angular velocity (ω): Rate of change of angular displacement.

Angular acceleration (α): Rate of change of angular velocity.

Equations (like linear motion):

• ω = ω₀ + αt
• θ = ω₀t + ½ αt²
• ω² = ω₀² + 2αθ

Relation between linear & angular quantities:

• Displacement: s = rθ
• Velocity: v = rω
• Acceleration: a = rα

👉 Example: A wheel (r = 0.5 m) rotates at ω = 10 rad/s → v = 5 m/s.

1.2 Kinetic Energy of Rotation

• Definition: Energy due to rotation of a rigid body.
• Formula: K.E. = ½ Iω²
• For rolling body: K.E. total = ½ Mv² + ½ Iω²

👉 Example: Solid sphere rolling → both translational + rotational K.E.

1.3 Moment of Inertia (M.I.) & Radius of GyrationMoment of Inertia (I):
Resistance of body to change in rotational motion.

Factors affecting I: mass, distribution of mass, axis of rotation.

Radius of Gyration (K):
Distance from axis where whole mass can be imagined concentrated to give same I.

Example: A thin ring about center → I = MR².

1.4 M.I. of Uniform Rod

• About center, ⟂ axis: I = (1/12) ML²
• About end, ⟂ axis: I = (1/3) ML²

👉 Example: Rod length = 2 m, mass = 3 kg → I(center) = 1 kg·m².

1.5 Torque & Angular AccelerationTorque (τ): Turning effect of a force.

• Relation: τ = Iα (rotational form of F = ma).

👉 Example: Force 10 N at 0.2 m, θ = 90° → τ = 2 N·m.

1.6 Work & Power in Rotation

• Work (W):
  W = τθ (like F·s in linear motion).
• Power (P):
  P = τω.

👉 Example: Torque = 4 N·m, θ = 2 rad → W = 8 J.

1.7 Angular Momentum & Conservation

• Angular Momentum (L):
  L = Iω = r × p.
• Conservation Law: If net external torque = 0, angular momentum remains constant.

👉 Example: Ice skater pulls arms → I↓, ω↑, but L constant.

Minor Topics & Definitions

• Rigid Body: Body whose shape & size do not change during motion.
• Axis of Rotation: Line about which body rotates.
• Right-hand rule: Curl fingers along rotation, thumb → direction of ω & α.
• Perpendicular Axis Theorem: For plane lamina, I_z = I_x + I_y.
• Parallel Axis Theorem: I = I_cm + Md².
• Rolling Motion: Combination of translation + rotation.
• Dimensional Formula of Torque: [ML²T⁻²].

Quick Formula Sheet (Last-Minute Revision):

Important Questions and Answers in Short

Q1. Define angular displacement.
👉Angle rotated by a body about an axis (in radian).

Q2. Define angular velocity (ω).
👉 Rate of change of angular displacement: ω = dθ/dt.

Q3. Define angular acceleration (α).
👉 Rate of change of angular velocity: α = dω/dt.

Q4. Write relations between linear and angular quantities.
👉 s = rθ, v = rω, a = rα.

Q5. Write equations of angular motion.
👉 ω = ω₀ + αt, θ = ω₀t + ½ αt², ω² = ω₀² + 2αθ.

Q6. Write expression for rotational kinetic energy.
👉 K.E. = ½ Iω².

Q7. Total K.E. of rolling body?
👉 K.E. = ½ Mv² + ½ Iω².

3. Moment of Inertia & Radius of Gyration

Q8. Define moment of inertia (I).
👉 Rotational inertia of a body, I = Σmr².

Q9. Define radius of gyration (K).
👉 Distance where whole mass is assumed concentrated to give same I.
Formula: I = MK².

Q10. State factors affecting moment of inertia.
👉 Mass, distribution of mass, and axis of rotation.

4. Moment of Inertia of Rod

Q11. M.I. of a uniform rod about center (perpendicular axis).
👉 I = (1/12)ML².

Q12. M.I. of a uniform rod about end (perpendicular axis).
👉 I = (1/3)ML².

5. Torque & Angular Acceleration

Q13. Define torque (τ).
👉 Turning effect of force, τ = rFsinθ.

Q14. State relation between torque & angular acceleration.
👉 τ = Iα.

6. Work & Power in Rotation

Q15. Work done in rotational motion.
👉 W = τθ.

Q16. Power in rotational motion.
👉 P = τω.

7. Angular Momentum

Q17. Define angular momentum (L).
👉 L = Iω = r × p.

Q18. State law of conservation of angular momentum.
👉 If external torque = 0, angular momentum remains constant.

Q19. Give one example of conservation of angular momentum.
👉 Ice skater pulling arms → I decreases, ω increases, L constant.

8. Minor Theorems & Facts

Q20. State perpendicular axis theorem.
👉 For plane lamina: I_z = I_x + I_y.

Q21. State parallel axis theorem.
👉 I = I_cm + Md² (d = distance between axes).

Q22. Define rigid body.
👉 Body whose shape & size do not change during motion.

Q23. Dimensional formula of torque.
👉 [ML²T⁻²].