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].
