Rotational Motion

1. Angular Momentum

Angular momentum:

L = r × p = r × mv

For a rigid body:

L = Iω

2. Torque & Work in Rotational Motion

Torque: τ = r × F

Work done:

W = τθ

Power: P = τω

3. Rotational Kinematics (Analogous to Linear Motion)

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

4. Moment of Inertia (I)

I = Σmr²

For continuous bodies: I = ∫r² dm

SI unit: kg m²

Radius of gyration: I = Mk²

5. Torque–Rotation Relation

τ = Iα

(Law of rotation)

6. Kinetic Energy of Rotation

KE_rot = ½ Iω²

7. Important Theorems

Parallel Axis Theorem:

I = I_CM + Md²

Perpendicular Axis Theorem:

I_z = I_x + I_y

(Applicable only to plane lamina)

8. Standard Moments of Inertia

• Ring (about centre, ⟂ plane): MR²
• Disc (about centre, ⟂ plane): ½MR²
• Disc (about diameter): ¼MR²
• Solid cylinder (about axis): ½MR²
• Hollow cylinder (about axis): MR²
• Solid sphere (about diameter): 2/5 MR²
• Hollow sphere (about diameter): 2/3 MR²
• Rod (about one end): ⅓ML²

9. Centre of Mass

r_CM = (Σm_ir_i) / Σm_i

F_ext = M a_CM

10. Rolling Motion (No Slipping)

v = ωR

Acceleration down an incline:

a = g sinθ / (1 + I / MR²)

For a solid disc:

a = (2/3) g sinθ

11. Conservation of Angular Momentum

If net external torque = 0,

Angular momentum remains constant.