ω = dθ/dt, α = dω/dt

Equations for constant α:
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ

Relation between linear & angular:
v = rω, a_t = rα, a_c = rω²

📌 Common MIs (about axis through CM)

• Rod (length L): I_CM = ML²/12
• Ring: I = MR²
• Disc: I = ½ MR²
• Solid sphere: I = ⅖ MR²
• Hollow sphere: I = ⅔ MR²

🔧 Theorems

Parallel axis: I = I_CM + Md²

Perpendicular axis (for lamina): I_z = I_x + I_y

For a system of particles, net external torque = dL/dt.

Work-energy: W_rot = ∫ τ dθ = ΔK_rot = ½ I ω² – ½ I ω₀².

For a particle: L = r × p = m(r × v).

Total KE = ½ M v_cm² + ½ I_cm ω² = ½ M v_cm² (1 + k²/R²) where k = radius of gyration.

Acceleration on incline: a = g sinθ / (1 + I/MR²). For disc: a = (2/3)g sinθ; for ring: a = (1/2)g sinθ.

💡 NEET TIPS & SHORTCUTS

• For rolling objects, the one with smallest I/MR² (disc, sphere) reaches bottom first on incline.
• In conservation of L, if moment of inertia changes, angular velocity changes inversely.
• Torque = rate of change of angular momentum; if torque is zero, L constant.
• Use parallel axis theorem to find I about any axis quickly.

⚠️ COMMON MISTAKES

• Using I = MR² for all bodies – memorize standard MIs.
• Forgetting direction in torque (use sign convention).
• Assuming friction always opposes motion – in pure rolling, static friction may act up or down incline.
• Applying conservation of angular momentum when external torque acts.

Rotational KE: K_rot = ½ Iω²

Torque: τ = Iα = r × F

Angular momentum: L = Iω

Rolling KE: K = ½ Mv² (1 + I/MR²)

Parallel axis: I = I_CM + Md²

Perpendicular axis (lamina): I_z = I_x + I_y