Rolling Without Slipping and Angular Momentum Conservation

• Rolling condition: v_cm = Rω — translation and rotation linked by no-slip constraint.
• Total KE of rolling body: KE = ½mv_cm² + ½Iω² = ½mv_cm²(1 + I/mR²).
• Angular momentum: L = Iω (rigid body), L = r × p (particle); SI unit kg·m²·s⁻¹.
• Conservation of angular momentum: If τ_ext = 0, then L = constant.
• Application: rolling on incline (acceleration depends on I), ice skater spin, planetary orbits.

Level 1 — Foundations

Verbal: A body rolls without slipping when the contact point has zero velocity relative to ground — rotation and translation are perfectly matched.

Rolling condition: v_cm = Rω; equivalently a_cm = Rα.

Total KE (rolling): KE_total = KE_translational + KE_rotational = ½mv² + ½Iω². Substituting ω = v/R: KE = ½mv²(1 + I/mR²).

KE fractions for common shapes:

Angular momentum: L = Iω for rotation; L = mvr (for particle moving in circle).

Conservation: If net external torque = 0, ΔL = 0 → L_initial = L_final.

Level 2 — JEE / NEET depth

Rolling on incline (smooth): Using energy conservation (no friction work in pure rolling): mgh = ½mv²(1 + I/mR²) → v = √(2gh / (1 + I/mR²)). Smaller I/mR² → faster rolling → solid sphere reaches bottom first.

Acceleration on incline: a = g sinθ / (1 + I/mR²). Solid sphere: a = 5g sinθ/7; Solid disc: a = 2g sinθ/3; Ring: a = g sinθ/2.

Condition for pure rolling on rough incline: Friction f = mI sinθ / (mR² + I) — friction causes rotation but does no net work.

Angular momentum of system: L_total = L_cm + L_about_cm = mv_cm r + I_cm ω (for rolling: both terms present).

Conservation examples:

• Ice skater pulls in arms: I decreases → ω increases (L constant).
• Diver in tucked position: smaller I → faster spin.
• Earth-Moon system: angular momentum conserved (Moon slowly recedes).

τ_ext = dL/dt: This is the fundamental rotational law (Newton's 2nd law, angular form).

Solid sphere rolling down incline

A solid sphere (M = 2 kg, R = 0.1 m) rolls without slipping from rest
down a 30° incline of length L = 5 m. Find speed at bottom and time taken.

Step 1 — Height: h = L sinθ = 5 × 0.5 = 2.5 m

Step 2 — For solid sphere: I = ²⁄₅ MR², so I/MR² = 2/5.

Step 3 — Speed at bottom (energy conservation):
v = √(2gh / (1 + I/MR²)) = √(2 × 10 × 2.5 / (1 + 2/5))
  = √(50 / 1.4) = √35.71 ≈ 5.98 m/s

Step 4 — Acceleration along incline:
a = g sinθ / (1 + I/MR²) = 10 × 0.5 / 1.4 = 5/1.4 ≈ 3.57 m/s²

Step 5 — Time from rest: v = at → t = v/a = 5.98/3.57 ≈ 1.67 s

Step 6 — Check with kinematics: v² = 2aL → v = √(2×3.57×5) = √35.7 ≈ 5.97 m/s ✓

Step 7 — KE partition:
KE_total = ½mv² = ½×2×35.7 = 35.7 J
KE_rot = (2/5)/(1+2/5) × 35.7 = (2/7) × 35.7 = 10.2 J (28.6%)

Ice skater angular momentum conservation

A skater spins at ω₀ = 2 rad/s with arms extended (I₀ = 4 kg·m²).
She pulls her arms in to I₁ = 1.5 kg·m². Find new ω.

Step 1 — No external torque (ice is frictionless for rotation here).
L = I₀ω₀ = I₁ω₁

Step 2 — Conservation:
4 × 2 = 1.5 × ω₁
ω₁ = 8/1.5 = 5.33 rad/s

Step 3 — KE check:
KE₀ = ½I₀ω₀² = ½×4×4 = 8 J
KE₁ = ½I₁ω₁² = ½×1.5×28.4 = 21.3 J
KE increased — muscle work done pulling arms in provides extra energy.

• State the rolling condition. What is the velocity of the contact point in pure rolling?
• A solid disc rolls at v_cm = 4 m/s (m=3 kg, R=0.2 m). Find total KE.
• Why does a solid sphere reach the bottom of an incline before a hollow sphere of same mass?
• A diver reduces I from 6 to 2 kg·m² during a somersault at ω=1 rad/s. Find new ω.
• Stretch: Compare the time for a ring and solid disc to roll down the same incline from rest.

NCERT Chapter 7 link: Rolling without slipping ties together translation (v_cm) and rotation (ω) through one constraint: v_cm = Rω. Angular momentum conservation is as powerful as linear momentum conservation — apply it whenever external torque is absent.

Exam connections: JEE tests: speed at bottom of incline for various shapes (sphere always fastest, ring always slowest); angular momentum conservation for spinning bodies changing I; total KE split between translation and rotation. Classic: a sphere, disc, and ring released from same height — sphere always wins.

Study strategy: Memorise the I/mR² ratio for each shape and the speed formula v = √(2gh/(1+I/mR²)). For L conservation problems, check first: is τ_net = 0? If yes, write L_i = L_f.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.