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Formula dump, flashcards, memory tricks. Use this the night before the exam.
1. Read the formula dump top to bottom (15 min). 2. Test yourself with flashcards (10 min). 3. Review mistake alerts (5 min). 4. Done. You're ready.
Complete Formula Dump
📍 Centre of Mass
🔄 Rotational Kinematics
🌀 Torque & Angular Momentum
⚖️ Moment of Inertia
| Body | Axis | I |
|---|---|---|
| Ring (R) | Centre ⊥ plane | MR² |
| Ring (R) | Diameter | MR²/2 |
| Disc (R) | Centre ⊥ plane | MR²/2 |
| Disc (R) | Diameter | MR²/4 |
| Solid Sphere (R) | Diameter | 2MR²/5 |
| Hollow Sphere (R) | Diameter | 2MR²/3 |
| Solid Cylinder (R) | Own axis | MR²/2 |
| Hollow Cylinder (R) | Own axis | MR² |
| Thin Rod (L) | Centre ⊥ length | ML²/12 |
| Thin Rod (L) | End ⊥ length | ML²/3 |
🎳 Rolling Motion
📊 k²/R² Quick Reference
Flashcards — Tap to Flip
Click the card to reveal the answer. Use arrow buttons to navigate.
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Shuffle cards or complete all 20 to finish the revision mode.
Never Forget Again
Smallest k²/R² wins. "Solid always beats Hollow of same type."
Pattern: hollow=2, solid=1, numerator (for spheres). "Two-fifths for solid sphere, two-thirds for hollow."
Curl fingers of right hand in direction of rotation → thumb points in direction of ω (angular velocity vector). Works for torque direction (curl from r to F → thumb = τ direction).
Note: disc COM = (2/3) × ring COM. "Disc is lower because mass is inside, not just on edge."
The word "perpendicular axis theorem" contains "flat" thinking: it ONLY works for flat (planar) objects. Think of a flat disc, flat ring, flat plate. NEVER for a ball or cylinder.
Solid sphere: (2/5)/(1 + 2/5) = 2/7. Solid cylinder: (1/2)/(1 + 1/2) = 1/3. Remember these two fractions cold.
"Perpendicular distance" = lever arm. Remember by analogy: a wrench works better when you push far from the bolt (larger lever arm). τ = F × (⊥ distance from axis).
When moment of inertia decreases (arms pulled in), angular velocity must increase to keep L = Iω constant. Think: "I decreases → ω increases." Works for any spinning system without external torque.
Last-Minute Mistake Alerts
These mistakes appear in 70% of student papers. Read each one carefully.
NEVER apply I_z = I_x + I_y to spheres or cylinders. It's only for flat (planar) bodies: disc, ring, rectangular plate. This mistake appears in 1 in every 3 JEE Main papers.
In rolling motion, different points have different velocities. v_contact = 0, v_cm = v, v_top = 2v. If a question asks "velocity of topmost point," the answer is 2v_cm, NOT v_cm. This is tested frequently.
In collision problems with a pivot: conserve L about the PIVOT (impulsive force acts there, but has zero torque about that point). In free-body collisions: conserve L about COM of the rod/body. Wrong axis = wrong answer = zero marks.
When mass redistributes (person walks on disc, bead slides on rod), I changes. Use τ = dL/dt, not τ = Iα. This is a conceptual trap in JEE Advanced.
In any perfectly inelastic collision (bullet embeds in disc/rod), angular momentum is conserved, kinetic energy is lost. Never say "KE is conserved" in embedding problems.
Hollow hemisphere COM = R/2. Solid hemisphere COM = 3R/8. These are different — and both are asked. Always check what type of hemisphere is specified.
Pure rolling requires static friction. If surface is frictionless, the body CANNOT roll — it can only slide or spin. In frictionless incline problems, use energy conservation without rotational KE (no rolling). If surface is rough, include rotational KE.
A particle moving in a straight line has non-zero angular momentum about a point NOT on the line of motion. L = mvd, where d = perpendicular distance from point to line. This is non-zero even though the particle doesn't rotate. Classic exam trap.