Physics notesMechanicsRotation
Independent topic page
Pick translation+rotation about COM (or fixed-axis) in 15 seconds, write τ=Iα and energy/angular momentum as needed; rolling without slipping is one constraint v=ωr that closes the system — expect joins with collisions and friction audits.
01. For a rigid body, K = ½ M V_COM² + ½ I_COM ω². About a fixed axis, K=½ I_axis ω² with I about that axis.
02. Parallel-axis: I_z = I_COM + Md² for axes parallel to a COM axis. Perpendicular-axis (planar lamina): I_z = I_x + I_y for z perpendicular to the plane.
03. Στ_ext = dL/dt. About a fixed axis or about COM, τ = Iα in the usual syllabus settings. Rolling without slipping: a = rα and v = ωr (signs by sense of rotation).
04. Angular momentum: L_COM = I_COM ω for rotation about a principal COM axis; about a point O, include M (R_COM × V_COM) as well. Collisions: impulsive torque changes L; friction impulse often needed for pure rolling to establish.
05. Advanced stems frequently chain: impulse → V_COM and ω → rolling check → energy after slipping stops. Write the sequence before algebra.
06. Rotation about a fixed axis: τ_ext = Iα with one consistent axis and sign. Rolling without slip couples v=ωr and a=αr—write the constraint before inventing friction magnitude.
07. Energy: ½Mv_cm²+½I_cmω² (or ½I_pω² about instantaneous axis of rotation). Torque–energy consistency beats memorising which friction does work (static friction on pure roll does no work if contact point is instantaneously at rest).
08. Angular momentum about a point: L=Iω for planar rotation about a fixed axis; for CM plus spin use L_cm + I_cmω. Conservation needs τ_ext≈0 about that point—check hinges and friction carefully.
Official · 2026
Official syllabus points for Rigid Body Rotation only — not the full Mechanics portion map. Portion hub: Mechanics · Mechanics analysis.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusMoment of inertia, torque, rolling, and angular momentum of rigid bodies.
Syllabus points
Do / check
2020–2026 paper weight
Topic match inside parent portion Mechanics (32.3% of Physics tags). Stats below are portion backdrop; matched subtopics show where Rigid Body Rotation sits.
Portion-wide type/difficulty mix lives on the Mechanics analysis — not repeated here so this page stays on Rotation.
01. Dominant subtopics: Rotation (21), Mixed (16), Projectile/Kinematics (10), Gravitation (7).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Physics stems that touch Rigid Body Rotation — practise the knowledge bridge, not only the single-chapter drill.
Step 1: relate arc lengths so the small disks roll without slipping on the large circumference (geometry of contact). Step 2: convert path radius R+r into orbital and spin rates (ω/51-type ratios). Step 3: close the remaining angular gap 2π − Δθ to get the meeting time τ.
Easy to use the wrong Δθ (2/51 vs 4/51) if relative rolling sense is mishandled.
Always draw contact velocities and write v = ωr for every rolling interface before algebra.
Step 1: apply angular impulse about the pivot C for the particle–disk collision. Step 2: convert post-collision ω into maximum rise of centre O (energy after collision). Step 3: compare mechanical energy before vs after to get the collision energy loss.
Choosing the wrong axis or mixing linear KE with rotational KE about C breaks both answers.
Linked collision–rotation: conserve angular momentum about the eventual pivot, then energy separately.
Step 1: use energy to relate ω and θ as the cylinder rolls on the edge. Step 2: write radial (centripetal) condition with normal → 0 at loss of contact. Step 3: solve for v = √(5gR/7) (or the option equivalent).
Combines rolling constraint, energy and N=0 in one geometric setting.
Memorise the ‘energy + N=0’ template for bodies leaving curved/edge surfaces.
Step 1: identify axis and moment of inertia for each stem. Step 2: apply torque = Iα or energy+constraint as needed. Step 3: refuse to reuse numbers from Q2 in Q3 without re-deriving.
Similar-looking rotation stems invite copy-paste mistakes.
Reset the free-body and axis choice on every new rotation question.
Step 1: finish the rotation NUM with a clear MOI/energy chain. Step 2: switch mental model to interference/diffraction for the next NUM. Step 3: protect calculator/decimal hygiene across both.
Topic switch mid-numerical section drains working memory.
In NUM blocks, write a one-line topic label before each solve.
Step 1: isolate the electromagnetic law needed (flux/induced emf/impedance). Step 2: compute the asked scalar. Step 3: consciously leave rotation kinematics behind so constants are not mixed.
Topic island inside a Mechanics-heavy NUM sea.
Underline the chapter keywords in each NUM stem before writing equations.
Why methods matter
Advanced rotation is COM+spin, smart axes and a rolling audit — assume pure rolling, solve f, then verify |f|≤μN.
One language covers rolling, toppling setup and free rigid collisions.
Do thisAlways write translation of COM and rotation about COM as two coupled equations.
One language covers rolling, toppling setup and free rigid collisions.
Cue impulse on billiard ball: J = M ΔV_COM and −J h = I Δω; then check v ? ωr.
Gives ω and V after impact without knowing friction impulse magnitude.
Do thisAbout the contact point during a quick collision on a rough surface, friction impulse may have no moment—use L conservation there if gravity impulse negligible.
Gives ω and V after impact without knowing friction impulse magnitude.
Rod falling flat hit: about bottom contact point, angular momentum just before equals just after if impulsive friction acts at that point.
Same stick-then-verify discipline as linear friction—critical on Advanced rolling stems.
Do thisAssume pure rolling, solve f, verify |f|≤μN; if not, redo with kinetic friction and a≠rα.
Same stick-then-verify discipline as linear friction—critical on Advanced rolling stems.
Steep incline: required f for pure rolling can exceed μmg cosθ ⇒ slipping while spinning.
Predicts whether spool rolls toward or away from pull before full algebra.
Do thisFor a pulled spool, take torque about contact point: pull line’s moment decides sense of roll tendency.
Predicts whether spool rolls toward or away from pull before full algebra.
Pull above critical angle/height: torque about contact can reverse intended roll direction.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Find I of a thin ring of mass M and radius R about a tangent in its plane.
This stem rewards early recognition of the moment of inertia and axis theorems model.
Model chosenUse perpendicular and parallel axis theorems.
Answer(3/2)MR²
Takeaway: Shift I with parallel axis after choosing the right central value.
A solid sphere rolls without slipping down an incline of angle 30°. Find acceleration. Take g=10 m/s².
This stem rewards early recognition of the rolling without slipping model.
Model chosena=g sinθ/(1+I/mR²).
Answer≈3.57 m/s²
Takeaway: Rolling inertia effectively increases mass in the denominator.
A uniform rod of mass M and length ℓ lies on a smooth horizontal table. A particle of mass m and speed v₀ strikes one end perpendicular to the rod and sticks. Find the angular speed of the system about the COM.
This Advanced-level variation couples hidden constraints with the angular momentum conservation model.
Model chosenConserve angular momentum about the final COM; kinetic energy is not conserved.
Answerω=m v₀ (ℓ/2−d)/I with d=m(ℓ/2)/(M+m), I=I_rod+Md²+m(ℓ/2−d)²
Takeaway: Inelastic stick collisions conserve L about COM, not mechanical energy.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likeI about a new axis, lamina, or composite body
Fast modelBuild I from known pieces; then parallel/perpendicular axis.
Recognition fixes the model before arithmetic begins.
Do not start blind expansion or casework before identifying the governing model.
Looks likePure rotation about fixed axis
Fast modelΣτ = Iα about the rotation axis.
Recognition fixes the model before arithmetic begins.
Do not start blind expansion or casework before identifying the governing model.
Looks likeSphere/cylinder/hoop on rough surface
Fast modelUse a = Rα and static friction ≤ μN; energy if no slip throughout.
Recognition fixes the model before arithmetic begins.
Do not start blind expansion or casework before identifying the governing model.
Looks likeCollision with rod, person on disc, or impulsive torque-free
Fast modelConserve L about a fixed/CM axis when external impulse torque is zero.
Recognition fixes the model before arithmetic begins.
Do not start blind expansion or casework before identifying the governing model.
Hall timing
Underline given data, unknowns and constraints. Name the mechanical model before recalling any formula; discard relations whose assumptions are already broken.
Draw axes, free-body or energy/impulse diagram. Write governing equations symbolically with consistent sign and reference.
Solve algebraically as far as possible; substitute numbers only once. Keep vectors, components and units on the rough sheet.
Check dimensions, limiting cases and physical sense. For multi-correct, test every option independently rather than forcing one narrative.
After the model is locked
τ_net = I α (fixed axis / about COM)
Rigid body angular acceleration
I = I_COM + Md²
Shift to parallel axis
I_z = I_x + I_y (planar lamina)
Lamina in xy-plane
v = ω r; a = α r (consistent sense)
Pure rolling constraint
K = ½Mv² + ½I_COM ω² = ½Mv²(1+k²/r²) if v=ωr, I=Mk²
Energy of pure rolling
a = g sinθ / (1+I/(Mr²))
Static friction, no slip down incline
L_O = R_COM×M V_COM + I_COM ω (vector)
About point O
rod COM ⊥: ML²/12; disc COM ⊥ plane: MR²/2; hoop COM: MR²; solid sphere: 2/5 MR²
Quick I values
Make it automatic