Rolling-Ring Bench

QMission brief — the question, verbatim

JEE (Advanced) 2026, Paper 1, Physics, Section 1 — four options, only one correct: +3 if only the correct option is chosen, 0 if unanswered, −1 in all other cases.

Q.1 Consider a large disk of radius R and two smaller disks, each of radius r = R/50, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation Δθ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities ω and 2ω while the large disk is held stationary. The time τ at which the smaller disks are again in contact is:

[Use sin(Δθ) = Δθ and ignore gravity.]

(A)τ = 51 × (2π − 4/51)/ω
(B)τ = 51 × (2π − 2/51)/3ω
(C)τ = 51 × (2π − 4/51)/3ω
(D)τ = 51 × (2π − 2/51)/ω

CONTACT ANGLE Δθ

Ω₁ + Ω₂

ωτ

The mystery. Two tiny rollers sit touching on the rim of a fixed disk and set off in opposite directions, one twice as fast as the other. How fast does each one's centre really go round, and how far must they travel before they touch again? The bench rolls them for real, times the meeting, and checks all four options.

01Interactive experiment

A rolling-ring bench: a clamped steel disk and two knurled rollers, with a magnifying lens riding on each roller. Press START EXPERIMENT: set the rollers touching → roll without slipping → the centres' circle → move apart → the long way round → they meet → the formula → audit the options → answer.

Ready

01 Contact02 No slip03 Centre circle 04 Apart05 Long way round06 Contact again 07 Formula08 Audit09 Final answer

Set your values or use the question defaults.

Answer——

step 1 of 7

Two rollers touch on the rim of a clamped disk. The question's values are loaded — press START EXPERIMENT.

✓ QUESTION VALUES LOADED

Customize experiment — optional

Instrument panel

stopwatch ωt

—

roller 1 moved

—

roller 2 moved

—

centres / 2r

—

spins · roller 1

—

spins · roller 2

—

Ω₁ · Ω₂ (×ω)

—

progress

0 %

option

—

The disk and the rollers are drawn at true scale (r = R/50); the lenses magnify each roller so you can see it roll. Time is measured in units of 1/ω, so ωt is a pure number.

Nothing needs to be chosen. Press START EXPERIMENT.

How far each centre has gone round

angles moved against ωt — they touch again when the two add up to 2π − 2Δθ

ωτ of each option against the bench

the stopwatch reading when they touched, and the four printed formulas

Result audit — the four options

(A)τ = 51 × (2π − 4/51)/ω

not yet examined

UNEVALUATED

(B)τ = 51 × (2π − 2/51)/3ω

not yet examined

UNEVALUATED

(C)τ = 51 × (2π − 4/51)/3ω

not yet examined

UNEVALUATED

(D)τ = 51 × (2π − 2/51)/ω

not yet examined

UNEVALUATED

📐Contact Geometer

🛞No-Slip Roller

🪙Coin-Paradox Breaker

↔️Relative-Motion Pro

🪤Trap Spotter

🏆Physics Master

Reasoning log — input → action → change → observation → conclusion

Nothing investigated yet.

Evidence table (0)

02Coin-rotation explorer

Roll one roller once round the fixed disk and count how many times it spins. Rolled along a straight track of the same length it would spin R/r times — round the outside of a circle it spins one extra time, because its centre travels on the bigger circle R + r. Change the ratio, drag the roller, or let it roll.

Ratio R/r: 4

Roll it round: 0°

03Detailed solution

The complete worked solution — geometry, the rolling constraint, the angular speeds and the time.

Understand the concept

Rolling without slipping

The point of the roller touching the fixed disk is momentarily at rest. So the centre moves at v = ωr and the top at 2v.

v_centre = ω r

The centre's circle is R + r

On the outside of the fixed disk the centre stays at distance R + r from O, so its angular speed about O is v/(R + r), not v/R.

Ω = ω r/(R + r) = ω/51

The coin-rotation paradox

A roller taken once round spins (R + r)/r = 51 times, not R/r = 50. The extra turn is the trip round the circle itself.

spins per lap = 1 + R/r

Relative angular motion

Moving in opposite directions, the angle between the centres changes at Ω₁ + Ω₂. The gap must grow from Δθ to 2π − Δθ.

ωτ = (2π − 2Δθ)/(Ω₁ + Ω₂)·ω

Step 1 — The angle between the centres when they touch

The two centres are 2r apart, each at distance R + r from O. With sin Δθ = Δθ the chord equals the arc.

(R + r) Δθ = 2r ⇒ Δθ = 2r/(R + r) = 2(R/50)/(51R/50) = 2/51 rad
Step 2 — Rolling without slipping sets each centre's speed

The contact point is at rest, so every point moves as if turning about it: centre v = ωr, top 2v. (Drawn on a straight track; on the rim the centre's path is the circle R + r.)

v₁ = ω r, v₂ = 2ω r
Step 3 — Angular speed of each centre about O

Ω₁ = v₁/(R + r) = ω r/(51 r) = ω/51, Ω₂ = 2ω r/(51 r) = 2ω/51

Using R instead of R + r would give ω/50 — the coin-rotation trap. The rollers roll on the outside, so the centre circle is the bigger one.
Step 4 — How far apart they must get

Start (filled): gap Δθ on one side. Meet (rings): the gap on the far side has closed to Δθ. The angle swept between them is 2π − 2Δθ.

(Ω₁ + Ω₂) τ = 2π − 2Δθ = 2π − 4/51

Step 5 — The time

Ω₁ + Ω₂ = ω/51 + 2ω/51 = 3ω/51
τ = (2π − 4/51) / (3ω/51) = 51 × (2π − 4/51)/3ω
ωτ = (320.442 − 4)/3 = 105.48 → option —

Option	ωτ	What it does
(A) 51(2π − 4/51)/ω	316.44	uses ω/51 for the relative angular speed — as if only one roller moved
(B) 51(2π − 2/51)/3ω	106.15	subtracts Δθ once — forgets that the far-side gap also ends at Δθ
(C) 51(2π − 4/51)/3ω	105.48	correct: sweep 2π − 2Δθ at 3ω/51
(D) 51(2π − 2/51)/ω	318.44	both mistakes

Traps: using R instead of R + r (50 in place of 51); taking a full 2π; forgetting that both rollers move.

Key takeaways

Rolling without slipping on a fixed surface: the contact point is at rest, so v_centre = ωr.
Rolling round the outside of a circle, the centre moves on radius R + r: Ω = ωr/(R + r). A roller spins 1 + R/r times per lap.
Two bodies moving round a circle in opposite directions close or open their angle at Ω₁ + Ω₂.
"In contact again" for bodies of finite size: the centres must be the contact angle apart on the other side — sweep 2π − 2Δθ, not 2π.
Small-angle step: the chord 2r equals the arc (R + r)Δθ when sin Δθ = Δθ; the exact 2 asin(1/51) differs by less than 10⁻⁵ rad here.

Independent verification

Symbolic. sympy derives Ω = ωr/(R + r), Δθ = 2r/(R + r) and τ from the rolling constraint; only option C simplifies to the same expression.
Time stepping. Each centre is moved by the no-slip condition alone (the contact point's velocity set to zero), with RK4 and the exact contact angle; the rollers touch again at ωτ = 105.4807, within 0.01 % of C.
Coin rotation. The integrated spin is 51 turns per lap, not 50; roller 2 turns twice as far as roller 1.
Controls. R/r = 10, 5 and equal or unequal speeds all follow (R + r)/r × (2π − 2Δθ)/(n₁ + n₂).
Solution website. MathonGo's published solution (their Q17 = Physics Q.1) gives (C) by the same steps.
This page. The bench computes τ from Ω and Δθ and, separately, by stepping the rollers until the centres are 2r apart; the answer is the option that matches.
Answer key. Checked against the official key printed with the paper: Q.1 → C.

04How to use this simulation

For a student opening it for the first time. The whole investigation runs by itself after one press.

STEP 1

Read the mission brief

Two small disks touching on a fixed disk roll apart at ω and 2ω. When do they touch again?

STEP 2

Check the values

The question's values are loaded. You do not need to change anything. The answer is shown above the experiment.

STEP 3

Optional: customize

Change R/r, the two speeds, the direction or the contact-angle rule. The lab says honestly when you have left the question. ↺ RESTORE QUESTION CONDITIONS puts everything back.

STEP 4

Start

Press ▶ START EXPERIMENT under the experiment screen. Nothing else to click (about a minute; SLOW MOTION or 2× changes the pace).

STEP 5

Watch the lenses

Each lens rides on a roller: the green dot is the contact point at rest, the orange arrows are the centre and top velocities. The stopwatch counts ωt.

STEP 6

The long way round

The rollers separate, go round in opposite directions and meet on the far side. The bench stops the clock when their centres are 2r apart again.

STEP 7

Explore

In the Coin-rotation explorer roll one roller round and count its spins. Then read the Detailed solution.

STEP 8

Experiment again

Press ▶ REPLAY EXPERIMENT, or change a value and press ⚙ RUN WITH CUSTOM VALUES. CLASSROOM MODE enlarges everything for a projector.

Classification

Exam	JEE Advanced 2026, Paper 1, Physics, Section 1, Q.1
Chapter	System of Particles and Rotational Motion
Topic	Rolling without slipping on a fixed circular surface - angular speed of the centre and relative angular motion
Difficulty	Moderate
Verification	script-verified (automated pipeline, no human review)
Answer	— run the experiment

Rolling-Ring Bench — when do the two rollers touch again?

QMission brief — the question, verbatim

01Interactive experiment