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Physics · Mechanics · Hard✓ Verified
A disc of mass $M$ and radius $R$ is connected to a wall by a spring of stiffness $k$ at its centre and rolls without slipping on a horizontal floor. Find the angular frequency of small oscillations.
Concept tested: SHM with rolling constraint: rotational inertia raises effective mass to $\frac{3M}{2}$
1Set up equations of motion
Spring force $= -kx$, friction $f$ at contact point. Translational: $-kx - f = M\ddot{x}$. Rotational about centre: $fR = \frac{MR^2}{2} \cdot \frac{\ddot{x}}{R}$, giving $f = \frac{M\ddot{x}}{2}$.
2Apply rolling constraint and solve
Substituting back: $-kx = M\ddot{x} + \frac{M\ddot{x}}{2} = \frac{3M}{2}\ddot{x}$. Therefore $\omega = \sqrt{\dfrac{2k}{3M}}$.
Common mistake:
Writing $\omega = \sqrt{k/M}$, ignoring rotational KE. For a rolling disc, effective mass is $\frac{3M}{2}$, not $M$.

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A feature-by-feature breakdown of what each tool actually offers.

FeatureJJEEniusDoubtnutChatGPT / Gemini
Instant step-by-step solution (text & image)
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Handles matrix-match, multi-correct & assertion-reason
Asks a follow-up to check you actually understood it
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Dual AI verification in action

JEEnius caught what other AIs missed

These are real JEE/NEET traps where ChatGPT and Gemini give the wrong answer. JEEnius's Chain-of-Verification catches them every time.

Physics: Pressure formula trap

A soap bubble of radius R is in equilibrium. What is the excess pressure inside the bubble?

ChatGPT / Gemini said

2T/R

JEEnius verified

4T/R

A soap bubble has TWO surfaces (inner + outer), so ΔP = 4T/R. A liquid drop has only one surface: ΔP = 2T/R. ChatGPT/Gemini routinely confuse the two.

Organic Chemistry: Substitution mechanism

A tertiary alkyl bromide reacts with NaOH in polar protic solvent (water/ethanol). What is the major product and mechanism?

ChatGPT / Gemini said

SN2: direct backside attack gives inversion product

JEEnius verified

SN1: carbocation intermediate, racemic mixture

Tertiary substrates block backside attack (steric hindrance) → SN2 impossible. Polar protic solvents stabilise carbocations → SN1 favoured. Common AI error: applying SN2 rules to tertiary substrates.

Algebra: GP constraint trap

Let a₁, a₂, a₃, … be a G.P. of increasing positive terms. If a₁a₅ = 28 and a₂ + a₄ = 29, then a₆ is equal to?

ChatGPT / Gemini said

64

JEEnius verified

√(a₁ · a₅) · r² = 2·√7 · (√7)² → a₆ = 56√7 ≈ 148.2

JEEnius verified step-by-step: a₁·a₅ = a₁²r⁴ = 28 and a₂+a₄ = a₁r(1+r²) = 29. Solving gives a₁ = 28/(a₁r⁴), the constraint 'increasing' eliminates the extraneous root that naive solvers accept.

Organic Chemistry: Markovnikov exception

Propene reacts with HBr in the presence of peroxide (ROOR). What is the major product?

ChatGPT / Gemini said

2-bromopropane (Markovnikov addition)

JEEnius verified

1-bromopropane (anti-Markovnikov via free radical mechanism)

Peroxides initiate a free-radical chain mechanism. Br• adds to the less-substituted carbon (more stable secondary radical intermediate) giving the anti-Markovnikov product. Without peroxide it is Markovnikov.

Physics: Mirror formula sign convention

An object is placed 20 cm in front of a concave mirror of focal length 15 cm. Find the image distance.

ChatGPT / Gemini said

Image at +60 cm (virtual, behind mirror)

JEEnius verified

Image at −60 cm (real, in front of mirror)

Using the New Cartesian sign convention: u = −20 cm, f = −15 cm (concave). Mirror formula: 1/v + 1/u = 1/f → 1/v = 1/(−15) − 1/(−20) = −1/60, so v = −60 cm. Real image in front. ChatGPT often drops the sign on f for concave mirrors.

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