Q.30 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 30 Solution

Topic: Atoms · Official NTA Booklet Code 45

Question

A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius r of the nth orbit and the particle's speed v in the orbit depend on n as

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Verified Against NTA Final Answer Key

Option (3) : r \propto n^{2/3}; \quad v \propto n^{1/3}

📚 Atoms Medium

📋 Given Parameters & Problem Setup

Force Law Constant magnitude central pulling force F (independent of r)

Theoretical Framework Bohr atomic quantization model

Target Variables Orbital radius r(n) and orbital speed v(n) scaling with quantum number n

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The 30-Second Exam Hall Shortcut: ⚡ Centripetal vs. Angular Momentum Power Matching

⏱️ Save ~45 seconds

F = m v² / r = const ⟹ v ∝ r^(1/2). Bohr condition: m v r = n ħ ⟹ r · r^(1/2) ∝ n ⟹ r^(3/2) ∝ n ⟹ r ∝ n^(2/3) and v ∝ n^(1/3)!

1 Condition 1 (Centripetal equilibrium): m v² / r = F (constant) ⟹ v² ∝ r ⟹ v ∝ r^(1/2).

2 Condition 2 (Bohr quantization): Angular momentum L = m v r = n (h / 2π) ⟹ v r ∝ n.

3 Substitute v ∝ r^(1/2): r^(1/2) · r ∝ n ⟹ r^(3/2) ∝ n ⟹ r ∝ n^(2/3).

4 Find speed scaling: v ∝ r^(1/2) ∝ (n^(2/3))^(1/2) = n^(1/3).

5 Result: r ∝ n^(2/3) and v ∝ n^(1/3).

6 Match Option (3) in under 20 seconds!

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Why You Can 100% Trust This Shortcut: Bohr’s Postulates applied to Generalized Central Potentials (NCERT Atoms).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Relate Speed and Radius via Centripetal Force

The particle of mass m moves in a circular orbit of radius r under the action of a constant inward radial force F. Equating centripetal force to F:

\frac{m v^2}{r} = F \implies v^2 = \frac{F}{m} r \implies v = \sqrt{\frac{F}{m}} \, r^{1/2} \propto r^{1/2}

• Because F and m are constant, orbital speed is proportional to the square root of radius: v ∝ r^(1/2).

Step 2

Apply Bohr’s Angular Momentum Quantization

According to Bohr’s postulate, the orbital angular momentum is an integral multiple of h / 2π:

L = m v r = n \frac{h}{2\pi} = n \hbar

• Substitute v ∝ r^(1/2) into the angular momentum equation:
• m (C r^(1/2)) r = n ħ ⟹ r^(3/2) ∝ n.

Step 3

Derive the Exponents for r and v

Solve for r and v as functions of the principal quantum number n:

r^{3/2} \propto n \implies r \propto n^{2/3}

• Now substitute r ∝ n^(2/3) back into the speed relation:
• v ∝ r^(1/2) ∝ (n^(2/3))^(1/2) = n^(1/3).
• Thus: r ∝ n^(2/3) and v ∝ n^(1/3), corresponding to Option (3).

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360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 A

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. If r ∝ n^(1/3), then mvr ∝ n^(1/3) · n^(1/3) = n^(2/3) ≠ n.

⚠️

Student Fallacy (Why candidates pick this trap): Failing to balance angular momentum product.

2 B

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Swapped exponents of r and v.

⚠️

Student Fallacy (Why candidates pick this trap): Inverting the relation between velocity and radius.

3 C

✓ 100% Correct

Verdict & Scientific Reason:

100% correct. v ∝ r^(1/2) and v·r ∝ n gives r ∝ n^(2/3) and v ∝ n^(1/3).

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Why this is airtight: None; textbook non-Coulombic Bohr orbit derivation.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Corresponds to Coulomb potential (where F ∝ 1/r² ⟹ r ∝ n² and v ∝ 1/n).

⚠️

Student Fallacy (Why candidates pick this trap): Conflating Coulomb 1/r² force with constant force.

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NTA Trap Alert: Assuming Standard Coulomb Orbit Results (r ∝ n²)

Students often memorize standard hydrogen atom results (r ∝ n² and v ∝ 1/n) from Coulomb potential F ∝ 1/r². Here the force is CONSTANT (F = const), so you MUST re-derive using v² ∝ r!

🛡️ How to avoid the -1 mark: Never use hydrogen formulas when the force law is modified; always set m v² / r = F_given.

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Atoms Standard Benchmark

Core Scientific Principle Tested:

"Bohr quantization condition mvr = nħ applied to arbitrary central force law F(r)."

Syllabus Focus: Bohr’s Model of the Atom

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

v² ∝ r ⟹ v·r = r^(3/2) ∝ n ⟹ r ∝ n^(2/3) and v ∝ n^(1/3)!

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NEET 2025 Physics Question 30 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Centripetal vs. Angular Momentum Power Matching

Step-by-Step Rigorous Workflow & Mathematical Proof

Relate Speed and Radius via Centripetal Force

Apply Bohr’s Angular Momentum Quantization

Derive the Exponents for r and v

360° Option Elimination Engine Key Feature

NTA Trap Alert: Assuming Standard Coulomb Orbit Results (r ∝ n²)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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