Q.29 Physics ⭐ 360° Option Elimination Active

NEET 2023 Physics Question 29 Solution

Topic: System of Particles and Rotational Motion · Official NTA Booklet Code G5

Question

The ratio of radius of gyration of a solid sphere of mass M and radius R about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

✓

Verified Against NTA Final Answer Key

Option (5) : Question Dropped / Bonus (Option 2 was closest intended: 3 : 5)

📚 System of Particles and Rotational Motion Easy

📋 Given Parameters & Problem Setup

Solid sphere mass & radius M, R (about central diameter)

Hollow sphere mass & radius M, R (about central diameter)

Moment of inertia solid sphere (I₁) (2/5) M R²

Moment of inertia hollow sphere (I₂) (2/3) M R²

Theoretical ratio k₁ : k₂ √(3/5) = √3 : √5

⚡

The 30-Second Exam Hall Shortcut: ⚡ Instant Radius of Gyration: k = √(I / M) ⟹ k ∝ √I

⏱️ Save ~60 seconds

k_solid = √(2/5) R and k_hollow = √(2/3) R ⟹ k₁/k₂ = √( (2/5) / (2/3) ) = √(3/5) = √3 : √5! Paper omitted the radical sign (printed 3:5), resulting in an official NTA Bonus!

1 Radius of gyration is defined by I = M k² ⟹ k = √(I / M).

2 For a solid sphere about its diameter: I₁ = (2/5) M R² ⟹ k₁ = √(2/5) R.

3 For a thin hollow sphere (spherical shell): I₂ = (2/3) M R² ⟹ k₂ = √(2/3) R.

4 Take the ratio: k₁ / k₂ = √(2/5) / √(2/3) = √( (2/5) × (3/2) ) = √(3/5).

5 Therefore, the exact physical ratio is √3 : √5.

6 The examination paper printed options without square roots: (1) 5:2, (2) 3:5, (3) 5:3, (4) 2:5.

7 Because √3 : √5 was missing, NTA DROPPED this question in the official final answer key and awarded full marks (+4) to all candidates!

🛡️

Why You Can 100% Trust This Shortcut: Rotational Dynamics · Moment of Inertia Standards (Radius of gyration depends on the square root of inertial distribution: k = √(I/M)).

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Fundamental Definition of Radius of Gyration

The radius of gyration k of a rigid body of mass M about a specified axis of rotation is the effective radial distance at which the entire mass could be concentrated without altering its moment of inertia:

I = M k^2 \implies k = \sqrt{\frac{I}{M}}

Step 2

Radius of Gyration for the Solid Sphere

For a uniform solid sphere of mass M and radius R about its own diametrical axis:

I_1 = \frac{2}{5} M R^2 \implies k_1 = \sqrt{\frac{\frac{2}{5} M R^2}{M}} = \sqrt{\frac{2}{5}}\, R

Step 3

Radius of Gyration for the Thin Hollow Sphere

For a thin hollow spherical shell of identical mass M and radius R about its diametrical axis:

I_2 = \frac{2}{3} M R^2 \implies k_2 = \sqrt{\frac{\frac{2}{3} M R^2}{M}} = \sqrt{\frac{2}{3}}\, R

Step 4

Evaluation of the Theoretical Ratio & Identification of Paper Typo

Evaluating the ratio k₁ / k₂:

\frac{k_1}{k_2} = \frac{\sqrt{\frac{2}{5}}\, R}{\sqrt{\frac{2}{3}}\, R} = \sqrt{\frac{2}{5} \times \frac{3}{2}} = \sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}} = \sqrt{3} : \sqrt{5}

Step 5

Official NTA Final Verdict (Question Dropped)

None of the four printed options contained the square root sign. The closest intended option was Option (2) 3:5, which is the ratio of k₁² : k₂² (or I₁ : I₂). Because of this clerical typographical omission, the National Testing Agency (NTA) dropped Question 29 and awarded full marks to all candidates.

\text{NTA Final Status: Dropped (Full Marks Awarded to All Candidates)}

⭐

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 (1)

✗ Incorrect

Verdict & Scientific Reason:

Reciprocal of (2/5) without square root.

⚠️

Student Fallacy (Why candidates pick this trap): Candidate inverted the inertia of the solid sphere.

2 (2)

✓ 100% Correct

Verdict & Scientific Reason:

Accepted under NTA bonus policy. Closest intended option representing (k₁/k₂)² = 3/5.

💡

Why this is airtight: Intended answer by question setter who forgot to take the square root of I₁/I₂.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

Inverted ratio representing hollow to solid sphere (I₂ / I₁ = 5/3).

⚠️

Student Fallacy (Why candidates pick this trap): Candidate computed hollow sphere to solid sphere instead of solid to hollow.

4 (4)

✗ Incorrect

Verdict & Scientific Reason:

Represents only the prefactor of the solid sphere.

⚠️

Student Fallacy (Why candidates pick this trap): Candidate selected the single coefficient of solid sphere.

⚠️

NTA Trap Alert: Forgetting the Square Root in Radius of Gyration: k = √(I/M)

The ratio of moments of inertia is I₁ / I₂ = (2/5) / (2/3) = 3/5. But the question asked for RADIUS OF GYRATION, which requires k = √(I/M) ⟹ k₁/k₂ = √(3/5). The exam setter committed this exact blunder!

🛡️ How to avoid the -1 mark: Never equate ratio of gyration radii (k₁/k₂) with ratio of moments of inertia (I₁/I₂). Remember k = √(I/M).

📖

Core NEET Syllabus Standard

National Eligibility cum Entrance Test (NEET UG) · System of Particles and Rotational Motion Standard Benchmark

Core Scientific Principle Tested:

"Radius of gyration k is defined by I = M k². For solid sphere k = √(2/5) R; for thin hollow sphere k = √(2/3) R."

Syllabus Focus: Radius of Gyration and Moment of Inertia

Preserved in Rationalized Curriculum Syllabus & Academic Standards ↗

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NEET 2023 Physics Question 29 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Instant Radius of Gyration: k = √(I / M) ⟹ k ∝ √I

Step-by-Step Rigorous Workflow & Mathematical Proof

Fundamental Definition of Radius of Gyration

Radius of Gyration for the Solid Sphere

Radius of Gyration for the Thin Hollow Sphere

Evaluation of the Theoretical Ratio & Identification of Paper Typo

Official NTA Final Verdict (Question Dropped)

360° Option Elimination Engine Key Feature

NTA Trap Alert: Forgetting the Square Root in Radius of Gyration: k = √(I/M)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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