Q.27 Physics ⭐ 360° Option Elimination Active

NEET 2026 Physics Question 27 Solution

Topic: System of Particles & Rotational Motion · Official NTA Booklet Code 50

Question

A solid sphere A of radius R and mass M is attached at a point to a smaller solid sphere B of radius r < R and mass m < M. Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of A is I_A and that calculated about a vertical axis passing through the centre of B is I_B. The difference I_A − I_B is:

✓

Verified Against NTA Final Answer Key

Option (2) : I_A - I_B = (m - M)(R + r)^2

📚 System of Particles & Rotational Motion Medium

📋 Given Parameters & Problem Setup

Sphere A Radius R, mass M

Sphere B Radius r < R, mass m < M (touching sphere A)

Center Distance d = R + r (horizontal separation between centers)

Axis A Vertical axis passing through center of sphere A (inertia I_A)

Axis B Vertical axis passing through center of sphere B (inertia I_B)

Target Quantity Difference I_A - I_B

⚡

The 30-Second Exam Hall Shortcut: ⚡ Self-Inertia Cancellation Shortcut

⏱️ Avoids computing (2/5)*M*R^2 and (2/5)*m*r^2.

Self-moments of inertia cancel out completely when subtracting!

1 About axis A: Sphere A has self-inertia I_A,self. Sphere B is at distance d = R + r: I_A = I_A,self + I_B,self + m*(R + r)^2.

2 About axis B: Sphere B has self-inertia I_B,self. Sphere A is at distance d = R + r: I_B = I_A,self + I_B,self + M*(R + r)^2.

3 Subtract I_A - I_B: Both self-moments of inertia cancel out!

4 I_A - I_B = m*(R + r)^2 - M*(R + r)^2 = (m - M)*(R + r)^2. Solved in 15 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Parallel Axis Theorem: I = I_cm + M_body * d^2. Since distance between centers is identically d = R + r in both cases, the difference depends ONLY on which mass is placed at distance d from the axis.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Calculate Moment of Inertia About Axis A Passing Through Center of Sphere A

Apply the parallel axis theorem to find total moment of inertia I_A:

I_A = I_{A,\text{cm}} + \left[ I_{B,\text{cm}} + m (R + r)^2 \right]

• Sphere A rotates about its diametrical centroidal axis: I_{A,cm} = (2/5) M R^2.
• Sphere B's center is at distance d = R + r from axis A: I_{B,about A} = (2/5) m r^2 + m (R + r)^2.

Step 2

Calculate Moment of Inertia About Axis B Passing Through Center of Sphere B

Similarly, apply the parallel axis theorem about vertical axis B:

I_B = I_{B,\text{cm}} + \left[ I_{A,\text{cm}} + M (R + r)^2 \right]

• Sphere B rotates about its diametrical centroidal axis: I_{B,cm} = (2/5) m r^2.
• Sphere A's center is at distance d = R + r from axis B: I_{A,about B} = (2/5) M R^2 + M (R + r)^2.

Step 3

Compute the Algebraic Difference I_A - I_B

Subtract equation 2 from equation 1:

I_A - I_B = \left[ I_{A,\text{cm}} + I_{B,\text{cm}} + m(R + r)^2 \right] - \left[ I_{A,\text{cm}} + I_{B,\text{cm}} + M(R + r)^2 \right]

• The centroidal terms I_{A,cm} and I_{B,cm} cancel out identically.
• I_A - I_B = m(R + r)^2 - M(R + r)^2 = (m - M)(R + r)^2.
• Matches Option (2) exactly.

⭐

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 Option (1)

✗ Incorrect

Verdict & Scientific Reason:

Sign reversal (this is I_B - I_A).

⚠️

Student Fallacy (Why candidates pick this trap): Computes I_B - I_A instead of I_A - I_B.

Note: Since m < M, I_A must be smaller than I_B, so I_A - I_B must be negative: (m - M) < 0.

2 Option (2)

✓ 100% Correct

Verdict & Scientific Reason:

100% mathematically exact; confirmed by Official NTA Final Answer Key.

💡

Why this is airtight: Correctly applies parallel axis theorem and preserves sign order.

Note: Exact rotational physics.

3 Option (3)

✗ Incorrect

Verdict & Scientific Reason:

Uses difference of radii (R - r).

⚠️

Student Fallacy (Why candidates pick this trap): Assumes the distance between centers is R - r instead of R + r.

Note: Touching spheres have center separation d = R + r.

4 Option (4)

✗ Incorrect

Verdict & Scientific Reason:

Ignores mass inequality.

⚠️

Student Fallacy (Why candidates pick this trap): Assumes symmetry makes both moments of inertia identical.

Note: Masses M and m are unequal, so placing mass M further from axis B makes I_B > I_A.

⚠️

NTA Trap Alert: The Sign Reversal Trap: (M - m) vs (m - M)

Because M > m, students naturally like to write (M - m) to keep things positive, incorrectly selecting Option (1).

🛡️ How to avoid the -1 mark: Physical sanity check: Sphere A has larger mass M. When rotating about axis B, that larger mass M is shifted to the outside (distance d), so I_B MUST be larger than I_A! Hence I_A - I_B must be negative: (m - M)(R + r)^2.

📖

Core NEET Syllabus Standard

NEET UG Physics Core Standards · System of Particles and Rotational Motion Standard Benchmark

Core Scientific Principle Tested:

"Parallel axis theorem: I = I_cm + M*d^2. For composite bodies, total moment of inertia is the scalar sum of all components about the shared axis."

Syllabus Focus: Moment of Inertia and Parallel Axis Theorem

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2026 Physics Question 27 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Self-Inertia Cancellation Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Calculate Moment of Inertia About Axis A Passing Through Center of Sphere A

Calculate Moment of Inertia About Axis B Passing Through Center of Sphere B

Compute the Algebraic Difference I_A - I_B

360° Option Elimination Engine Key Feature

NTA Trap Alert: The Sign Reversal Trap: (M - m) vs (m - M)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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