Q.25 Physics ⭐ 360° Option Elimination Active

NEET 2026 Physics Question 25 Solution

Topic: Oscillations & SHM · Official NTA Booklet Code 50

Question

A cylindrical cork of uniform density floats in a liquid of density ρ₁. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density ρ₂, then the similar oscillation has time period 2T. The value of ρ₂/ρ₁ is:

✓

Verified Against NTA Final Answer Key

Option (4) : rho_2 / rho_1 = 1/4

📚 Oscillations & SHM Medium

📋 Given Parameters & Problem Setup

Floating Body Cylindrical cork of uniform mass m and cross-section A

Liquid 1 Density rho_1, oscillation period T

Liquid 2 Density rho_2, oscillation period 2T

Target Ratio rho_2 / rho_1

⚡

The 30-Second Exam Hall Shortcut: ⚡ Inverse Square Period Scaling

⏱️ Bypasses full differential equation setup.

Time period of floating oscillations is inversely proportional to sqrt(density)

1 Restoring force for vertical displacement y: F = -(A * rho * g) * y ==> Spring constant k = A * rho * g.

2 Time period of SHM: T = 2pi * sqrt(m / k) = 2pi * sqrt(m / (A * rho * g)).

3 Therefore, T is inversely proportional to sqrt(rho) ==> rho is proportional to 1 / T^2.

4 Form the ratio: rho_2 / rho_1 = (T_1 / T_2)^2 = (T / 2T)^2 = (1/2)^2 = 1/4.

5 Solved in 15 seconds!

🛡️

Why You Can 100% Trust This Shortcut: A denser liquid exerts a stiffer restoring buoyant force per unit depression (larger k = A*rho*g), causing faster oscillations (smaller period). Doubling the period requires reducing the restoring constant 4-fold, so density must be 1/4.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Formulate Equation of Motion for Floating Cylinder SHM

Consider a cylinder of mass m and cross-sectional area A floating in equilibrium in a liquid of density rho. When depressed downward by distance y from equilibrium, additional buoyant force provides a restoring force:

F_{\text{restoring}} = -\Delta F_B = - (A \cdot y) \rho g = -(A \rho g) y

• Because restoring force is directly proportional to displacement y and directed toward equilibrium, the motion is Simple Harmonic Motion (SHM).
• Effective restoring force constant: k = A * rho * g.

Step 2

Derive Time Period in Terms of Liquid Density

Apply the standard harmonic oscillator time period formula:

T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{A \rho g}} \implies T \propto \frac{1}{\sqrt{\rho}}

• Mass m and area A are constant since the same cork is transferred between liquids.
• Squaring gives: T^2 \propto \frac{1}{\rho} \implies \rho \propto \frac{1}{T^2}.

Step 3

Calculate the Density Ratio rho_2 / rho_1

Substitute T_1 = T and T_2 = 2T into the inverse square relationship:

\frac{\rho_2}{\rho_1} = \left(\frac{T_1}{T_2}\right)^2 = \left(\frac{T}{2T}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}

• Confirms rho_2 / rho_1 = 1/4.
• Matches Option (4) 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:

Inverted density scaling.

⚠️

Student Fallacy (Why candidates pick this trap): Assumes T is proportional to sqrt(rho), leading to rho_2 / rho_1 = (2T/T)^2 = 4.

Note: Denser liquids oscillate FASTER, not slower.

2 Option (2)

✗ Incorrect

Verdict & Scientific Reason:

Linear direct proportionality error.

⚠️

Student Fallacy (Why candidates pick this trap): Assumes T ~ rho directly.

Note: Time period involves square root of inertia over restoring force.

3 Option (3)

✗ Incorrect

Verdict & Scientific Reason:

Forgot to square the period ratio.

⚠️

Student Fallacy (Why candidates pick this trap): Takes rho_2 / rho_1 = T_1 / T_2 = 1/2 without squaring.

Note: T scales with 1/sqrt(rho), meaning rho scales with 1/T^2.

4 Option (4)

✓ 100% Correct

Verdict & Scientific Reason:

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

💡

Why this is airtight: Flawless evaluation of inverse square period relationship.

Note: Exact physics.

⚠️

NTA Trap Alert: The Direct Proportionality Trap (Thinking Denser = Slower)

Students often intuitively assume that a thicker/denser liquid "slows down" the oscillation, picking Option (1) or (2).

🛡️ How to avoid the -1 mark: Remember: Liquid density creates the RESTORING spring force! A denser liquid pushes back harder (larger k), making the system oscillate FASTER (smaller T).

📖

Core NEET Syllabus Standard

NEET UG Physics Core Standards · Oscillations Standard Benchmark

Core Scientific Principle Tested:

"Time period of vertical oscillations of a floating cylinder: T = 2pi * sqrt(m / (A*rho*g)) = 2pi * sqrt(h_submerged / g)."

Syllabus Focus: Simple Harmonic Motion of Floating Bodies

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2026 Physics Question 25 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Inverse Square Period Scaling

Step-by-Step Rigorous Workflow & Mathematical Proof

Formulate Equation of Motion for Floating Cylinder SHM

Derive Time Period in Terms of Liquid Density

Calculate the Density Ratio rho_2 / rho_1

360° Option Elimination Engine Key Feature

NTA Trap Alert: The Direct Proportionality Trap (Thinking Denser = Slower)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Oscillations

Spring-Mass System with Slowly Leaking Mass (Variable Mass SHM)

Ratio of Amplitudes for Identical Masses with Equal Maximum Speeds

Amplitude and Time Period of an SHM Equation

Time Period Variation with Mass and Length of a Simple Pendulum

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