Q.1 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 1 Solution

Topic: Mechanical Properties of Fluids · Official NTA Booklet Code 45

Question

Consider a water tank shown in the figure. It has one wall at x = L and can be taken to be very wide in the z direction. When filled with a liquid of surface tension S and density ρ, the liquid surface makes angle θ₀ (θ₀ << 1) with the x-axis at x = L. If y(x) is the height of the surface then the equation for y(x) is: (take θ(x) = sin θ(x) = tan θ(x) = dy/dx, g is the acceleration due to gravity)

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Verified Against NTA Final Answer Key

Option (2) : \frac{d^2y}{dx^2} = \frac{\rho g}{S} y

📚 Mechanical Properties of Fluids Challenging

📋 Given Parameters & Problem Setup

Fluid Properties Surface tension S, Density ρ

Boundary Condition Wall at x = L, liquid makes small angle θ₀ ≪ 1 with x-axis

Small Angle Approximation θ(x) = sin θ(x) = tan θ(x) = dy/dx

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The 30-Second Exam Hall Shortcut: ⚡ Dimensional & Physical Balance Shortcut

⏱️ Save ~90 seconds

Laplace Pressure S(d²y/dx²) = Hydrostatic Column Pressure ρgy ⟹ d²y/dx² = (ρg/S) y!

1 Recognize the physical balance: Pressure jump across the curved liquid-air interface is Laplace excess pressure ΔP = S/R.

2 In differential calculus for small slopes, curvature 1/R = d²y/dx².

3 This pressure difference supports the hydrostatic liquid head of height y: ΔP = ρgy.

4 Equate: S(d²y/dx²) = ρgy ⟹ d²y/dx² = (ρg/S) y.

5 Dimensional check: [d²y/dx²] = L⁻¹; [(ρg/S) y] = (L⁻²) × (L) = L⁻¹. Option (2) is the only dimensionally and physically sound equation!

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Why You Can 100% Trust This Shortcut: Young-Laplace Equation & Fluid Statics (NCERT Physics · Mechanical Properties of Fluids: ΔP = 2T/R).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Young-Laplace Equation for 1D Cylindrical Meniscus

Because the tank is very wide in the z-direction, the liquid surface is cylindrical (curvature in z-direction is zero). The Laplace excess pressure across a surface with radius of curvature R is governed by surface tension S.

\Delta P = \frac{S}{R}

• For a surface profile y(x) with small slope (dy/dx ≪ 1), the curvature simplifies to 1/R ≈ d²y/dx².

Step 2

Hydrostatic Pressure Balance

At any horizontal position x, the atmospheric pressure acts above the surface. At the flat baseline level (y = 0), the pressure is atmospheric. The hydrostatic pressure difference between the liquid at height y(x) and the flat surface is ρgy.

\Delta P = \rho g y(x)

• Equilibrium requires excess Laplace pressure to balance the gravitational head.

Step 3

Formulation of the Governing Differential Equation

Equating the Laplace pressure to the hydrostatic pressure head yields the second-order linear differential equation governing the meniscus profile.

S \frac{d^2y}{dx^2} = \rho g y \implies \frac{d^2y}{dx^2} = \frac{\rho g}{S} y \implies \text{Option (2)}

• This is the classical capillary rise meniscus differential equation with characteristic capillary length λ_c = √(S/ρg).

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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. Hydrostatic pressure depends on vertical elevation y, not horizontal coordinate x.

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Student Fallacy (Why candidates pick this trap): Confusing horizontal coordinate x with vertical height y in hydrostatic pressure.

2 B

✓ 100% Correct

Verdict & Scientific Reason:

100% scientifically and dimensionally exact. Directly balances Laplace curvature pressure with hydrostatic head ρgy.

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Why this is airtight: None; rigorous first-principles derivation.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. The RHS is a constant equal to the inverse capillary length (1/λ_c), missing the variable dependence on y.

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Student Fallacy (Why candidates pick this trap): Confusing the inverse capillary length scale with the differential equation itself.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Laplace excess pressure relates directly to surface curvature (second derivative d²y/dx²), not slope (dy/dx).

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Student Fallacy (Why candidates pick this trap): Failing to recognize that curvature is a second derivative.

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NTA Trap Alert: Coordinate Variable Confusion (x vs. y)

Students frequently rush and pick Option (1) because x is on the horizontal axis. But hydrostatic pressure is caused by GRAVITY acting vertically: P = ρgy. Thus the restoring force is proportional to y, not x.

🛡️ How to avoid the -1 mark: Always ask: "Which coordinate determines hydrostatic pressure?" Answer is vertical elevation y!

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Mechanical Properties of Fluids Standard Benchmark

Core Scientific Principle Tested:

"Laplace pressure across a curved interface ΔP = S/R balances hydrostatic liquid head ρgy."

Syllabus Focus: Surface Tension and Capillarity

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Laplace Curve = Hydrostatic Head: S·y'' = ρg·y!

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NEET 2025 Physics Question 1 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Dimensional & Physical Balance Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Young-Laplace Equation for 1D Cylindrical Meniscus

Hydrostatic Pressure Balance

Formulation of the Governing Differential Equation

360° Option Elimination Engine Key Feature

NTA Trap Alert: Coordinate Variable Confusion (x vs. y)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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