Q.24 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 24 Solution

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

Question

A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled in it) has small area A. It is filled with a gas of density ρ and takes a spherical shape of radius R. When the gas is allowed to flow freely out of it, its radius r changes from R to 0 (zero) in time T. If the speed v(r) of gas coming out of the balloon depends on r as rᵃ and T ∝ S^α A^β ρ^γ R^δ then

✓

Verified Against NTA Final Answer Key

Option (3) : a = -1/2, \alpha = -1/2, \beta = -1, \gamma = 1/2, \delta = 7/2

📚 Mechanical Properties of Fluids Challenging

📋 Given Parameters & Problem Setup

Surface Tension S

Outlet Hole Area A

Gas Density ρ

Initial Radius R (collapsing to r = 0 in time T)

Given Scaling Relations v(r) ∝ r^a and T ∝ S^α A^β ρ^γ R^δ

⚡

The 30-Second Exam Hall Shortcut: ⚡ Bernoulli & Volume Integration Power Law

⏱️ Save ~90 seconds

ΔP ∝ S/r ⟹ v = √(2ΔP/ρ) ∝ r^(-1/2) ⟹ a = -1/2! Volume rate dV/dt = 4πr²(dr/dt) = -Av ⟹ r^(5/2) dr ∝ dt ⟹ T ∝ R^(7/2) ⟹ δ = 7/2!

1 Excess pressure in spherical balloon: ΔP ∝ S / r.

2 Torricelli/Bernoulli efflux velocity: v = √(2ΔP / ρ) ∝ √(S / ρr) = (S/ρ)^(1/2) r^(-1/2).

3 Comparing with v(r) ∝ r^a: a = -1/2. Eliminates Options (1) & (4)!

4 Volume rate of deflation: -dV/dt = A v(r) ⟹ -4πr² (dr/dt) = A (S/ρ)^(1/2) r^(-1/2).

5 Separate variables: -r² · r^(1/2) dr = -r^(5/2) dr ∝ A (S/ρ)^(1/2) dt.

6 Integrate from R to 0: ∫ r^(5/2) dr = [ r^(7/2) / (7/2) ] ∝ R^(7/2).

7 Thus time T ∝ R^(7/2) A^(-1) S^(-1/2) ρ^(1/2) ⟹ δ = 7/2, β = -1, α = -1/2, γ = 1/2.

8 Matches Option (3) uniquely!

🛡️

Why You Can 100% Trust This Shortcut: Fluid Dynamics & Surface Tension Scaling (NCERT Physics · Mechanical Properties of Fluids).

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Determine Efflux Velocity Exponent a

The excess pressure inside a spherical elastic balloon of instantaneous radius r with effective surface tension S is given by Laplace’s law:

\Delta P = \frac{4S}{r} \propto \frac{S}{r}

• Applying Bernoulli’s theorem between the high-pressure gas inside the balloon and the atmospheric exhaust outside through the small opening of area A:
• ΔP = (1/2) ρ v² ⟹ v(r) = √(2ΔP / ρ) ∝ √(S / (ρ r)) = (S / ρ)^(1/2) r^(-1/2).
• Comparing with v(r) ∝ r^a directly gives: a = -1/2.

Step 2

Formulate the Differential Equation for Deflation

By conservation of mass / volume continuity, the rate of decrease of the balloon volume equals the volume flux exiting through the aperture:

-\frac{dV}{dt} = A \cdot v(r) \implies -\frac{d}{dt}\left(\frac{4}{3}\pi r^3\right) = -4\pi r^2 \frac{dr}{dt} = A \sqrt{\frac{2(4S)}{\rho}} r^{-1/2}

• Rearranging to separate variables r and t:
• -r^2 \cdot r^{1/2} dr = -r^{5/2} dr = \left(\frac{A}{4\pi}\sqrt{\frac{8S}{\rho}}\right) dt.

Step 3

Integrate to Find Exponents of Time T

Integrate from the initial radius r = R at t = 0 to r = 0 at t = T:

\int_0^R r^{5/2} dr = \left[ \frac{r^{7/2}}{7/2} \right]_0^R = \frac{2}{7} R^{7/2} \propto A S^{1/2} \rho^{-1/2} T

• Solving for total deflation time T:
• T ∝ S^(-1/2) A^(-1) ρ^(1/2) R^(7/2).
• Comparing with T ∝ S^α A^β ρ^γ R^δ:
• α = -1/2, β = -1, γ = 1/2, δ = 7/2.
• Along with a = -1/2, this corresponds uniquely to Option (3).

⭐

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. Assumed v ∝ √r instead of 1/√r, and wrong radius exponent δ.

⚠️

Student Fallacy (Why candidates pick this trap): Inverting pressure dependence on radius (thinking ΔP ∝ r).

2 B

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Missed the r² factor in dV = 4πr² dr, resulting in δ = 5/2 instead of 7/2.

⚠️

Student Fallacy (Why candidates pick this trap): Failing to include the 4πr² spherical volume Jacobian.

3 C

✓ 100% Correct

Verdict & Scientific Reason:

100% exact. a = -1/2 from Bernoulli efflux and δ = 7/2 from integrating r^(5/2) dr.

💡

Why this is airtight: None; masterclass fluid dynamics derivation.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Exponent a = +1/2 is wrong; area exponent β must be -1, not -1/2.

⚠️

Student Fallacy (Why candidates pick this trap): Confusing orifice area with radius.

⚠️

NTA Trap Alert: Efflux Area vs. Sphere Volume Integration

Students often get confused by the exponent of R. Since volume is r³, dV/dt has r². Multiplying r² by v (which is r^(-1/2)) gives r^(5/2). Integrating r^(5/2) yields r^(7/2), so δ MUST be 7/2!

🛡️ How to avoid the -1 mark: Always trace powers: r² × r^(-1/2) = r^(5/2) ⟹ Integral gives r^(7/2) ⟹ δ = 7/2.

📖

Core NEET Syllabus Standard

NEET Physics Core Syllabus · Mechanical Properties of Fluids Standard Benchmark

Core Scientific Principle Tested:

"Laplace pressure ΔP ∝ 1/r generates efflux velocity v ∝ r^(-1/2); volume integration scales as R^(7/2)."

Syllabus Focus: Torricelli’s Law and Surface Tension Pressure

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Laplace v ∝ r^(-1/2) ⟹ r²·v = r^(5/2) ⟹ Integrates to R^(7/2): Exponent is 7/2!

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NEET 2025 Physics Question 24 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Bernoulli & Volume Integration Power Law

Step-by-Step Rigorous Workflow & Mathematical Proof

Determine Efflux Velocity Exponent a

Formulate the Differential Equation for Deflation

Integrate to Find Exponents of Time T

360° Option Elimination Engine Key Feature

NTA Trap Alert: Efflux Area vs. Sphere Volume Integration

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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