Q.45 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 45 Solution

Topic: Waves · Official NTA Booklet Code 45

Question

A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:

✓

Verified Against NTA Final Answer Key

Option (2) : f

📚 Waves Easy

📋 Given Parameters & Problem Setup

Initial Pipe Geometry Open organ pipe of length L, open at both ends

Initial Fundamental Frequency f in air

Dipping Condition Dipped vertically in water drum to half its length (L/2)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Open to Half-Closed Pipe Invariance Rule

⏱️ Save ~30 seconds

Open pipe of length L: f = v / (2L). Closed pipe of length L/2: f' = v / [4(L/2)] = v / (2L) = f! Frequency remains completely unchanged: f' = f!

1 Initial state: Pipe open at both ends with length L. Fundamental frequency is f = v / (2L).

2 When dipped half into water: Water surface acts as a rigid boundary (closed end).

3 The remaining air column has length L' = L / 2, open at top and closed at bottom (closed pipe).

4 Fundamental frequency of a closed pipe of length L' is: f' = v / (4 L').

5 Substitute L' = L / 2: f' = v / [4 (L / 2)] = v / (2L) = f.

6 The frequency is identically f!

7 Select Option (2) in under 10 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Acoustic Standing Waves in Organ Pipes (NCERT Physics · Waves: f = (2n-1)v/(4L)).

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Fundamental Frequency of an Open Pipe

For an organ pipe of length L open at both ends, displacement antinodes form at both open ends. The fundamental mode corresponds to a half wavelength in the pipe:

L = \frac{\lambda}{2} \implies \lambda = 2L \implies f = \frac{v}{\lambda} = \frac{v}{2L}

• v is the speed of sound in air.

Step 2

Determine Geometry After Dipping in Water

When the pipe is immersed in water to half its length (L/2), the water surface provides a rigid closed boundary at the bottom, while the top remains open. The vibrating air column has length:

L' = L - \frac{L}{2} = \frac{L}{2}

• The system is now a closed organ pipe of effective length L / 2.

Step 3

Calculate Fundamental Frequency of the Resulting Closed Pipe

In a pipe closed at one end and open at the other, a node forms at the water surface and an antinode at the open end:

L' = \frac{\lambda'}{4} \implies \lambda' = 4 L' = 4 \left(\frac{L}{2}\right) = 2L

• The new fundamental frequency is:
• f' = \frac{v}{\lambda'} = \frac{v}{2L} = f.
• Thus the fundamental frequency remains unchanged at f, matching Option (2).

⭐

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. Halving the length does not halve the frequency because boundary conditions changed from open to closed.

⚠️

Student Fallacy (Why candidates pick this trap): Thinking halving effective length halves frequency.

2 B

✓ 100% Correct

Verdict & Scientific Reason:

100% correct. f' = v / (4(L/2)) = v / (2L) = f.

💡

Why this is airtight: None; classic organ pipe invariance.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Corresponds to third harmonic or arithmetic error.

⚠️

Student Fallacy (Why candidates pick this trap): Confusing fundamental mode with first overtone.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Would be the frequency if the pipe remained open at both ends with length L/2.

⚠️

Student Fallacy (Why candidates pick this trap): Treating the immersed pipe as still open at both ends (v / (2(L/2)) = 2f).

⚠️

NTA Trap Alert: Treating Water as an Open End (2f Trap)

Students often reason: "Length is halved, so frequency doubles to 2f" (Option 4). They forget that water creates a CLOSED END, which introduces a factor of 4 instead of 2 in the denominator, exactly cancelling the halving of length!

🛡️ How to avoid the -1 mark: Remember: Water surface = CLOSED boundary (Node)! Closed pipe formula is v / 4L'.

📖

Core NEET Syllabus Standard

NEET Physics Core Syllabus · Waves Standard Benchmark

Core Scientific Principle Tested:

"Open pipe: f = v / 2L; Closed pipe: f = v / 4L'. At L' = L/2, both frequencies are identical."

Syllabus Focus: Standing Waves in Organ Pipes (Open and Closed)

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Half length (×2) + Closed end (/2) = Exact Cancel! Frequency stays f!

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NEET 2025 Physics Question 45 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Open to Half-Closed Pipe Invariance Rule

Step-by-Step Rigorous Workflow & Mathematical Proof

Fundamental Frequency of an Open Pipe

Determine Geometry After Dipping in Water

Calculate Fundamental Frequency of the Resulting Closed Pipe

360° Option Elimination Engine Key Feature

NTA Trap Alert: Treating Water as an Open End (2f Trap)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Waves

Harmonic Modes & Node Counts in Open vs Closed Organ Pipes

Fundamental Harmonics of Open vs Closed Organ Pipes

Beat Frequency and String Tension: f_beat = |f_A - f_B|

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