Q.7 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 7 Solution

Topic: Work, Energy and Power · Official NTA Booklet Code 45

Question

A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v₀ as shown in figure. If the string gets slack at some point P making an angle θ from the horizontal, the ratio of the speed v of the bob at point P to its initial speed v₀ is:

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Verified Against NTA Final Answer Key

Option (4) : \left(\frac{\sin\theta}{2 + 3\sin\theta}\right)^{1/2}

📚 Work, Energy and Power Medium

📋 Given Parameters & Problem Setup

String Length l

Initial Velocity at Bottom v₀

Point P Orientation String makes angle θ above the horizontal

Critical Condition String gets slack (Tension T = 0 at point P)

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The 30-Second Exam Hall Shortcut: ⚡ Slack Ratio Formula

⏱️ Save ~60 seconds

At slack: v² = gl sin θ, and energy gives v₀² = v² [ (2 + 3 sin θ) / sin θ ] ⟹ v / v₀ = [ sin θ / (2 + 3 sin θ) ]^(1/2)!

1 Radial force balance at angle θ above horizontal: T + mg sin θ = m v² / l.

2 At slack condition (T = 0): v² = g l sin θ ⟹ g l = v² / sin θ.

3 Height from bottom: h = l + l sin θ = l(1 + sin θ).

4 Conservation of Energy: v₀² = v² + 2gh = v² + 2(v² / sin θ)(1 + sin θ) = v² [ (2 + 3 sin θ) / sin θ ].

5 Ratio: v / v₀ = [ sin θ / (2 + 3 sin θ) ]^(1/2).

6 Directly matches Option (4)!

🛡️

Why You Can 100% Trust This Shortcut: Fundamental Mechanics · Work, Energy and Power (Vertical Circular Motion Energy Conservation: v_min = √(5gR)).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Radial Force Balance at Point P

Point P is in the upper quadrant, where the string makes an angle θ above the horizontal. The component of gravity acting radially inward towards the suspension point O is mg sin θ.

T + mg \sin\theta = \frac{m v^2}{l}

• When the string becomes slack, tension vanishes: T = 0.
• Therefore, mg sin θ = m v² / l ⟹ v² = g l sin θ.

Step 2

Conservation of Mechanical Energy

Taking the bottom-most point as reference potential energy (PE = 0), the height of point P is h = l + l sin θ. Conservation of mechanical energy between the bottom and point P gives:

\frac{1}{2} m v_0^2 = \frac{1}{2} m v^2 + m g l (1 + \sin\theta) \implies v_0^2 = v^2 + 2 g l (1 + \sin\theta)

• Express gl in terms of v²: gl = v² / sin θ.

Step 3

Calculate the Speed Ratio

Substitute gl into the energy equation:

v_0^2 = v^2 + 2 \left(\frac{v^2}{\sin\theta}\right) (1 + \sin\theta) = v^2 \left[ 1 + \frac{2 + 2\sin\theta}{\sin\theta} \right] = v^2 \left[ \frac{2 + 3\sin\theta}{\sin\theta} \right]

• Inverting to find the ratio v / v₀:
• v / v₀ = ( sin θ / (2 + 3 sin θ) )^(1/2), matching Option (4).

⭐

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. Omits the initial velocity dependence and energy conservation denominator.

⚠️

Student Fallacy (Why candidates pick this trap): Ignoring work done by gravity.

2 B

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Missing the sin θ term in the numerator.

⚠️

Student Fallacy (Why candidates pick this trap): Setting v² = gl instead of gl sin θ.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Angle θ is defined with horizontal, so radial component of gravity is mg sin θ, not mg cos θ.

⚠️

Student Fallacy (Why candidates pick this trap): Confusing angle with vertical vs angle with horizontal.

4 D

✓ 100% Correct

Verdict & Scientific Reason:

100% exact. Rigorously derived from radial dynamic equilibrium at T = 0 and mechanical energy conservation.

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Why this is airtight: None; flawless mechanics derivation.

⚠️

NTA Trap Alert: Angle Reference Axis (Horizontal vs. Vertical)

The figure explicitly defines θ with the HORIZONTAL. If θ were defined with the vertical, the radial component would be cos θ. Because it is with the horizontal, radial gravity component is sin θ.

🛡️ How to avoid the -1 mark: Check the diagram: angle θ is between string and horizontal dotted line ⟹ radial gravity is mg sin θ.

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Work, Energy and Power Standard Benchmark

Core Scientific Principle Tested:

"Slack occurs when centripetal acceleration is provided entirely by the radial component of gravity (T = 0)."

Syllabus Focus: Motion in a Vertical Circle

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Slack at θ: Gravity takes over (v² = gl sin θ) ⟹ Ratio = √(sin θ / (2 + 3 sin θ))!

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NEET 2025 Physics Question 7 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Slack Ratio Formula

Step-by-Step Rigorous Workflow & Mathematical Proof

Radial Force Balance at Point P

Conservation of Mechanical Energy

Calculate the Speed Ratio

360° Option Elimination Engine Key Feature

NTA Trap Alert: Angle Reference Axis (Horizontal vs. Vertical)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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