IIT / JEE ADVANCED – TOUGH PROBLEMS

Oscillations & Simple Harmonic Motion (SET–2)

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Problem 1: Piecewise Spring Constant (Classic JEE Trap)

A particle of mass m is attached to a spring. For extension, spring constant = k. For compression, spring constant = 4k. The particle oscillates with very small amplitude. Find the time period.

Solution:

For small oscillations, motion depends on restoring force near mean position. Effective spring constant:

k_eff = (k + 4k)/2 = 5k/2

Time period:

T = 2π √(m / k_eff) = 2π √(2m / 5k)

Answer: T = 2π √(2m / 5k)

Examiner Thinking: Students wrongly use either k or 4k instead of averaging.

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Problem 2: SHM + Energy Conservation

A particle in SHM has maximum speed v_max. Find the speed when displacement is A/√3.

Solution:

v = ω√(A² − x²)

x = A/√3 ⇒ x² = A²/3

v = ωA √(1 − 1/3) = ωA √(2/3)

Since v_max = ωA,

v = v_max √(2/3)

Answer: v = v_max √(2/3)

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Problem 3: Time from Energy Condition

In SHM, find the time taken for kinetic energy to change from E/4 to E/2.

Solution:

KE = E(1 − x²/A²)

For KE = E/4:

1 − x₁²/A² = 1/4 ⇒ x₁²/A² = 3/4 ⇒ x₁ = (√3/2)A

For KE = E/2:

1 − x₂²/A² = 1/2 ⇒ x₂²/A² = 1/2 ⇒ x₂ = A/√2

Use phase method:

x = A cosθ

cosθ₁ = √3/2 ⇒ θ₁ = π/6 cosθ₂ = 1/√2 ⇒ θ₂ = π/4

Δθ = π/4 − π/6 = π/12

Δt = (T / 2π)(π / 12) = T / 24

Answer: T / 24

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Problem 4: Moving Support (Advanced)

A mass attached to a spring oscillates vertically. If the support of the spring moves upward with constant velocity, the nature of oscillation is:

Answer: Still SHM

Reason:

Constant velocity does not introduce acceleration. Restoring force remains proportional to displacement.

Hidden Concept: Acceleration of support matters, not velocity.

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Problem 5: SHM Validity Test

Which force can produce SHM for small oscillations?

• A) F = −kx
• B) F = −kx³
• C) F = −k sin x
• D) F = −k/x

Correct Answer: C

Because for small x, sin x ≈ x ⇒ restoring force ∝ x

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Problem 6: Acceleration Comparison

Compare acceleration at displacement A/2 and A/√2.

Solution:

a = ω²x

a₁ / a₂ = (A/2) / (A/√2) = √2 / 2

Answer: a(A/2) < a(A/√2)

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SET–2 FINAL TAKEAWAYS

• Small oscillation ≠ full motion
• Average stiffness near mean position
• Energy method > equations
• Phase saves time in JEE Advanced
• Velocity of support does not break SHM

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IIT / JEE Tough Problems – SET–2 Completed ✅

📚 IIT–JEE Physics Complete Library

Concept Mastery • PYQs • Strategy • Revision
One-stop structured learning hub

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🔹 Simple Harmonic Motion (SHM) — Core Series

Coverage: Concepts → PYQs → Advanced Thinking → Exam Readiness

• SHM – Final Exam Day Checklist
• SHM Part 2
• SHM Part 3
• SHM Part 4
• SHM Part 5
• SHM Part 6
• SHM Part 7
• SHM Part 8
• SHM Part 9
• SHM Part 10
• SHM Part 11
• SHM Part 12
• SHM Part 13
• SHM Part 14
• SHM Part 15
• SHM Part 16
• SHM Part 17
• SHM Part 18
• SHM Part 19
• SHM Part 20
• SHM Part 21
• SHM Part 22
• SHM Part 23
• SHM Part 24
• SHM Part 25
• SHM Part 26
• SHM Part 27
• SHM Part 28
• SHM Part 29

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🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)

These pages are placed separately for continuity & reference.

30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56

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🔹 Supporting & Mega Libraries

• 📘 IIT–JEE Physics Mega Library
• 🧠 How IIT Toppers Think
• 🌐 Learning Resource Hub
• 📢 WhatsApp Channel

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