IIT / JEE ADVANCED – EXTREME TOUGH PROBLEMS
Oscillations & Simple Harmonic Motion (SET–4)
Problem 1: SHM in Non-Inertial Frame (Angular Acceleration)
A mass m attached to a spring of spring constant k is placed on a smooth table. The table rotates with constant angular acceleration α. Find the nature of motion for small oscillations.
Solution:
In rotating frame, pseudo forces include:
- Centrifugal force ∝ r
- Coriolis force ∝ velocity
Coriolis force does not affect restoring force. Centrifugal force shifts equilibrium position.
Restoring force about new mean is still proportional to displacement.
Answer: Motion remains SHM with same time period
Examiner Insight: Only forces proportional to displacement affect SHM frequency.
Problem 2: SHM with Slowly Varying Spring Constant
In a spring–mass system, the spring constant varies slowly with time as: k(t) = k₀(1 − βt), where βt ≪ 1. How does the time period change?
Solution:
T(t) = 2π √(m / k(t))
As k decreases with time, T increases gradually.
Answer: Time period increases slowly with time
JEE Trap: Students assume time period constant because change is slow.
Problem 3: Maximum Power Transfer in SHM
At what position during SHM is instantaneous power maximum?
Solution:
Power P = Fv = (kx)(ω√(A² − x²))
To maximize P, differentiate or use symmetry:
Maximum power occurs at:
x = A / √2
Answer: x = A / √2
Elite Insight: Power depends on both force and velocity → balance point.
Problem 4: SHM with Time-Dependent Force
A particle is subjected to force: F = −kx + F₀ sin(ωt). Describe the long-term motion.
Solution:
The system shows:
- Natural SHM
- Forced oscillation
After transients die out, steady-state motion dominates.
Answer: Forced oscillation with driving frequency
Problem 5: Limiting Case Analysis (Advanced)
For SHM, consider ω → 0. What does motion approach?
Solution:
ω → 0 ⇒ restoring force → 0
System approaches uniform motion.
Answer: Uniform motion
IIT Trick: Always test limiting cases to verify reasoning.
Problem 6: Energy Loss per Cycle (Damping)
In lightly damped SHM, energy lost per cycle is proportional to:
- A) Amplitude
- B) Velocity
- C) Square of amplitude
- D) Time period
Correct Answer: C
Energy ∝ A², hence loss ∝ A².
SET–4 FINAL TAKEAWAYS
- Pseudo forces often shift mean, not frequency
- Slow variation ≠ no variation
- Power maximization uses balance logic
- Limiting cases reveal physics instantly
- Damping affects energy faster than time period
IIT / JEE Tough Problems – SET–4 Completed ✅
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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
🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)
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