IIT / JEE ADVANCED – TOUGH PROBLEMS
Oscillations & Simple Harmonic Motion
Problem 1: Energy + Phase Trap (JEE Advanced)
A particle executes SHM with amplitude A and angular frequency ω. At a certain instant, the ratio of kinetic energy to potential energy is 3 : 1. Find the phase at that instant.
Solution:
Let KE : PE = 3 : 1 ⇒ KE = 3E/4, PE = E/4
But, PE / E = x² / A² ⇒ x² / A² = 1/4 ⇒ x = A/2
Using x = A cosθ ⇒ cosθ = 1/2 ⇒ θ = π/3
Answer: Phase = π/3 rad
IIT Insight: Energy ratio → square relation → phase, not time equation.
Problem 2: Variable Spring (Conceptual Killer)
A spring has force constant k for extension and 2k for compression. A mass attached to it oscillates with small amplitude. Find the time period.
Solution:
Time period depends on restoring force near mean. Effective spring constant:
keff = (k + 2k)/2 = 3k/2
T = 2π √(m / keff) = 2π √(2m / 3k)
Answer: T = 2π √(2m / 3k)
Examiner Trap: Students wrongly use k or 2k directly.
Problem 3: Time Calculation without Equations
Find the time taken by a particle executing SHM to move from displacement A/2 to A/√2.
Solution:
Use phase method:
x = A cosθ A/2 ⇒ cosθ₁ = 1/2 ⇒ θ₁ = π/3 A/√2 ⇒ cosθ₂ = 1/√2 ⇒ θ₂ = π/4
Δθ = π/3 − π/4 = π/12
Δt = (T / 2π) × (π / 12) = T / 24
Answer: T / 24
Topper Method: Never use x(t) equation for time problems → use phase.
Problem 4: SHM + Uniform Motion (Path Question)
A particle performs SHM along x-axis and uniform motion along y-axis. Find the nature of the path.
Solution:
x = A sinωt y = vt
This represents an ellipse.
Answer: Elliptical path
Problem 5: Damped SHM Concept
In damped oscillations, which of the following remains constant?
- Amplitude
- Total energy
- Time period (small damping)
- Maximum velocity
Correct Answer: Time period
JEE Trap: Energy and amplitude reduce, but time period stays almost same.
Problem 6: Direction of Acceleration (Concept Test)
At any point during SHM, acceleration is always directed:
Answer: Towards mean position
Because a = −ω²x
FINAL JEE TAKEAWAYS
- Energy ratios → square relations
- Phase method saves huge time
- Restoring force decides SHM
- Mean & extreme rules never fail
- Most mistakes are conceptual, not mathematical
IIT / JEE Tough Problems 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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