Oscillations & Simple Harmonic Motion
Phase 1 – Page 3 | Advanced Ratio & Mixed Concept Problems
Problem 18: Ratio of Maximum Quantities
For a particle executing SHM with amplitude A and angular frequency ω, find the ratio of maximum velocity to maximum acceleration.
Solution:
Maximum velocity:
vmax = ωA
Maximum acceleration:
amax = ω²A
Ratio:
vmax : amax = 1 : ω
Problem 19: Ratio of Kinetic Energies
Find the ratio of kinetic energies of a particle in SHM at displacements x = A/3 and x = A/2.
Solution:
KE ∝ (A² − x²)
At x = A/3:
KE₁ ∝ A² − A²/9 = 8A²/9
At x = A/2:
KE₂ ∝ A² − A²/4 = 3A²/4
Ratio:
KE₁ : KE₂ = 32 : 27
Problem 20: Ratio of Times (IIT Favourite)
In SHM, find the ratio of time taken by the particle to move from x = 0 to x = A/2 and from x = A/2 to x = A.
Solution:
Displacement equation:
x = A sin(ωt)
For x = A/2:
sin(ωt₁) = 1/2 ⇒ ωt₁ = π/6
For x = A:
ωt₂ = π/2
Required ratio:
t₁ : (t₂ − t₁) = (π/6) : (π/3) = 1 : 2
Problem 21: Mean Square Velocity
Find the mean square velocity of a particle executing SHM.
Solution:
Mean square velocity:
⟨v²⟩ = ½ ω²A²
(This is an IIT Advanced result.)
Problem 22: Phase-Based Trap
At what phase does the particle have equal velocity and acceleration (numerically)?
Solution:
v = ωA cosθ
a = ω²A sinθ
Equating magnitudes:
ωA cosθ = ω²A sinθ
⇒ tanθ = 1/ω
Answer: θ = tan⁻¹(1/ω)
Problem 23: Energy vs Time Concept
How many times does kinetic energy become zero in one complete SHM?
Solution:
- KE = 0 at extreme positions
- Two extremes in one oscillation
Answer: 2 times
Problem 24: Misleading Velocity Question
A particle passes mean position with velocity v. What is its velocity at the same point in the opposite direction?
Solution:
Magnitude remains same, direction reverses.
Answer: Velocity = −v
Problem 25: Identifying Correct Graph
Which graph correctly represents velocity vs displacement in SHM?
- A) Straight line
- B) Circle
- C) Ellipse
- D) Parabola
Answer: C
Because:
v² + ω²x² = ω²A²
IIT-Level Thinking Summary (Phase 1 – Page 3)
- Ratios save time
- Use trigonometric form wisely
- Energy-based shortcuts are powerful
- Graph questions test understanding, not memory
Phase 1 – Page 3 Completed Successfully ✅
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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
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🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)
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