Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 6 | Phase, Phase Difference & Time Relations

1. What is Phase in SHM?

Phase describes the state of motion of a particle at any instant. It tells us:

Where the particle is
Which direction it is moving
How fast it is moving

For SHM:

Phase = (ωt + φ)

ω → angular frequency
t → time
φ → initial phase

Key Idea:
Two particles can have same displacement but different phase.

2. Initial Phase (φ)

Initial phase decides the starting position and direction of motion at t = 0.

φ = 0 → starts from mean position
φ = π/2 → starts from extreme position
φ = π → starts from opposite extreme

Exam Trap:
Never assume φ = 0 unless stated.

3. Phase Difference (Δφ)

Phase difference is the difference between phases of two SHMs.

Δφ = ωΔt

Where Δt is time difference between the two motions.

Δφ = 0 → particles move together
Δφ = π → particles move oppositely
Δφ = π/2 → one leads the other by quarter cycle

4. Phase Relations Between x, v, and a

In SHM:

Velocity leads displacement by π/2
Acceleration leads velocity by π/2
Acceleration is π out of phase with displacement

Must Remember:
Acceleration always opposes displacement.

5. Time Taken to Move Between Positions

Time taken to move from one displacement to another is found using phase.

If particle moves from x₁ to x₂:

x = A sinθ

Time taken:

Δt = (Δθ / 2π) × T

θ → phase angle
T → time period

IIT Favourite:
Time from mean to extreme = T/4

6. Special Time Results (Must Memorize)

Mean → Extreme = T/4
Extreme → Mean = T/4
Mean → Mean (same direction) = T
Mean → Mean (opposite direction) = T/2
Extreme → Extreme (same) = T

Shortcut:
Think in quarter cycles, not equations.

7. Direction Matters in Time Calculation

Same positions can be reached at different times depending on direction.

Approaching mean → velocity maximum
Leaving mean → velocity maximum (opposite sign)

Exam Trap:
Always check direction of motion.

8. How IIT Uses Phase Concepts

Multi-particle SHM questions
Synchronization problems
Time comparison questions

9. Why Phase Decides Rank

Minimum calculation
Maximum conceptual clarity
Fast option elimination

Topper Rule:
Never rush phase questions—visualize motion first.

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