Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 2 | Mathematical Description of SHM

1. Displacement in SHM

In Simple Harmonic Motion, the displacement of a particle from the mean position changes continuously with time.

The general equation of displacement is:

x = A sin(ωt + φ)
or
x = A cos(ωt + φ)

A → Amplitude (maximum displacement)
ω → Angular frequency
t → Time
φ → Phase constant

Important:
Sine or cosine depends only on the choice of time origin.

2. Amplitude (A)

Amplitude is the maximum displacement of the particle from its mean position.

Always positive
Depends on initial conditions
Does NOT affect time period

Exam Trap:
Amplitude affects energy, not time period.

3. Phase, Phase Angle & Phase Difference

Phase: The quantity (ωt + φ) represents the phase of motion.

Phase Angle: It tells the state of motion (position + direction) at a given time.

Phase Difference: Difference in phase between two SHMs.

Δφ = ωΔt

Same phase → move together
π phase difference → opposite motion

4. Velocity in SHM

Velocity is the rate of change of displacement.

Expression:

v = dx/dt = Aω cos(ωt + φ)

Alternative form (more useful in numericals):

v = ω√(A² − x²)

Maximum velocity at mean position
Zero velocity at extreme positions

At mean position: v_max = Aω

5. Acceleration in SHM

Acceleration is proportional to displacement and always directed towards mean position.

a = −ω²x

Maximum acceleration at extremes
Zero acceleration at mean position

Heart of SHM:
Acceleration ∝ −Displacement

6. Relation between Displacement, Velocity & Acceleration

When x = 0 → v = max, a = 0
When x = max → v = 0, a = max
Velocity and acceleration are not in same direction

Exam Insight:
Never assume velocity and acceleration are perpendicular or parallel without checking.

7. Graphical Understanding (Conceptual)

x–t graph → sine curve
v–t graph → cosine curve
a–t graph → negative sine curve

Each graph is phase-shifted with respect to the other.

8. Why This Page is Very Important

Most IIT/JEE questions start from equations here
Graph questions depend on this understanding
Foundation for energy in SHM

Student Tip:
If equations confuse you, redraw motion on a number line and imagine direction.

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