Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 7 | Damped, Forced Oscillations & Resonance

1. Why Real Oscillations Do Not Last Forever

In real life, oscillations gradually stop due to energy loss. This loss occurs because of:

Air resistance
Friction
Internal resistance

Such oscillations are called Damped Oscillations.

2. Damped Oscillations

A damped oscillation is one in which the amplitude decreases with time.

x = A e^-bt sin(ωt)

A → initial amplitude
b → damping constant
ω → angular frequency

Important:
Amplitude decreases exponentially, not linearly.

3. Types of Damping (Very Important)

① Light Damping

Amplitude decreases slowly
Oscillatory motion continues

② Critical Damping

System returns to equilibrium in minimum time
No oscillation

③ Heavy (Over) Damping

System returns slowly
No oscillation

Exam Favorite:
Car shock absorbers use critical damping.

4. Forced Oscillations

When an external periodic force is applied to a system, it undergoes forced oscillations.

Example:

Pushing a swing periodically
AC supply in electrical circuits

The frequency of oscillation equals the driving frequency, not natural frequency.

5. Resonance (Most Important Topic)

Resonance occurs when:

Driving frequency = Natural frequency

At resonance:

Amplitude becomes maximum
Energy transfer is maximum

Key Result:
Resonance produces maximum response.

6. Sharpness of Resonance

Sharpness depends on damping:

Low damping → sharp resonance
High damping → broad resonance

Quality factor (Q) measures sharpness:

Q = ω / Δω

7. Resonance Curve (Conceptual)

X-axis → frequency
Y-axis → amplitude
Peak occurs at resonance frequency

IIT Insight:
Graph-based resonance questions are common.

8. Practical Examples of Resonance

Radio tuning
Breaking of glass by sound
Bridge collapse due to marching soldiers

9. Common Mistakes Students Make

Confusing natural frequency with driving frequency
Ignoring damping effect
Assuming infinite amplitude

Reality Check:
Amplitude is always finite due to damping.

10. Why This Page Is Rank-Defining

Direct IIT/JEE conceptual questions
Application-based problems
Easy scoring if concepts are clear

Stage 1 – Page 7 Completed Successfully ✅

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