Resonance in Physics – Definition, Conditions & Applications | Entrance Exam Notes

RESONANCE — SHORT NOTES (Entrance Exam Point of View)

1. Definition

Resonance occurs when the frequency of an external periodic force matches the natural frequency of a system.

At resonance:

• Amplitude of oscillation becomes maximum

• System absorbs maximum energy from the external force

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2. Natural Frequency

For a simple harmonic oscillator:

$${\omega }_0=\sqrt{\frac{k}{m}}$$

Where
(k) = spring constant
(m) = mass

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3. Resonance Condition

$$\omega ={\omega }_0$$

where
ω = frequency of driving force
ω0 = natural frequency of the system

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4. Amplitude at Resonance

For a damped driven oscillator:

$$A_{max}=\frac{F_0}{2m\beta {\omega }_0}$$

Amplitude depends on:

• Amplitude depends on:

  • Driving force amplitude $F_0$

  • Mass $m$
    

  • Damping constant $b$ through $\beta = \frac{b}{2m}$

  • Natural frequency ${\omega }_{\mathrm{0}}$

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5. Effect of Damping

• Low damping → very high amplitude at resonance (tall, sharp peak)

• High damping → small amplitude, broad peak

• Critical/over damping → resonance becomes weak or negligible

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6. Phase Relationship

Phase difference (ϕ) between displacement and driving force:

At resonance:

$$\varphi ={90}^{\circ }$$

Below resonance → small phase lag
Above resonance → phase approaches $180^\circ$

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7. Energy Absorption

At resonance:

• Power absorbed is maximum

• The system oscillates with maximum energy

Power delivered:

$$P=F_0{v}_{max}\cos \varphi$$

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8. Quality Factor (Q-Factor)

Measures sharpness of resonance peak:

$$Q=\frac{{\omega }_0}{2\mathrm{\beta <span\; class="vlist-s"></span>}}$$

High Q
→ sharp resonance
→ low damping
→ more energy stored per cycle

Low Q
→ broad resonance
→ high damping

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9. Resonance Curve

Amplitude vs driving frequency graph:

• Peak at $\omega = \omega_0$
  

• Width of the curve depends on damping

• Sharper peak = higher Q

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10. Important Examples of Resonance

• Tuning fork sounding loudly near matching frequency

• Radio tuning circuits (LC circuits)

• Buildings/bridges vibrating during earthquakes

• Microwave ovens (specific frequency absorption)

• Swing pushed at exact natural frequency

• Musical instruments (air columns, strings)

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11. Common Entrance Exam Questions

• Condition for resonance in damped oscillation

• Amplitude at resonance

• Phase lag at resonance

• Effect of damping on resonance curve

• Q-factor and bandwidth

• Real-life examples

• Resonance in LCR circuits (JEE focus)