Damped & Forced Oscillations – Types, Formulas, Resonance | Entrance Exam Physics Notes

DAMPED AND FORCED OSCILLATIONS — SHORT NOTES

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1. Damped Oscillations

Definition

Oscillations whose amplitude decreases with time due to resistive forces (air resistance, friction, viscous force) are called damped oscillations.

The damping force is generally:

$$F_d=-bv$$ where b is the damping constant.

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Types of Damping

1. Underdamping

• Oscillations continue with gradually decreasing amplitude

• Most common in real life

• Displacement:

$$x(t)=A{e}^{-\beta t}\cos ({\omega }^{\mathrm{\prime }}t+\varphi )$$

where

$$\beta =\frac{b}{2m},{\omega }^{\mathrm{\prime }}=\sqrt{{\omega }_0^2-{\beta }^2}$$

2. Critical Damping

• System returns to equilibrium fastest without oscillation

• Used in car shock absorbers, door closers

• Condition:

$$b=2\sqrt{mk}$$

3. Overdamping

• System returns slowly to equilibrium

• No oscillation

• Condition:

$$b>2\sqrt{mk}$$

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Natural Frequency vs Damped Frequency

• Natural frequency (no damping):

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

• Damped frequency:

$${\omega }^{\mathrm{\prime }}=\sqrt{{\omega }_0^2-{\beta }^2}$$

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2. Energy in Damped Oscillations

Total energy decreases exponentially:

$$E(t)=E_0{e}^{-2\beta t}$$

Reason: Work done against damping force converts mechanical energy → heat.

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3. Forced Oscillations

Definition

Oscillations produced by an external periodic force:

$$F(t)=F_0\sin (\omega t)$$

The system eventually oscillates with frequency of the applied force (not natural frequency).

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Steady-State Solution

For a damped oscillator under periodic driving:

$$x(t)=A(\omega )\sin (\omega t+\varphi )$$

Amplitude of forced oscillation:

$$A(\omega )=\frac{F_0\mathrm{/}m}{\sqrt{({\omega }_0^2-{\omega }^2{)}^2+(2\beta \omega {)}^{\mathrm{2}}}}$$

Amplitude depends on:

• Driving frequency

• Damping constant

• Mass & stiffness

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4. Phase Difference

The forced oscillation lags behind driving force by phase:

$$\tan \varphi =\frac{2\beta \omega }{{\omega }_0^2-{\omega }^{\mathrm{2}}}$$

• At low ω → small phase lag

• At resonance → phase difference = 90°

• At high ω → phase difference → 180°

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5. Resonance (VERY IMPORTANT FOR EXAMS)

Definition

When the driving frequency = natural frequency:

$$\omega ={\omega }_{\mathrm{0}}$$

Amplitude becomes maximum:

$$A_{max}=\frac{F_0}{2m\beta {\omega }_{\mathrm{0<span\; class="vlist-s"></span>}}}$$

Key Points

• Amplitude ∝ 1/damping

• Lesser damping → higher resonance peak

• Occurs at:

  • Radio tuning

  • Microwave ovens

  • Musical instruments

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6. Quality Factor (Q-Factor)

Measures sharpness of resonance.

$$Q=\frac{{\omega }_0}{2\beta }=\frac{\sqrt{\frac{k}{m}}}{\frac{b}{m}}=\frac{\sqrt{mk}}{\mathrm{b}}$$

Higher Q = sharp resonance peak
Lower Q = broad resonance (more damping)

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7. Common Entrance Exam Problems

• Finding damped frequency

• Critical damping condition

• Energy decay in SHM

• Resonance frequency & amplitude

• Phase difference at different frequencies

• Q-factor and its effect

• Forced oscillation amplitude calculations