Chapter 7: Oscillations

• Oscillation and Vibration

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  • Oscillation: Repetitive back-and-forth motion about an equilibrium position.
  • Vibration: A rapid oscillation.

• Simple Harmonic Motion (SHM)

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  • Special type of oscillatory motion.
  • Restoring force is directly proportional to displacement from equilibrium and acts opposite to displacement.
  • $F = -kx$.
  • Acceleration is directly proportional to displacement and opposite to it.
  • $a = -\omega^2 x$.

  • Characteristics of SHM

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    • Acceleration is always directed towards the mean position.
    • Velocity is maximum at mean position, zero at extreme positions.
    • Acceleration is zero at mean position, maximum at extreme positions.
    • Period and frequency are independent of amplitude (for ideal SHM).

  • Displacement in SHM

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    • $x = x_o \sin(\omega t + \phi)$ (if starting from equilibrium).
    • $x = x_o \cos(\omega t + \phi)$ (if starting from extreme).
    • $x_o$: amplitude (maximum displacement).
    • $\omega$: angular frequency.
    • $\phi$: phase angle.

  • Velocity in SHM

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    • $v = \omega \sqrt{x_o^2 - x^2}$.
    • Maximum velocity at mean position ($x=0$): $v_{max} = \omega x_o$.
    • Zero velocity at extreme positions ($x=\pm x_o$).

  • Acceleration in SHM

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    • $a = -\omega^2 x$.
    • Maximum acceleration at extreme positions ($x=\pm x_o$): $a_{max} = \omega^2 x_o$.
    • Zero acceleration at mean position ($x=0$).

• Mass-Spring System

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  • Period: $T = 2\pi\sqrt{\frac{m}{k}}$.
  • Frequency: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$.
  • Angular frequency: $\omega = \sqrt{\frac{k}{m}}$.
  • $m$: mass.
  • $k$: spring constant.

• Simple Pendulum

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  • Period: $T = 2\pi\sqrt{\frac{L}{g}}$.
  • Frequency: $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$.
  • Angular frequency: $\omega = \sqrt{\frac{g}{L}}$.
  • $L$: length of pendulum.
  • $g$: acceleration due to gravity.
  • Valid for small angles of oscillation (typically less than $10^\circ$).

  • Factors Affecting Period

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    • Length of pendulum (T $\propto \sqrt{L}$).
    • Acceleration due to gravity (T $\propto \frac{1}{\sqrt{g}}$).
    • Independent of mass of bob and amplitude (for small angles).

• Energy in SHM

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  • Total mechanical energy is conserved (in absence of damping).

  • Kinetic Energy (KE)

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    • $KE = \frac{1}{2}m\omega^2(x_o^2 - x^2)$.
    • Maximum at mean position ($x=0$): $KE_{max} = \frac{1}{2}m\omega^2 x_o^2$.
    • Zero at extreme positions ($x=\pm x_o$).

  • Potential Energy (PE)

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    • $PE = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2$.
    • Maximum at extreme positions ($x=\pm x_o$): $PE_{max} = \frac{1}{2}kx_o^2 = \frac{1}{2}m\omega^2 x_o^2$.
    • Zero at mean position ($x=0$).

  • Total Energy (TE)

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    • $TE = KE + PE = \frac{1}{2}m\omega^2 x_o^2$.
    • Constant throughout SHM.
    • Proportional to square of amplitude ($x_o^2$).

• Damped Oscillations

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  • Oscillations where amplitude decreases over time due to dissipative forces (e.g., friction, air resistance).
  • Energy is lost from the system.

• Forced Oscillations and Resonance

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  • Forced Oscillations: An external periodic force drives the oscillation.
  • Resonance: Occurs when the frequency of the driving force matches the natural frequency of the oscillating system.
  • At resonance, amplitude becomes maximum.

  • Examples of Resonance

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    • Shattering of glass by sound.
    • Tacoma Narrows Bridge collapse.
    • Tuning a radio.
    • Microwave ovens.

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