OSCILLATIONS

A. Simple Harmonic Motion (SHM)

\[ F = -kx \]

Properties & Key Points:

  • Simple Harmonic Motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
  • \( F \): Restoring force.
  • \( k \): Spring constant.
  • \( x \): Displacement from the equilibrium position.
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B. Displacement in SHM

\[ x(t) = A \cos(\omega t + \phi) \]

Properties & Key Points:

  • The displacement in SHM is described as a sinusoidal function of time.
  • \( x(t) \): Displacement at time \( t \).
  • \( A \): Amplitude (maximum displacement).
  • \( \omega \): Angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase constant.
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C. Velocity in SHM

\[ v(t) = -A \omega \sin(\omega t + \phi) \]

Properties & Key Points:

  • The velocity in SHM is the rate of change of displacement with respect to time.
  • \( v(t) \): Velocity at time \( t \).
  • \( A \): Amplitude.
  • \( \omega \): Angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase constant.
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D. Acceleration in SHM

\[ a(t) = -A \omega^2 \cos(\omega t + \phi) \]

Properties & Key Points:

  • The acceleration in SHM is the rate of change of velocity with respect to time and is proportional to the displacement.
  • \( a(t) \): Acceleration at time \( t \).
  • \( A \): Amplitude.
  • \( \omega \): Angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase constant.
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E. Angular Frequency

\[ \omega = \sqrt{\frac{k}{m}} \]

Properties & Key Points:

  • Angular frequency is the rate of change of the phase of the oscillation and is related to the frequency of oscillation.
  • \( \omega \): Angular frequency.
  • \( k \): Spring constant.
  • \( m \): Mass of the object.
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F. Period and Frequency

\[ T = \frac{2\pi}{\omega} \quad \text{and} \quad f = \frac{1}{T} \]

Properties & Key Points:

  • The period is the time taken for one complete oscillation, and the frequency is the number of oscillations per unit time.
  • \( T \): Period.
  • \( f \): Frequency.
  • \( \omega \): Angular frequency.
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G. Energy in SHM

\[ E = \frac{1}{2} k A^2 \]

Properties & Key Points:

  • The total mechanical energy in SHM remains constant and is the sum of potential energy and kinetic energy.
  • \( E \): Total mechanical energy.
  • \( k \): Spring constant.
  • \( A \): Amplitude.
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H. Kinetic Energy in SHM

\[ K.E. = \frac{1}{2} m v^2 \]

Properties & Key Points:

  • The kinetic energy is the energy due to the motion of the object and depends on its velocity.
  • \( K.E. \): Kinetic energy.
  • \( m \): Mass of the object.
  • \( v \): Velocity.
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I. Potential Energy in SHM

\[ P.E. = \frac{1}{2} k x^2 \]

Properties & Key Points:

  • The potential energy is stored in the spring or system due to its displacement from the equilibrium position.
  • \( P.E. \): Potential energy.
  • \( k \): Spring constant.
  • \( x \): Displacement from equilibrium.
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J. Damped Harmonic Motion

\[ x(t) = A_0 e^{-\gamma t} \cos(\omega' t + \phi) \]

Properties & Key Points:

  • Damped harmonic motion occurs when the oscillations gradually decrease in amplitude due to friction or other resistive forces.
  • \( x(t) \): Displacement at time \( t \).
  • \( A_0 \): Initial amplitude.
  • \( \gamma \): Damping coefficient.
  • \( \omega' \): Damped angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase constant.
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K. Forced Oscillations

\[ x(t) = \frac{F_0}{m} \frac{1}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2 \gamma \omega)^2}} \]

Properties & Key Points:

  • Forced oscillations occur when an external force is applied to an oscillator, often causing resonance at a particular frequency.
  • \( x(t) \): Displacement at time \( t \).
  • \( F_0 \): Amplitude of the external force.
  • \( m \): Mass of the object.
  • \( \omega_0 \): Natural frequency.
  • \( \omega \): Frequency of the external force.
  • \( \gamma \): Damping coefficient.
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L. Resonance

\[ A_{\text{max}} = \frac{F_0}{m \gamma} \]

Properties & Key Points:

  • Resonance occurs when the frequency of the applied force matches the natural frequency of the system, leading to a large amplitude of oscillations.
  • \( A_{\text{max}} \): Maximum amplitude at resonance.
  • \( F_0 \): Amplitude of the external force.
  • \( m \): Mass of the object.
  • \( \gamma \): Damping coefficient.
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