Oscillation 🌏🕜

• Periodic Motion

  A motion that repeats itself along a definite path in a definite or fixed interval of time
• Periodic Motion

  • Swinging of pendulum
  • Motion of handle of a clock
  • Rotation of the Earth around the sun (revolution)
  • Movement of a fan
  • Motion of electrons around the nucleus
• Oscillatory Motion


  A motion where an object vibrates back and forth (up and down) repeatedly through equilibrium position in a definite interval of time
• All oscillatory motions are periodic but all periodic motions are not oscillatory
• Simple Harmonic Motion (SHM)


  Type of oscillatory motion where the acceleration of the object is directly proportional to the displacement and directed towards the equilibrium position
• Characteristict of Simple Harmonic Motion

  • Restoring force is directly proportional to displacement
  • Acceleration is directly proportional to restoring force
  • Both acceleration and restoring force are directed towards the mean position
  • The path of oscillation is back and forth
  • The amplitude should be constant
• Amplitude (A)

  Maximum displacement of an oscillating particle from mean position
• Time Period (T)

  Time taken to complete one oscillation
• Frequency (f)

  Number of oscillations made per second
• Phase (φ)

  Position and direction of motion of the particle at any instant
• Phase Constant (φ)

  Initial phase or initial condition of the motion
• Angular Frequency (ω)

  Rate at which an object oscillates or completes a cycle in a periodic motion
• D = √(A^2 + B^2)
• tan(θ) = B/A
• Uniform Circular Motion

  Motion of a particle moving on the circumference of a circle with uniform angular velocity
• Kinetic Energy

  1/2 mv^2
• Total Energy in SHM is constant
• Free Oscillation

  Oscillation of a body with its own natural frequency without external force
• Forced/Driven Oscillation

  Oscillation of a body due to a strong periodic external force with a frequency different from the natural frequency
• Resonance


  Phenomenon of increasing amplitude of oscillation when the frequency of driving force is close to the natural frequency of oscillation
• What is forced oscillation?

  A system is said to be executing forced oscillation if it oscillates with a frequency different from its natural frequency due to an external periodic force.
• How does an external periodic force affect a body's oscillation?

  It influences the body's oscillation, leading to forced oscillation.
• What happens to the amplitude of an oscillator in forced oscillation?

  The amplitude decreases due to damping forces but remains constant due to energy gained from the external source.
• What occurs to the amplitude of oscillation in forced oscillation?

  • Damping occurs in the amplitude of oscillation.
  • Amplitude remains constant due to external energy supplied by the system.
• What is the role of the driving force in forced oscillation?

  The driving force causes the body to oscillate, overriding the natural frequency.
• What is the relationship between driving frequency and natural frequency in forced oscillation?

  In forced oscillation, the driving frequency takes over the natural frequency of the oscillating body.
• What is the formula for driven angular frequency?

  The driven angular frequency is given by $\omega_d =$$2\pi f_d$, where $f_d$ is the driven frequency.
• Give an example of forced oscillation.

  A person swinging on a swing with someone pushing it is an example of forced oscillation.
• What is the equation for the external driving force of amplitude $F_o$ that varies periodically?


  The equation is $F =$$F_o \cos(\omega_d t)$, where $\omega_d$ is the driving angular frequency.
• What is the restoring force in forced oscillation?


  The restoring force is given by $F_R =$$-kx$.
• What is the damping force in forced oscillation?

  The damping force is given by $F_d =$$-bv$.
• What is the total force acting on a body in forced oscillation?

  • Total force is given by $F_T =$$F_R +$$F_d +$$F$.
  • The equation of motion is $ma =$$-kx - bv +$$F_o \cos(\omega_d t)$.
• What is the differential equation of forced oscillation?

  The differential equation is $m\frac{d^2x}{dt^2} +$$b\frac{dx}{dt} +$$kx =$$F_o \cos(\omega_d t)$.
• What is the solution to the differential equation of forced oscillation?

  The solution is given by $A =$ \frac{F_o}{\sqrt{m^2(\omega^2 - \omega_d^2)^2 + \omega_d^2 b^2}}^{1/2}.
• What do the variables in the solution of the differential equation represent?

  $F_o$ is the amplitude of the driving force, $b$ is the damping constant, $m$ is the mass of the oscillating body, $\omega_d$ is the driving angular frequency, and $\omega$ is the natural angular frequency.
• What does the equation $x =$$A \cos(\omega_d t)$ represent?


  This equation represents the displacement of the system due to the driving force.
• What does $A$ represent in the equation $x =$$A \cos(\omega_d t)$?


  $A$ represents the amplitude of forced oscillation.