Periodic and Oscillatory Motion

Periodic and Oscillatory Motion

• A motion that repeats itself after a fixed interval of time is called periodic motion.
• Oscillatory motion is a specific type of periodic motion where an object moves back and forth around a central point called the equilibrium position.
• The movement of the object is driven by a restoring force that pushes or pulls it back towards the equilibrium position.
• This force increases as the object moves further away from the equilibrium position.
• Oscillatory motion can be linear or circular.

Types of Oscillatory Motion

• Examples of linear oscillation include:

  • Mass-spring systems: A mass attached to a spring oscillates back and forth.
  • Fluid column in a U-tube: A column of fluid in a U-shaped tube oscillates up and down.
  • Floating cylinder: A cylinder floating in a liquid oscillates vertically.
  • Body dropped in a tunnel along earth diameter: A body dropped through a tunnel passing through the center of the earth would oscillate between the surface and the opposite side of the earth.
  • Strings of musical instruments: Vibrating strings on musical instruments produce sound through oscillations.

• Examples of circular oscillation include:

  • Simple pendulum: A mass hanging from a string swings back and forth.
  • Solid sphere in a cylinder: A solid sphere rolling without slipping inside a cylinder oscillates in a circular path.
  • Circular ring suspended on a nail: A ring suspended on a nail oscillates in a circular motion.
  • Balance wheel of a clock: The balance wheel in a mechanical clock oscillates to regulate timekeeping.
  • Rotation of the earth around the sun: The Earth's orbit around the sun is a type of circular oscillation.

Oscillatory Systems

• An oscillatory system is composed of an object that exhibits to-and-fro motion around an equilibrium point due to a restoring force.
• Oscillatory systems can be mechanical or non-mechanical.
• Mechanical Oscillatory Systems: In this type, the object itself changes its position. Examples include the oscillations of a mass-spring system, a pendulum, and a body dropped through a tunnel.
• Non-mechanical Oscillatory Systems: These involve periodic variations in a physical property rather than the position of an object. For example, variations in electric current in an oscillating circuit or fluctuations in pressure in a medium carrying sound waves.

Simple Harmonic Motion (SHM)

• This is the simplest type of oscillatory motion.
• It occurs when the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
• The mathematical formula for SHM restoring force is: F=-kx, where F is the restoring force, x is the displacement, and k is the force constant.
• SHM is described by a sinusoidal function. This sinusoidal motion can be expressed as x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
• The angular frequency is related to the period (T) and frequency (f) of the motion by the following equations: ω = 2πf = 2π/T.

Damped Oscillations

• Real oscillations often experience damping, where the amplitude decreases over time.
• This decrease is caused by energy dissipation due to factors like friction or air resistance.
• The damped oscillation can be represented by the equation x(t) = Ae^(-βt) cos(ω1t + φ), where β is the damping coefficient and ω1 is the damped angular frequency.
• Underdamped Oscillations: For a damping coefficient β that is less than the natural angular frequency ω0, the system oscillates with gradually decreasing amplitude.
• Overdamped Oscillations: When β is greater than ω0, the system returns to equilibrium without oscillating.
• Critically Damped Oscillations: In this case, β equals ω0, and the system returns to equilibrium in the shortest possible time without oscillating.

Diffraction of Light

• Light bends when it passes around an obstacle, particularly when the size of the obstruction is comparable to the wavelength of the light.
• This phenomenon, called Diffraction, results in the spreading of light into the geometric shadow of the obstacle.
• Diffraction can be classified as Fresnel diffraction or Fraunhofer diffraction.

Fresnel Diffraction

• Occurs when either the source of light, the screen, or both are a finite distance from the obstacle.
• No lenses are used to make the rays converge or parallel.
• The incident wavefront can be cylindrical or spherical.

Fraunhofer Diffraction

• This occurs when the light source and the screen are at an infinite distance from the obstacle.
• Lenses are used to make the incident light rays parallel and then converge the diffracted rays on the screen.
• The incident wavefront is planar.
• Examples of Fresnel Diffraction: Diffraction at the narrow, straight edge of an obstacle
• Examples of Fraunhofer Diffraction: Diffraction due to a single slit.

Fraunhofer Diffraction Due to a Single Slit

• When a beam of light passes through a narrow slit, the light diffracts, creating an interference pattern of alternating bright and dark bands on a screen.
• The central bright fringe is the most intense and is located directly opposite the slit.
• The position of the minima (dark bands) is determined by the formula e sin(θ) = mλ, where e is the slit width, θ is the angle of diffraction, m is the order of the minima (1, 2, 3, ...), and λ is the wavelength of light.
• The position of the maxima (bright bands) is determined by the formula e sin(θ) = (m + 1/2)λ.