Oscillations (1)

PHY-101: Waves & Oscillations

Periodic Motion

  • Definition: Motion that repeats itself at regular intervals.

    • Examples:

      • Earth's motion around the Sun

      • Hands of a clock (hour, minute)

      • Motion of planets

Oscillation

  • Definition: A periodic back-and-forth motion where the object moves half of its time period in one direction and the other half in the opposite direction.

    • Examples:

      • Motion of a simple pendulum

      • Vibration of a tuning fork

      • Motion of a spring

Simple Harmonic Motion (SHM)

  • Definition: Oscillation where the oscillating body experiences a restoring force proportional to its displacement from its equilibrium position.

    • System is referred to as Simple Harmonic Oscillator.

    • Examples:

      • Motion of a spring

      • Oscillation of a floating cylinder

      • Swinging of a child on a playground

      • Simple pendulum

Differential Equation of SHM

  • For a mass m executing SHM with restoring force F acting at displacement x:

    • Formula:

      • F = -kx

      • where k is the force constant.

    • According to Newton’s 2nd law of motion:

      • F = ma

Solution of the Differential Equation

  • Form:

    • d²x/dt² = -ω²x

    • where,

      • ω² = k/m

    • Integration leads to:

      • x(t) = A sin(ωt + δ)

    • Parameters of the oscillation are explained below.

Parameters of a Simple Harmonic Oscillator

Amplitude

  • Maximum displacement from the mean position.

  • Formula:

    • Displacement is maximum when sin(ωt + δ) = ±1.

    • Therefore, the maximum displacement is ±A.

Displacement

  • The position of the mass as a function of time:

  • Formula:

    • x(t) = A sin(ωt + δ)

Velocity

  • Derivative of displacement with respect to time:

  • Formula:

    • v(t) = Aω cos(ωt + δ)

    • This shows that v is maximum at x = 0 (equilibrium position) and zero at x = ±A (maximum displacements).

Acceleration

  • Derivative of velocity with respect to time:

  • Formula:

    • a(t) = -ω²x

    • Indicates that acceleration is directed towards the equilibrium position.

Time Period and Frequency

  • Time Period (T): Time taken for one complete oscillation.

  • Formula (derived from SHM solution):

    • x = A sin(ωt + δ)

    • T = 2π/ω

  • Frequency (f): Number of oscillations per second

  • Formula:

    • f = 1/T = ω/2π

Phase and Phase Constant

  • Phase (φ): The term (ωt + δ) in the oscillator's equations, determining displacement and direction of motion.

  • Phase Constant (δ): Defines the initial condition of the motion, e.g. the position at t = 0.

Graphical Representation

  • Displacement, velocity, and acceleration can be visualized graphically as sinusoidal functions.

    • Displacement: y = A sin(ωt + δ)

    • Velocity: v = Aω cos(ωt + δ)

    • Acceleration: a = -Aω² sin(ωt + δ)

Simple Harmonic Oscillation Examples

  • Spring mass system

  • Simple pendulum

  • LC circuit

  • Torsional pendulum

Special Systems

Simple Pendulum

  • Consists of a light string supporting a small mass.

  • The governing equations based on forces acting on the mass indicate it undergoes SHM.

LC Circuit

  • Capacitor charged and quickly discharged through an inductance, producing oscillations similar to SHM.

Torsional Pendulum

  • Disk oscillates about a fixed point, demonstrating angular SHM.

Damped Oscillations

  • Real-world oscillators experience damping from friction or air resistance.

  • Example: Pendulum experiences decaying oscillations over time.

Forced Oscillations and Resonance

  • Forced Oscillator: Damped oscillator subjected to periodic force.

  • Resonance: The phenomenon where the amplitude of oscillation increases dramatically when an external force matches the natural frequency of the system.