Oscillation Notes

Oscillations

Introduction

  • Oscillations are repeating vibrations of a quantity around its equilibrium position.
  • The term originates from the Latin verb meaning "to swing."
  • A restoring force pushes or pulls an object back to its central point after displacement, causing it to oscillate around this equilibrium.

Periodic Motion

  • Periodic motion is motion repeated in equal time intervals.
  • Period (T): Time for one complete repetition or cycle.
  • Frequency (f): Number of periods per unit time.
  • Examples:
    • Rocking chair
    • Bouncing ball
    • Vibrating tuning fork
    • Swing in motion

Spring Oscillator

  • A spring oscillator (spring-mass system) is a mechanical system where a mass attached to a spring oscillates around an equilibrium point.
  • This oscillation is a form of simple harmonic motion.
  • The spring exerts a restoring force proportional to the displacement from equilibrium.

Simple Pendulum

  • A simple pendulum consists of a mass (pendulum bob) suspended from a fixed point by a string.
  • It exemplifies periodic motion with a regular back-and-forth swing.
  • Key Parameters:
    • T: Period (time for one complete swing, in seconds).
    • f: Frequency (number of complete swings per second, in Hertz).
    • L: Length (pendulum length from pivot to center of mass, in meters).
    • g: Acceleration due to gravity (approximately 9.8m/s29.8 m/s^2 on Earth).

Physical Pendulum

  • A physical pendulum is a rigid body oscillating about a fixed horizontal axis under gravity's influence.
  • It accounts for the object's size and mass distribution, unlike the simple pendulum.
  • Key Parameters:
    • T: Period (time for one complete swing, in seconds).
    • f: Frequency (number of complete swings per second, in Hertz).
    • I: Moment of inertia of the object about the pivot point (in kg⋅m2kg \cdot m^2).
    • m: Mass of the object (in kg).
    • g: Acceleration due to gravity (approximately 9.8m/s29.8 m/s^2 on Earth).
    • d: Distance from the pivot point to the center of mass (in meters).

Difference Between Simple and Physical Pendulum

  • Simple Pendulum:
    • Mass concentrated at one point (the bob).
    • Motion is easily predictable.
  • Physical Pendulum:
    • A whole object swings on a pivot.
    • Mass is spread out, affecting the swing based on shape and weight distribution.

Frequency and Period of Objects in Periodic Motion

  • Period (T): Time to complete one full cycle (in seconds).
  • Frequency (f): Number of complete cycles per second (in Hertz).

Spring Oscillator Parameters

  • T: Period (time for one complete oscillation, in seconds).
  • f: Frequency (number of complete oscillations per second, in Hertz).
  • m: Mass of the object attached to the spring (in kg).
  • k: Spring constant (stiffness of the spring, in N/m).

Oscillation Explained

  • Oscillation is a repetitive motion around a central value or equilibrium point, typically occurring at regular intervals.
  • Examples:
    • Swinging pendulum (gravity restores equilibrium).
    • Vibrating guitar strings (producing sound through repeated motion).

Simple Harmonic Motion (SHM)

  • Simple Harmonic Motion (SHM) is a special oscillation where the restoring force is: Directly proportional to the displacement from the equilibrium position. Always directed toward the equilibrium position.
  • The farther the object moves from the center, the stronger the restoring force.

SHM Examples

  • Playground swing (gravity acts as the restoring force).
  • Metronome (gravity restores the center position).

Conditions Necessary for SHM

  • Restoring Force Proportional to Displacement: The force pulling back increases with distance from equilibrium.
  • Restoring Force Acts in the Opposite Direction: The force always points toward equilibrium.
  • Minimal Damping: Ideally, no energy loss, so the motion continues without fading.
  • Stable Equilibrium: The object returns to its original position when slightly disturbed.

SHM Graphs

  • Displacement, velocity, and acceleration graphs follow wave-like patterns, peaking at different times.
  • Displacement graph: distance from the center position.
  • Velocity graph: how fast it’s moving.
  • Acceleration graph: strength of the push or pull.

Equations of SHM

  • F=−kxF = -kx
  • F=maF = ma
  • −kx=md2xdt2-kx = m \frac{d^2x}{dt^2}
  • x(t)=xmcos⁡(ωt+ϕ)x(t) = x_m \cos(\omega t + \phi)
  • v(t)=−ωxmsin⁡(ωt+ϕ)v(t) = -\omega x_m \sin(\omega t + \phi)
  • a(t)=−ω2xmcos⁡(ωt+ϕ)a(t) = -\omega^2 x_m \cos(\omega t + \phi)

Hooke’s Law

  • Hooke's Law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.
    • F=kxF = kx
      • F = Force
      • k = Spring Constant
      • x = Displacement from equilibrium

Force and Motion Relationship

  • F=ma=−kxF = ma = -kx
  • a=−kxma = -\frac{kx}{m}

Freeze-Frames

  • A freeze-frame is a snapshot of a moving system at one instant in time, showing positions, forces, and velocities as if motion is paused.
  • It helps analyze what's happening right at that moment without considering past or future motion.

Energy in SHM

  • Potential energy and velocity are zero at extreme points and greatest at x=0.
  • Kinetic energy and acceleration are zero at x = 0 and greatest at extreme points.

Displacement at Time T

  • x(t)=xmcos⁡(ωt+ϕ)x(t) = x_m \cos(\omega t + \phi)
    • x(t)x(t) = Displacement at time t
    • xmx_m = Amplitude
    • ω\omega = Angular frequency
    • tt = Time
    • ϕ\phi = Phase constant or phase angle

Angular Frequency

  • ω\omega is the angular frequency of the motion.
  • The position x(t) returns to its initial value at the end of a period (T), i.e., at time T + T.

Velocity at Time T

  • The velocity varies in magnitude and direction.
  • It's momentarily zero at extreme points and maximum through the central point.
  • v(t)=dx(t)dt=−ωxmsin⁡(ωt+ϕ)v(t) = \frac{dx(t)}{dt} = -\omega x_m \sin(\omega t + \phi)

Acceleration at Time X

  • The acceleration varies because the cosine function varies with time, between +1 and -1.

Equations for a Spring-Mass System Summary

  • Hooke's Law: F=−kxF = -kx
  • Angular Frequency: ω=km\omega = \sqrt{\frac{k}{m}}
  • Frequency: f=ω2π=12πkmf = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
  • Period: T=1f=2πmkT = \frac{1}{f} = 2\pi \sqrt{\frac{m}{k}}
  • Position: x=Acos⁡(ωt+ϕ)x = A \cos(\omega t + \phi)
  • Velocity: v=−Aωsin⁡(ωt+ϕ)v = -A\omega \sin(\omega t + \phi)
  • Acceleration: a=−Aω2cos⁡(ωt+ϕ)a = -A\omega^2 \cos(\omega t + \phi)

Damped and Forced Oscillation

  • Damped oscillations: Amplitude decreases over time due to energy loss.
  • Forced Oscillation: An oscillating system is driven by an external periodic force.

Types of Damping

  • Underdamped: Oscillations occur with gradually decreasing amplitude; the system oscillates multiple times before coming to rest.
  • Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
  • Overdamped: The system returns to equilibrium slowly without oscillating; takes longer than critically damped systems.

Sample Problems

  • A block of mass m=680gm = 680 g is fastened to a spring with k=65N/mk = 65 N/m. The block is Pulled a distance x=11cmx=11 cm from its equilibrium position at x=0x=0 on a frictionless surface and released from rest at t=0t=0.
    • What is the angular frequency of the resulting motion?
    • What is the frequency of the motion?
    • What is the period of the motion?
  • A 2.00−kg2.00-kg block is placed on a frictionless surface. A spring with a force constant k=32.00N/mk=32.00N/m is attached to the block, and the opposite end of the spring is attached to a wall. The spring can be compressed or extended. The equilibrium position is marked as x=0.00mx=0.00m. Work is done on the block, pulling it out to x=+0.02mx=+0.02m. The block is released from rest and oscillates between x=+0.02mx=+0.02m and x=−0.02mx=−0.02m. The period of the motion is 1.57s1.57 s.
  • A 2kg2 kg mass is hung vertically from a spring, causing it to stretch by 0.1m0.1 m. What is the spring constant (k) of the spring?
  • A 0.5kg0.5 kg mass is attached to a spring with a spring constant of 200N/m200 N/m. Find the frequency and period of the resulting simple harmonic motion.
  • A mass-spring system oscillates with an amplitude of 0.3m0.3 m and a spring constant of 50N/m50 N/m. If the mass is 1kg1 kg, what is the maximum speed of the mass during oscillation?
  • A 0.25kg0.25 kg object is attached to a spring with a spring constant of 100N/m100 N/m and displaced by 0.1m0.1 m from equilibrium. What is the total mechanical energy of the system, assuming no damping?
  • A 1kg1 kg mass is hanging from a spring and set into vertical oscillation. The spring constant is 150N/m150 N/m. How far does the mass move from the equilibrium position if its initial speed is 2m/s2 m/s at equilibrium?
  • A particle undergoes simple harmonic motion with a maximum displacement (amplitude) of 0.2m0.2m and an angular velocity of 4rad/s4 rad/s. At time t=1.5st=1.5s, calculate the following:
    • Instantaneous position
    • Instantaneous velocity
    • Instantaneous acceleration. Assume the motion starts from maximum displacement