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 $9.8 m/s^2$ 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 $kg \cdot m^2$).
  • m: Mass of the object (in kg).
  • g: Acceleration due to gravity (approximately $9.8 m/s^2$ 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

• $F = -kx$
• $F = ma$
• $-kx = m \frac{d^2x}{dt^2}$
• $x(t) = x_m \cos(\omega t + \phi)$
• $v(t) = -\omega x_m \sin(\omega t + \phi)$
• $a(t) = -\omega^2 x_m \cos(\omega t + \phi)$

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.
  • $F = kx$
    • F = Force
    • k = Spring Constant
    • x = Displacement from equilibrium

Force and Motion Relationship

• $F = ma = -kx$
• $a = -\frac{kx}{m}$

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

• $x(t) = x_m \cos(\omega t + \phi)$
  • $x(t)$ = Displacement at time t
  • $x_m$ = Amplitude
  • $\omega$ = Angular frequency
  • $t$ = Time
  • $\phi$ = Phase constant or phase angle

Angular Frequency

• $\omega$ 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.
• $v(t) = \frac{dx(t)}{dt} = -\omega x_m \sin(\omega t + \phi)$

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: $F = -kx$
• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$
• Frequency: $f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
• Period: $T = \frac{1}{f} = 2\pi \sqrt{\frac{m}{k}}$
• Position: $x = A \cos(\omega t + \phi)$
• Velocity: $v = -A\omega \sin(\omega t + \phi)$
• Acceleration: $a = -A\omega^2 \cos(\omega t + \phi)$

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