PHYS 1111 – Introductory Physics – Mechanics, Waves, Thermodynamics

PHYS 1111 – Introductory Physics – Mechanics, Waves, Thermodynamics

The University of Georgia, Fall 2025
Lecture 30 (Oct 27, 2025)

Key Takeaways from Last Time

• Typical questions discussed include:
  • Describe Simple Harmonic Motion (SHM) with amplitude, frequency, period.
  • Connect SHM to previously learned concepts such as springs, Uniform Circular Motion (UCM), pendulums.
  • Connect properties of SHM to energy.
• Main topics of study:
  • Periodic Motion
  • Simple Harmonic Motion
  • Mass on a Spring & UCM
  • Energy and SHM

Quiz Questions

Question 1

• A mass oscillates on a horizontal spring with period $T = 2.0 ext{ s}$.
• If the amplitude of the oscillation is doubled, the new period will be:
  • A. 1.0 s
  • B. 1.4 s
  • C. 2.0 s
  • D. 2.8 s
  • E. 4.0 s

Question 2

• A block of mass $m$ oscillates on a horizontal spring with period $T = 2.0 ext{ s}$.
• If a second identical block is glued to the top of the first block, the new period will be:
  • A. 1.0 s
  • B. 1.4 s
  • C. 2.0 s
  • D. 2.8 s
  • E. 4.0 s

Question 3

• Two identical blocks oscillate on different horizontal springs. Which spring has the larger spring constant?
  • A. The red spring
  • B. The blue spring
  • C. Both are the same.
  • D. There’s not enough information to tell.

Question 4

• A block oscillates on a very long horizontal spring. A graph shows the block’s kinetic energy as a function of position. What is the spring constant?
  • A. 1 N/m
  • B. 2 N/m
  • C. 4 N/m
  • D. 8 N/m

Today's Agenda

1. Damped and Driven Oscillations
2. Waves

Damped Oscillations

• An oscillation that runs down and stops is termed a damped oscillation.
• Example Application:
  • Shock absorbers in cars and trucks are heavily damped springs.
  • The vehicle’s vertical motion after hitting a rock or pothole is a damped oscillation.
• Energy dissipation occurs due to non-conservative forces such as friction or air resistance.
• A simple linear model describes this dissipation of energy:

Characteristics of Damped Oscillations

• In most cases, a non-conservative force decreases the amplitude of the oscillation, which is typically proportional to the speed.
• Losing energy results in decreased amplitude, subsequently affecting the frequency, termed the "natural frequency."

Decay of Amplitude in Damped Oscillators

• The maximum displacement declines over time as:
  • This slow decay is indicated as $x_{max}$, which creates an envelope that bounds the rapid oscillations.
• Various envelopes correspond to different values of the damping constant $b$.

Limits of Damped Oscillations

• Angular frequency can be used to classify oscillators:
  • Underdamped:
  • System oscillates while the amplitude decays exponentially.
  • Critically damped:
  • Damping results in the system returning to equilibrium as quickly as possible without oscillating.
  • Overdamped:
  • Similar to critical damping, the system may overshoot equilibrium but does not oscillate, reaching equilibrium over a longer period.

Driven Oscillations and Resonance

• An oscillating system naturally oscillates at a frequency $f_0$.
• When subjected to a periodic external force with driving frequency $f<em>{ext}$, the system's oscillation amplitude is low if $f</em>{ext}$ greatly differs from $f_0$.
  • As $f<em>{ext}$ approaches $f</em>0$, the amplitude rises dramatically.

External Frequency and Resonance

• A response curve plots the amplitude of a driven oscillator as a function of the external frequency.
  • Near the natural frequency, the system strongly oscillates.

Damping in Driven Oscillations

• The amplitude in driven oscillations is affected by the degree of damping in the system.
• As the damping constant decreases, the resonance amplitude becomes higher and narrower.

Waves

• Definition: A wave is a disturbance traveling through a medium.
• Mechanical waves are governed by Newton’s laws.
  • The medium produces an elastic restoring force when deformed, enabling wave propagation.
• Important Notes:
  • Mechanical waves transfer energy and momentum without transferring mass.
  • Examples: water waves, sound waves, seismic waves.
• Understanding waves involves the concept of traveling waves: organized disturbances traveling with a defined wave speed.

Types of Waves

Transverse Waves

• A transverse wave is characterized by the medium's displacement being perpendicular to the wave's direction of motion.

Longitudinal Waves

• A longitudinal wave features displacement along the wave's direction of motion.
• Example: sound waves, where compression and rarefaction occur longitudinally.

Identifying Types of Waves

Sound Waves

• Characteristics:
  • Sound waves are compression waves and longitudinal in nature.
  • In sound waves, the density and pressure of the medium oscillate (e.g., air).
• Analogy: Comparable to the compression and stretching of a Slinky.

Water Waves

• Water waves can exhibit characteristics of both transverse and longitudinal waves.
  • As waves propagate outward, the crests and troughs form concentric circles.

Sinusoidal Waves and Pulses

• Waves can take various forms as disturbances in a medium.

Wave Pulse

• Definition: A single, localized disturbance traveling through the medium with a defined speed and duration.

Sinusoidal Wave

• Definition: A continuous, repeating wave with regular oscillation, often characterized by a sine or cosine function.
  • Commonly found in sound, light, and water waves, closely related to SHM.

Wave Properties

• Characterization of a wave:
  • Wave speed: Rate of travel through a medium.
  • Wavelength: Distance over which the wave pattern repeats.
  • Period: Time taken for one wavelength to pass a point.
  • Frequency: Number of oscillations per second.
  • Amplitude: Maximum displacement from the equilibrium position.
• These quantities are interrelated, influencing the speed of a wave depending on the medium.

Example: Propagating Wave in a Rope

• A transverse mechanical wave propagates in the positive x-direction through a rope.
  • At $t=0.0 ext{ s}$ (dotted line) and $t=3.0 ext{ s}$ (solid line), the wave's height and its properties can be analyzed.
• Tasks:
  • Determine the wavelength and amplitude of the wave.
  • Find the propagation velocity of the wave.
  • Calculate the period and frequency of the wave.

Describing Sinusoidal Waves

• Model for a sinusoidal wave identifies it as:
  • This representation connects SHM, but substitutes sine for cosine during wave travel in positive x-direction with constant velocity $v$ moving a distance $vt$ in time $t$.
• Mathematical components:
  • Wave number:
  • Angular frequency:

Mathematical Representation of Waves

• The wave function, denoted as $ext{wavefunction}$, serves as a solution to complicated wave equations (scope outside of this course).
• Connections to SHM include accelerations and speeds in the y-direction:
  • Wave number:
  • Angular frequency:

Summary of Oscillatory Behavior and Waves

• Damped and Driven Behavior
• Classification of Waves
  • Longitudinal Waves
  • Transverse Waves
• Properties of Waves
• Wave function construction and understanding.