Oscillations About Equilibrium (Chapter 13) – PHYS 1021 Study Notes

Oscillations About Equilibrium

13-1 Periodic Motion

Periodic motion refers to any motion that repeats itself at regular intervals. This type of motion is fundamental in physics and is observed in many natural and engineered systems.

• Definition: Periodic motion is motion that repeats itself over and over.

• Examples: A swing moving back and forth, the beating of a heart, or the ticking of a clock.

• Complete Cycle: One full repetition of the motion, returning to the starting point, is called a cycle or oscillation.

• Period (T): The time required for one complete cycle. SI unit: seconds (s).

• Frequency (f): The number of cycles per unit time. SI unit: hertz (Hz) or s-1.

Key Equations:

• Period: $T = \text{time for one cycle}$

• Frequency: $f = \frac{1}{T}$

Example: If a frog takes 18 seconds to complete 10 oscillations, the period is $T = \frac{18\ \text{s}}{10} = 1.8\ \text{s}$ and the frequency is $f = \frac{1}{1.8\ \text{s}} \approx 0.56\ \text{Hz}$.

13-2 Simple Harmonic Motion (SHM)

Simple harmonic motion is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

• Definition: Simple harmonic motion (SHM) is periodic motion under a linear restoring force.

• Classic Example: A mass attached to a spring with spring constant $k$.

• Hooke's Law: $F = -kx$ (restoring force is proportional and opposite to displacement $x$).

• Amplitude (A): The maximum displacement from equilibrium.

• Position as a function of time: $x = A \cos(\omega t)$, where $\omega$ is the angular frequency.

• Angular frequency: $\omega = 2\pi f = \frac{2\pi}{T}$

Example: An air-track cart attached to a spring oscillates with a period of 2.4 s and amplitude 0.10 m. Its position at various times can be calculated using the position equation above.

13-3 Connections Between Uniform Circular Motion and SHM

There is a deep connection between uniform circular motion and simple harmonic motion. The projection of uniform circular motion onto one axis produces SHM.

• Angular speed: $\omega = \frac{2\pi}{T}$

• Linear speed: $v = \omega R$

• Position in SHM: $x = A \cos(\omega t)$

• Velocity in SHM: $v = -A\omega \sin(\omega t)$

• Acceleration in SHM: $a = -A\omega^2 \cos(\omega t)$

• Maximum values: $v_{\text{max}} = A\omega$, $a_{\text{max}} = A\omega^2$

Summary: Velocity is maximum at equilibrium ($x=0$), zero at extremes ($x=\pm A$). Acceleration is maximum at extremes, zero at equilibrium.

13-4 The Period of a Mass on a Spring

The period of oscillation for a mass-spring system depends on the mass and the spring constant.

• Newton's Second Law: $ma = -kx$

• Period of a mass-spring system: $T = 2\pi \sqrt{\frac{m}{k}}$

• Applies to both horizontal and vertical spring systems (with appropriate equilibrium position).

Example: A 0.260-kg mass on a vertical spring with a period of 1.12 s. The stretch and period can be analyzed using the above formula.

13-5 Energy Conservation in Oscillatory Motion

In an ideal SHM system (no friction), the total mechanical energy is conserved and is the sum of kinetic and potential energies.

• Total energy: $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant}$

• At maximum displacement ($x=\pm A$), all energy is potential: $E = \frac{1}{2}kA^2$

• At equilibrium ($x=0$), all energy is kinetic: $E = \frac{1}{2}mA^2\omega^2$

• Maximum kinetic and potential energies are equal: $K_{\text{max}} = U_{\text{max}} = E$

13-6 What is an Angle in Radians?

Angles in SHM are measured in radians, a dimensionless unit defined by the ratio of arc length to radius.

• Definition: $\theta = \frac{s}{r}$, where $s$ is arc length and $r$ is radius.

• Radians are essential for trigonometric calculations in SHM.

13-6 The Pendulum

A simple pendulum exhibits SHM for small angles. The restoring force is provided by gravity.

• Restoring force: $F = mg \sin \theta$

• For small angles, $\sin \theta \approx \theta$ (in radians).

• Period of a pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$

• The period is independent of mass and amplitude (for small oscillations).

Example: A grandfather clock pendulum with a period of 2.00 s. The length can be found using the period formula.

Conceptual Question: To make a slow clock run faster, raise the weight to shorten the pendulum and decrease the period.

13-7 Damped Oscillations

Real oscillatory systems experience damping, where energy is lost over time due to resistive forces.

• Damping force: $F = -bv$, where $b$ is the damping coefficient.

• Underdamped: System oscillates with decreasing amplitude.

• Critically damped: System returns to equilibrium as quickly as possible without oscillating.

• Overdamped: System returns to equilibrium without oscillating, but more slowly than critical damping.

• Amplitude decay (underdamped): $A = A_0 e^{-bt/2m}$

Example: Shock absorbers in cars are designed to be critically damped for quick return to equilibrium.

13-8 Driven Oscillations and Resonance

When an external periodic force drives an oscillator, the system can reach a steady-state amplitude that depends on the driving frequency.

• Natural frequency ($f_0$): The frequency at which a system oscillates when not driven or damped.

• Resonance: When the driving frequency matches the natural frequency, the amplitude is maximized.

• Resonance curve: Shows amplitude as a function of driving frequency for different damping levels.

• Key equations:

  • For a mass-spring: $f_0 = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

  • For a pendulum: $f_0 = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$

• Damped driven oscillator: Amplitude depends on the relationship between driving and natural frequencies.

Example: If a pendulum's amplitude decreases as the driving frequency increases past a certain value, the resonance frequency is between the two tested frequencies.

Type of Oscillation	Restoring Force	Period Formula	Depends on
Mass-Spring (horizontal/vertical)	$F = -kx$	$T = 2\pi \sqrt{\frac{m}{k}}$	Mass, Spring Constant
Pendulum (small angle)	$F = -mg\theta$	$T = 2\pi \sqrt{\frac{L}{g}}$	Length, Gravity

Additional info: These notes are based on lecture slides and cover all major subtopics of Chapter 13: Oscillations About Equilibrium, including definitions, equations, examples, and conceptual questions relevant for college-level physics.