# Lecture 17: Oscillations and Simple Harmonic Motion

## Physics I — Mechanics & Thermodynamics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Define simple harmonic motion (SHM) and identify systems that exhibit it
2. Write and interpret the equations of motion for SHM (position, velocity, acceleration)
3. Relate the parameters amplitude, angular frequency, period, and phase constant
4. Analyze the energy of a simple harmonic oscillator
5. Solve problems involving mass-spring systems and simple pendulums
6. Describe damped and driven oscillations qualitatively

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## Lecture Content

### I. Oscillatory Motion — Overview

**Oscillation** is any motion that repeats itself periodically about an equilibrium position. The key parameters describing oscillatory motion are the **amplitude** (A), the maximum displacement from equilibrium; the **period** (T), the time for one complete cycle; the **frequency** (f), the number of cycles per second with f = 1/T and units of Hz; and the **angular frequency** (omega), defined as omega = 2 pi f = 2 pi / T with units of rad/s.

**Simple harmonic motion (SHM)** is the simplest and most fundamental type of oscillation. It occurs whenever the restoring force is proportional to the displacement, as described by Hooke's law F = -kx for a spring. The resulting motion is sinusoidal.

### II. Equations of SHM

The position of an object undergoing SHM varies sinusoidally with time: x(t) = A cos(omega t + phi), where A is the amplitude, omega is the angular frequency, and phi is the phase constant that determines the initial position. The velocity is the time derivative of position: v(t) = -A omega sin(omega t + phi), reaching its maximum value v_max = A omega at the equilibrium position. The acceleration is a(t) = -A omega^2 cos(omega t + phi) = -omega^2 x(t), with maximum magnitude a_max = A omega^2 occurring at maximum displacement.

The defining characteristic of SHM is that the acceleration is always proportional to and opposite in direction to the displacement: a = -omega^2 x. This relationship is both the signature and the mathematical definition of simple harmonic motion.

<image>Three vertically aligned graphs showing one full cycle of SHM. Top: x(t) = A cos(omega t), a cosine wave with amplitude A and period T labeled. Middle: v(t) = -A omega sin(omega t), shifted by a quarter period relative to x(t). Bottom: a(t) = -A omega^2 cos(omega t), in antiphase with x(t). Vertical dashed lines connect key points: when x is maximum, v = 0 and a is at its negative maximum; when x = 0, v is at its maximum magnitude and a = 0. These relationships are annotated with brief explanatory notes.</image>

### III. Mass-Spring System

A mass m attached to a spring with spring constant k on a frictionless surface provides the prototypical example of SHM. The equation of motion F = -kx gives ma = -kx, yielding a = -(k/m)x. Comparison with the general SHM equation identifies omega = sqrt(k/m), the period T = 2 pi sqrt(m/k), and the frequency f = (1/2pi) sqrt(k/m).

A remarkable feature of SHM is that the period does **not depend on the amplitude** A. A larger amplitude means the object must travel farther, but it also moves faster, and these two effects exactly compensate. For a vertical spring, the equilibrium position shifts downward by Delta L = mg/k, but oscillation about this new equilibrium is still SHM with the same angular frequency.

Springs can be combined in different ways. Springs in parallel have an effective constant k_eff = k_1 + k_2, producing a stiffer system with higher frequency. Springs in series have 1/k_eff = 1/k_1 + 1/k_2, producing a softer system with lower frequency.

### IV. The Simple Pendulum

A point mass m on a massless string of length L, swinging through small angles, approximates SHM. The restoring force is F = -mg sin(theta), which for small angles (where sin theta is approximately equal to theta in radians) becomes F approximately equal to -mg theta. Since the arc length is s = L theta, this gives F approximately equal to -(mg/L)s, analogous to a spring with effective constant k = mg/L. The angular frequency is omega = sqrt(g/L) and the period is T = 2 pi sqrt(L/g).

The period depends on the string length L and the local gravitational acceleration g, but **not on the mass or the amplitude** (for small angles). For large angles, the motion remains periodic but is no longer simple harmonic, and the period increases with amplitude. The **physical pendulum**, an extended body oscillating about a pivot, has period T = 2 pi sqrt(I/(mgh)), where I is the moment of inertia about the pivot and h is the distance from the pivot to the center of mass.

### V. Energy in SHM

For a mass-spring system, the kinetic energy is K = (1/2)mv^2 = (1/2)kA^2 sin^2(omega t + phi), the potential energy is U = (1/2)kx^2 = (1/2)kA^2 cos^2(omega t + phi), and the total energy is E = K + U = (1/2)kA^2, which remains constant throughout the motion. Energy oscillates continuously between kinetic and potential forms. At the equilibrium position (x = 0), all energy is kinetic. At the turning points (x = +/- A), all energy is potential. The total energy is proportional to the **square of the amplitude**, and the time-averaged kinetic and potential energies are each equal to E/2.

<image>Panel A: A mass on a spring shown at three positions — maximum compression (all PE, no KE), equilibrium (all KE, no PE), and maximum extension (all PE, no KE). Energy bar charts (KE and PE bars) are shown at each position, with total E constant. Panel B: A graph showing K(t) and U(t) as functions of time over two periods. K and U are both cos-squared and sin-squared curves that are complementary — when one is maximum, the other is zero. The total E = K + U is a constant horizontal line at (1/2)kA^2.</image>

### VI. Damped Oscillations

Real oscillators lose energy to friction or drag, causing the amplitude to decrease over time. In **damped SHM**, the damping force is proportional to velocity and opposes motion: F_damping = -b v. The equation of motion becomes m d^2x/dt^2 + b dx/dt + kx = 0, with solution x(t) = A_0 e^(-bt/2m) cos(omega' t + phi). The amplitude decays exponentially as A(t) = A_0 e^(-bt/2m), and the frequency shifts slightly to omega' = sqrt(omega_0^2 - (b/2m)^2).

Three regimes of damping exist. The **underdamped** case (b < 2m omega_0) produces oscillations with gradually decaying amplitude. **Critical damping** (b = 2m omega_0) returns the system to equilibrium as quickly as possible without oscillating. The **overdamped** case (b > 2m omega_0) produces a slow, non-oscillatory return to equilibrium.

### VII. Driven Oscillations and Resonance

A **driven (forced) oscillator** has an external periodic force applied: F_ext = F_0 cos(omega_d t). After initial transients die out, the system oscillates at the **driving frequency** omega_d. **Resonance** occurs when the driving frequency matches the natural frequency, omega_d = omega_0 = sqrt(k/m). At resonance, the amplitude becomes very large, limited only by damping. The amplitude as a function of driving frequency has a peak at omega_0, which is tall and narrow for low damping and shorter and broader for high damping.

Resonance is ubiquitous in physics and engineering. Pushing a swing at its natural frequency, the wind-driven collapse of the Tacoma Narrows Bridge, nuclear magnetic resonance in MRI, and shattering a wine glass with sound at its resonant frequency are all manifestations of this phenomenon.

<image>A graph of steady-state amplitude vs. driving frequency omega_d for a driven damped oscillator. Multiple curves are shown for different damping coefficients b. For low damping, the peak is tall and narrow, centered at the natural frequency omega_0. For higher damping, the peak is shorter and broader. The natural frequency omega_0 is marked with a vertical dashed line. An annotation explains that at resonance (omega_d = omega_0), maximum energy transfer occurs from the driver to the oscillator.</image>