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# Oscillations

The study of collective phenomena—more is different (but also the same).

Simple triatomic molecules, such as \(\ce{CO2},\) consist of three bonded atoms arranged in a line. We will model the three atoms classically as point masses interacting via springs. We assume the molecule has two atoms of one species and one of another, like \(\ce{CO2}.\)

The motion of a mass connected to a spring is oscillatory. An oscillator is said to be **coupled** to another when its acceleration depends on its own position and also the positions of **other oscillators**. To which other masses is the left atom coupled?

In this quiz, by constructing a flexible solution proposal (an **ansatz**), we will discover oscillating solutions to the equations of motion describing two coupled oscillators.

Let's first recall the general solution for an isolated harmonic oscillator, like a pendulum in the small-angle approximation.

The equation of motion for \(\theta\) is the equation for simple harmonic motion \[\ddot{\theta} = -\omega_0^2 \theta,\] where \(\omega_0\) is the natural frequency of the oscillator.

What is a general solution to this equation, which describes the resulting motion from any initial condition?

**Normal modes** are a unique set of oscillatory states in which all objects in a system vibrate harmonically at the same frequency.

Thinking in terms of normal modes greatly simplifies our analysis of systems of linearly coupled oscillators. However, the normal modes of a system are not immediately obvious—they're hidden in the governing equations.

The key to finding them rests on a defining property of simple harmonic motion.

Suppose you twice measure the **oscillation period** of a simple pendulum

A) after you pull it back through an angle of \(\theta_0\) and release it from rest;

B) after you pull it back through an angle of \(\theta_0\) and push it gently as you let go.

How do your measurements compare? Assume the small-angle approximate is valid during both subsequent oscillations.

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