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Oscillations

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

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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.

Master the problem solving skills of Classical Mechanics.

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