Moments of Inertia: Examples

Michael Fowler

Molecules

The moment of inertia of the hydrogen molecule was historically important. It's trivial to find: the nuclei (protons) have 99.95% of the mass, so a classical picture of two point masses a fixed distanceapart gives  In the nineteenth century, the mystery was that equipartition of energy, which gave an excellent account of the specific heats of almost all gases, didn't work for hydrogen -- at low temperatures, apparently these diatomic molecules didn't spin around, even though they constantly collided with each other. The resolution was that the moment of inertia was so low that a lot of energy was needed to excite the first quantized angular momentum state, . This was not the case for heavier diatomic gases, since the energy of the lowest angular momentum state  is lower for molecules ... Read more »

Motion of a Rigid Body: the Inertia Tensor

Michael Fowler

Definition of Rigid

We're thinking here of an idealized solid, in which the distance between any two internal points stays the same as the body moves around. That is, we ignore vibrations, or strains in the material resulting from inside or outside stresses. In fact, this is almost always an excellent approximation for ordinary solids subject to typical stresses -- obvious exceptions being rubber, flesh, etc. Following Landau, we'll usually begin by representing the body as a collection of particles of different massesheld in their places  by massless bonds. This approach has the merit that the dynamics can be expressed cleanly in terms of sums over the particles, but for an ordinary solid we'll finally take a continuum limit, replacing the finite sums over the constituent particles by integrals over a continuous mass distribution.

Rotation of a Body about a Fixed Axis

As a preliminary, let's look at a body firmly attached to a rod fixed in space, and rotating with angular veloci ... Read more »

Anharmonic Oscillators

Michael Fowler

Introduction

Landau (para 28) considers a simple harmonic oscillator with added small potential energy terms . We'll simplify slightly by dropping theterm, to give an equation of motion

(We'll always takepositive, otherwise only small oscillations will be stable.)

We'll do perturbation theory (following Landau):

(Standard practice in most books would be to write  with the superscript indicating the order of the perturbation -- we're following Landau's notation, hopefully reducing confusion...)

We take as the leading term

... Read more »

 Motion in a Rapidly Oscillating Field: the Ponderomotive Force

Michael Fowler

Introduction

Imagine first a particle of massmoving along a line in a smoothly varying potential , so  Now add in a rapidly oscillating force, not necessarily small, acting on the particle:

where  are in general functions of position. This force is oscillating much more rapidly than any oscillation of the particle in the original potential, and we'll assume that the position of the particle as a function of time can be written as a sum of a "slow motion"  and a rapid oscillation ,

... Read more »

 Parametric Resonance

Michael Fowler

Introduction

(Following Landau para 27)

A one-dimensional simple harmonic oscillator, a mass on a spring,

has two parameters,  and  For some systems, the parameters can be changed externally (an example being the length of a pendulum if at the top end the string goes over a pulley).

We are interested here in the system's response to some externally imposed periodic variation of its parameters, and in particular we'll be looking at resonant response, meaning large response to a small imposed variation.

Note first that imposed variation in the mass term is easily dealt with, by simply redefining the time variable to , meaning 

Then

... Read more »