Basics of Mechanics

Notes on motion and forces. I once read a book called "Nothing but Motion". I can't recall if the book was even coherent (it was a long time ago) but the idea has stayed with me, that the only thing we seem to experience is change. The development of a systematic method to study change came about surprisingly late in human history. Actually, that really only refers to the modern form of that method since previous generations developed a vast amount of knowledge leading up to the systematic study of motion, but it took a conceptual leap to bring us to where we are now. This tutorial or primer serves as my notes on the study of classical mechanics. It is mostly based on the notes I took during grad school, but the beauty of the information age is that not even information is static, so this will change as my understanding of mechanics is improved. Oh, of course, I should apologize to those that are seeing this before it is anywhere near completion, but then, I'm not sure completion is even an option. Enjoy.

• Physical Measurements
• Units & Dimensions
• Sidenote: Precision & Accuracy
• Vectors
• Newton's Laws
• Particle Motion
• One Dimensional Motion
• Uniform Motion
• Two Dimensional Motion
• Projectiles
• Pendulums
• Circular Motion
• Angular Momentum
• Torque
• Groups of Particles
• Center of Mass
• Rigid Bodies
• Analytical Approach
• Lagrangian
• Hamiltonian
• Conservation Principles
• Action

Physical Measurements

Physics is all about basic physical quantities such as length, mass, and time. We "understand" these basic quantities as comparisons to standards which is how we meaasure them. We compare lengths or "distances" to a standard length such as a meter or an inch for example by determining how many of these standard lengths laid end to end will reach the length or distance we want to measure. We begin by assuming that our standard is constant and divisible into an infinitely fine fraction of that standard. Later, we might attempt to account for or compensate for all the possible deviations from such assumptions, for example the variation of our standard or measuring instrument due to variation in such things as temperature, pressure, the sag of a tape measure or the tape measure's variation from the actual standard. But, that will be part of much later discussion. For now, let's look at the other two basic quantities mentioned above. When we measure mass, we are usually comparing the effect of a gravitational field (like the one we all live in here on earth) on the mass we want to measure to the effect of that same field on a standard mass, a gram or a kilogram or "stone" for instance. In the U.S. we often skip from mass directly to the effect of the gravitational field and measure in lbs (pounds). 1 stone of mass will experience 14 lbs of force due to gravity on the surface of the earth. Finally, time, such as the time between two events, is measured as a comparison to a number of cycles of something we think of as cyclical with a constant frequency, such as the swing of a pendulum, or the cycles of a watch spring, or the vibrations of a quartz crystal, or more recently, the standard measure of time, the second [s], is defined as the length of time for about 9 billion oscillations of the caesium atom.

There are good reasons to worry about our definitions of basic physical quantities, for example: Inertial vs gravitational mass; Is length actually a basic quantity or is the speed of light more properly basic so that length should be compared to this in some way; And, is time or frequency the more basic concept. For now, we'll assume the traditional basic quantities:

Physics is also concerned about derivations of these basic quantities. Area for example, is a length multiplied by another length, Volume is an area multiplied by another length. Speed is a rate of change of length or "a change of length divided by a change of time", Acceleration is a rate of change of speed or a "change of speed divided by a change of time", Force is an acceleration imparted to a given mass, Momentum is a mass times a velocity, (so that force may also be thought of as rate of change of momentum). Going further, Work is force times a length, and Energy is potential work in the form of a momentum times a velocity or some other variation of the same form. But, we will have to distinguish between "potential" energy and "kinetic" energy which will finally lead us to Lagrangian and Hamiltonian formulations of the science of mechanics. We could even look at such complications as Temperature as the average "kinetic" engergy of a collection of particles, but we will save that for a discussion of statistical mechanics and thermal physics, Let's just pretend I didn't even mention temperature or electric charge, or photons, or color charge, or spin or subatomic particles or quantum field theory and just summarize the things we will talk about in the study of mechanics.

Units & Dimensions

Sidenote: Precision & Accuracy

Vectors

Newton's Laws

Particle Motion

One Dimensional Motion

Uniform Motion

Two Dimensional Motion

Projectiles

Pendulums

Circular Motion

Angular Momentum

Torque

Groups of Particles

Center of Mass

Rigid Bodies

Analytical Approach

Lagrangian

Hamiltonian

Conservation Principles

Action