Table Of ContentFeynman Simplified
1B: Harmonic Oscillators,
& Thermodynamics
Everyone’s Guide
to the
Feynman Lectures on Physics
by
Robert L. Piccioni, Ph.D.
Second Edition
Copyright © 2016
by
Robert L. Piccioni
Published by
Real Science Publishing
3949 Freshwind Circle
Westlake Village, CA 91361, USA
Edited by Joan Piccioni
All rights reserved, including the right of
reproduction in whole or in part, in any form.
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www.guidetothecosmos.com
Everyone’s Guide to the
Feynman Lectures on Physics
Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics.
Caltech Professor and Nobel Laureate Richard Feynman was the greatest scientist since Einstein. I
had the amazing opportunity to learn physics directly from the world’s best physicist. He had an
uncanny ability to unravel the most complex mysteries, reveal underlying principles, and profoundly
understand nature. No one ever presented introductory physics with greater insight than did Richard
Feynman. He taught us more than physics — he taught us how to think like a physicist.
But, the Feynman Lectures are like “sipping from a fire hose.” His mantra seemed to be: No
Einstein Left Behind. He sought to inspire “the more advanced and excited student”, and ensure “even
the most intelligent student was unable to completely encompass everything.”
My goal is to reach as many eager students as possible and bring Feynman’s genius to a wider
audience. For those who have struggled with the Big Red Books, and for those who were reluctant to
take the plunge, Feynman Simplified is for you.
Physics is one of the greatest adventures of the human mind — everyone can enjoy exploring nature.
To further Simplify this adventure, I have written an eBook explaining all the math needed to master
Feynman physics. Click here for more information about Feynman Simplified 4A: Math for
Physicists.
Additionally, an index for the Feynman Simplified series is in progress, with the latest edition
available for free here.
This Book
Feynman Simplified: 1B covers about a quarter of Volume 1, the freshman course, of The Feynman
Lectures on Physics. The topics we explore include:
Harmonic Oscillators, Resonances, and Transients
Kinetic Theory of Gases
Statistical Mechanics
Thermodynamics
Feynman Simplified makes Feynman’s lectures easier to understand without watering down his
brilliant insights. I have added reviews of key ideas at the end of each chapter and at the end of each
major section.
Feynman Simplified is self-contained; you need not go back and forth between this book and his.
But, for those who wish to read both, I provide extensive cross-references: V1p12-9 denotes his
Volume 1, chapter 12, page 9. If, for example, you have trouble with Feynman’s description of
reversible machines in Volume 1 page 4-2, simply search Feynman Simplified for V1p4-2. Some
material is presented in a different sequence — the best way to divide topics for one-hour lectures is
not necessarily the best way to present them in a book.
Many major discoveries have been made in the last 50 years; Feynman Simplified augments and
modifies his lectures as necessary to provide the best explanations of the latest developments. Links
to additional information on many topics are included.
To find out about other eBooks in the Feynman Simplified series, click HERE.
I welcome your comments and suggestions. Please contact me through my WEBSITE.
If you enjoy this eBook please do me the great favor of rating it on Amazon.com or BN.com.
Table of Contents
Chapter 12: Harmonic Oscillators
Chapter 13: Resonances
Chapter 14: Transients & Linear Systems Review
Chapter 15: Kinetic Theory of Gases
Chapter 16: Statistical Mechanics
Chapter 17: Brownian Motion
Chapter 18: Kinetic Theory At Equilibrium
Chapter 19: Kinetic Theory Near Equilibrium
Chapter 20: “Newton, We Have A Problem!”
Chapter 21: Thermodynamic Laws
Chapter 22: Thermodynamic Applications
Chapter 23: Irreversibility & Entropy
Chapter 24: Thermodynamics Review
Chapter 12
Harmonic Oscillators
In V1p21-1 Feynman says:
“In the study of physics…a strange thing occurs again and again: the equations which appear in
different fields of physics, and even in other sciences, are often almost exactly the same, so that
many phenomena have analogs in these different fields. So the study of a phenomenon in one
field may permit an extension of our knowledge in another field.”
“The harmonic oscillator, which we are about to study, has close analogs in many other fields;
although we start with a mechanical example of a weight on a spring…we are really studying a
certain differential equation. This equation appears again and again in physics and other
sciences…[including] charge flowing back and forth in an electrical circuit; the vibrations of a
tuning fork which is generating sound waves; the analogous vibrations of the electrons in an
atom, which generate light waves;…a thermostat trying to adjust a temperature;…the growth of a
colony of bacteria…; foxes eating rabbits eating grass, and so on.”
The equations governing all these phenomena are called linear differential equations with constant
coefficients. The words “linear" and “constant” make these the simplest of all differential equations
— a good place to start. The form of equations of this type is:
f(t) = a x + a dx/dt + … + a dnx/dtn
0 1 n
where t is the independent variable, x is the dependent variable, all a are constant, and n is the
j
order of this linear differential equation. This is called a differential equation because it contains
derivatives.
The Harmonic Oscillator
The simplest mechanical example of behavior governed by a linear differential equation is a mass on
a spring, illustrated in Figure 12-1. Here m is the mass, x is the vertical height, and x = 0 is the
equilibrium height, the height at which that mass can rest motionless on this spring.
Figure 12-1 Mass on a Spring
We will assume the spring is ideal, perfectly elastic, and obeys Hooke’s law, which means the force F
exerted by the spring is given by:
F = – k x
Here, k is the spring constant and x is the displacement from the equilibrium position. The minus sign
signifies that the force opposes the displacement: if the mass moves to +x, the spring’s force is
directed toward –x, and vice versa. The differential equation is then:
F = ma = –kx
d2x/dt2 = –(k/m) x
From Chapter 6, we recall two related functions, sine and cosine, whose second derivatives are
proportional to minus themselves. Specifically,
d2 (sinωt) /dt2 = –ω2 sinωt
d2 (cosωt) /dt2 = –ω2 cosωt
Indeed, the sine and cosine functions are identical except for a phase shift: sin(ø) = cos(ø–π/2).
Either function fits our need; using sine will require less writing if the mass is at x=0 at t=0, while
using cosine will require less writing if the velocity of the mass is zero at t=0.
Let’s pick the cosine, and start with a mass that is stationary at displacement x=A, and is released at
t=0.
x = A cosωt
d2x/dt2 = –ω2 x
ω2 = k/m
Note that A could have any value and satisfy the same equation. This is what we mean by a linear
differential equation. The series of terms in our original differential equation had derivatives of
various orders, but each was proportional to x0 or x1, not x2 or √x or any other power. So if we
multiply x by any constant A, Ax will satisfy all the same linear differential equations as does x. This
is not true for the independent variable t: if we divide t by 2, dx/dt will double and d2x/dt2 will
quadruple — t/2 will not satisfy all the same equations as does t.
Due to the cosine term, x oscillates up and down. The mass will start at x=A, drop through x=0, all
the way down to x=–A, stop there for an instant, rise again, pass through x=0, and return to a
momentary pause at x=A.
That cycle will repeat indefinitely. A is called the amplitude of the oscillation. The period of the
oscillation is how much time is required for the mass to complete one full cycle; that time is t = 2π/ω
= 2π√(m/k), since the cosine function repeats every 2π radians. For a given m and k, the period of
oscillation never changes; hence the name harmonic.
Note that the mass oscillates with the same period regardless of the amplitude of oscillation. If we
compress the spring twice as much, the force doubles, the acceleration doubles, the distance traveled
in one second doubles, which exactly balances the fact that the mass has twice as far to travel to
complete its cycle.
The factors that determine the oscillation period are m and k. A greater mass is harder to move; it
slows the motion and lengthens the period. Quadrupling the mass doubles the period, due to the
square root. Conversely, a stronger spring has a larger k, exerting a greater force and reducing the
period. Quadrupling k halves the period.
There is one more “knob” to play with: a phase shift. We saw earlier that subtracting π/2 from the
argument of the cosine function transforms it into the sine function. We can add or subtract any
constant angle ø. The equation:
x = A cos(ωt+ø)
allows us to describe any starting position and velocity, what we call initial conditions. This
equation can be expanded according to the usual rules of trigonometry:
x = (Acosø) cosωt – (Asinø) sinωt
Depending on application and personal preference, one can write this equation in any of the following
equivalent ways:
x = C cosωt + B sinωt
x = A cos(ωt+ø)
x = A cos(ω [t–t] )
0
In the first equation above √(C2+B2) = A the amplitude of oscillation. In all three, ω is called the
angular frequency of oscillation, which is measured in radians per second. If j complete oscillations
occur per second, ω equals 2π j.
Initial Conditions V1p21-4
Any specific linear differential equation can be solved by many different equations. We noted earlier
that their linearity ensures that Ax solves any equation that is solved by x. We also noted that sine
functions solve the mass-on-spring equation when the starting height is zero, while cosines solve the
equation when the starting velocity is zero.
The three forms of the solution listed above are each able to describe any mass moving on any ideal
spring, with any starting position and velocity. What we need to do next, therefore, is to learn how to
connect these initial conditions to the adjustable constants in our three solutions.
Let’s take the first solution as an example and take its time derivative:
x(t) = C cosωt + B sinωt
v(t) = –Cω sinωt + Bω cosωt
These two equations provide the position and velocity at any time t. The spring determines the
acceleration at any time t:
a(t) = –k x(t) / m
Let’s assume we know x(0) and v(0), the initial conditions at time t=0. We can then calculate C and B.
C = x(0)
B = v(0) / ω
Similarly, for the second solution:
x(t) = A cos(ωt+ø):
v(t) = –Aω sin(ωt+ø)
if x(0) equals 0:
ø = π/2
A = –v(0) / ω
if x(0) is not 0:
tanø = –v(0) / [ω x(0)]
A = x(0) / cosø
For the third solution:
x(t) = A cos(ω[t–t])
0
v(t) = –Aω sin(ω[t–t])
0
if x(0) equals 0:
t = π / 2ω
0
A = v(0) / ω
if x(0) is not 0:
