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Multi-Atwood Machine Assembly

How to Solve a Multi-Atwood Machine Assembly

Introduction The figure on the right shows a “double-double” Atwood machine with three ideal pulleys and four masses.  All pulleys are released from rest simultaneously.  Which of the choices below describes the angular motion of the top pulley P after some time has elapsed and why? It rotates clockwise with increasing angular speed. It rotates…

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Lambert W Function in Finance

The Lambert W Function in Finance

Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…

Why Division by Zero is a Bad Idea

A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs: What if it was allowed? Then…
digital audio guide

Digital Filtering and Exact Reconstruction of Digital Audio

Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function The first link in my insights article has…
digital audio intro

Introduction to Digital Audio

IntroductionFirst, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this…
geometric series

Series in Mathematics: From Zeno to Quantum Theory

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…
Epsilontic limits and continuity

Epsilontic – Limits and Continuity

Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I…

The Poor Man’s Milli-Ohm Meter

Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article,…
Differential Equation Systems and Nature

Differential Equation Systems and Nature

Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural…
calc precalc

Beginners Guide to Precalculus, Calculus and Infinitesimals

Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this.In the US, they go through…
Variable Mass Systems

How to Apply Newton’s Second Law to Variable Mass Systems

Introduction The applicability of Newton's second law in the oft-quoted "general form"  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$…
what are numbers

What Are Numbers?

Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going…
world of algebras

Introduction to the World of Algebras

Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…

Why ChatGPT Is Not Reliable

I'll start with the simple fact: ChatGPT is not a reliable answerer of questions.To try to explain why from scratch would be a heavy lift, but fortunately,…

What Are Infinitesimals – Simple Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
quantum information

How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics

In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit…

What Are Infinitesimals – Advanced Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
Carl Sagan

Opinion: When Pro Scientists Explain Using Pop Science

Abstract There is so much to say about the many endeavors by professional scientists to explain to us the world. The list is long: Carl Sagan, Harald…
art of integration

The Art of Integration

Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…
teaching physics

A Lesson In Teaching Physics: You Can’t Give It Away

A central principle of Physics Forums regarding homework help is not to provide solutions on demand but to guide students along a path to the answer.  The…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments…
Measure Internal Resistance of Battery

How to Measure Internal Resistance of a Battery

Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery - typically an AA or D…
lie group physics

When Lie Groups Became Physics

Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
white dwarfs

Why There Are Maximum Mass Limits for Compact Objects

In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Implications

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder…
tensors relativity

What Are Tensors and Why Are They Used in Relativity?

If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational…
twin paradox

When Discussing the Twin Paradox: Read This First

This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads here on…
gravity collapse

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the…
object slide down ramp physics

Subtleties Overlooked in Friction Questions: Object Slides Down Ramp

Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ' on the ramp,…
math classifications

Classification of Mathematics by 42 Branches

 I often read questions about our classification scheme that we use on to sort posts by science fields and subjects, what has…

Reduction of Order For Recursions

This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…
history of numbers

Counting to p-adic Calculus: All Number Systems That We Have

An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim…
evariste galois

Évariste Galois and His Theory

 * Oct. 25th, 1811  † May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave…
definition differences

Yardsticks to Metric Tensor Fields

I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…

Programming an ATmega8A using Arduino

If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino projects permanent,…

P vs. NP and what is a Turing Machine (TM)?

P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and…
quantum computers 101

Quantum Computing for Beginners

Introduction to Quantum Computing This introduction to quantum computing is intended for everyone and especially those who have no knowledge of this relatively…
gauss law misconceptions

A Physics Misconception with Gauss’ Law

Introduction It is relatively common to see the following type of argument: The surface area is ##A## and the enclosed charge is ##Q##. The electric…
model magnet

How to Model a Magnet Falling Through a Conducting Pipe

Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features…
model magnet

How to Model a Magnet Falling Through a Solenoid

Introduction Modeling a magnet realistically is a task best done numerically.  Even the simplified model of two separated disks with uniform surface…
why can we jump

How Can We Jump When the Ground Does No Work?

It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the ground, the ground does…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
magnetism current

Symmetry Arguments and the Infinite Wire with a Current

Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric…
cpu programming

Parallel Programming on a CPU with AVX-512

This article is the second of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
programming gpu

Parallel Programming on an NVIDIA GPU

This article is the first of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
hypercube integral

A Novel Technique of Calculating Unit Hypercube Integrals

Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first…
Riemann Hypothesis

The Extended Riemann Hypothesis and Ramanujan’s Sum

Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All…

The Amazing Relationship Between Integration And Euler’s Number

We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…
mermin device quantum entanglement

Superdeterminism and the Mermin Device

Superdeterminism as a way to resolve the mystery of quantum entanglement is generally not taken seriously in the foundations community, as explained in…
relativity rotated graph

Relativity on Rotated Graph Paper (a graphical motivation)

(based on and…
probabilities virus testing

Probabilistic Factors Involved in Disease and Virus Testing

Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,…