# Physics Tutorials

Here contain all the expert written technical physics tutorials for all physics areas. These are technical how-to articles that focus on teaching you a specific skill or how to solve a specific problem. From classic mechanics to general relativity. Self study and classroom strategy. Learn something new today step by step!

### An Alternate Approach to Solving 2 Dimensional Elastic Collisions

/
Introduction This article follows on from the previous on an alternate approach to solving collision problems. In that article we determined the equal…

### How to Recognize Split Electric Fields

/
Introduction In a previous Insight, A New Interpretation of Dr. Walter Lewin’s Paradox, I introduced the fact that there are two kinds of E fields. …

/

### An Alternative Approach to Solving Collision Problems

/
Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles…

### Maxwell's Equations in Magnetostatics and Solving with the Curl Operator

/
Introduction: Maxwell's equation in differential form ## \nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}  ##  with ## \dot{\vec{E}}=0…

### Learn the Basics of Dimensional Analysis

/
As a university teacher and as a PF member, I have often noted that students are largely unaware of or not using dimensional analysis to help them in their…

### The Classical Limit of Quantum Mechanical Commutator

/
The Classical Limit of Commutator (without fancy mathematics) Quantum mechanics occupies a very unusual place among physical theories: It contains classical…

### Precession in Special and General Relativity

/
The Absolute Derivative In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which…

### A Formal Definition of Large-Scale Isotropy

/
This Insight is part of my attempt to develop a formal definition of 'large-scale isotropy', a concept that is fundamental to most cosmology, but that…

### Fermi-Walker Transport in Kerr Spacetime

/
In the last two posts in this series, we developed some tools for looking at Fermi-Walker transport in Minkowski spacetime and then applied them in Schwarzschild…

### Fermi-Walker Transport in Schwarzschild Spacetime

/
In the first post in this series, we introduced the concepts of frame field, Fermi-Walker transport, and the "Fermi derivative" of a frame field, and developed…

### Fermi-Walker Transport in Minkowski Spacetime

/
This is the first of several posts that will develop some mathematical machinery for studying Fermi-Walker transport. In this first post, we focus on Minkowski…

### An Example of Servo-Constraints in Mechanics

/
Servo-constraint was invented by Henri Beghin in his PhD thesis in 1922. For details see the celebrated monograph in rational mechanics by Paul Appell.To…

### Orbital Mechanics in Unity Game Engine for Augmented Reality

/
In this post I’ll go over implementing basic orbital mechanics simulations in the Unity game engine as well as an approach to scaling the simulation…

### A New Interpretation of Dr. Walter Lewin's Paradox

/
Much has lately been said regarding this paradox which first appeared in one of W. Lewin's MIT lecture series on ##{YouTube}^{(1)}##.  This lecture was…

### Calculating the Spin of Black Hole Sagittarius A*

/
This Insight takes a look at how it is possible to calculate the spin of Sagittarius A*, the supermassive black hole at the centre of the Milky Way using…

### Solving Einstein's Field Equations in Maxima

/
A few months ago, pervect pointed me to a post by Chris Hillman which is an introduction to the usage of Maxima for General Relativity. Maxima is a free…

### Rindler Motion in Special Relativity: Rindler Coordinates

/
Our destination In our last article, Hyperbolic Trajectories, we derived some facts about the trajectory of a rocket that is undergoing constant (proper)…

### Rindler Motion in Special Relativity: Hyperbolic Trajectories

/
Introduction: Why Rindler Motion? When students learn relativity, it's usually taught using inertial (constant velocity) motion. There are lots of reasons…

### Statistical Mechanics: Equilibrium Systems

/
This is the first of a multi-part series of articles intended to give a concise overview of statistical mechanics and some of its applications. These articles…

### The Homopolar Generator: An Analytical Example

/
Introduction It is surprising that the homopolar generator, invented in one of Faraday's ingenious experiments in 1831, still seems to create confusion…

### An Integral Result from Parseval's Theorem

/
Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…

### Relativity Using the Bondi K-calculus

/
Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity…

### Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity

/
Traditional presentations of special relativity place emphasis on "velocity", which of course has an important physical interpretation... carried over…

### Frames of Reference: Linear Acceleration View

/
My previous Insight, Frames of Reference: A Skateboarder's View, explored mechanical energy conservation as seen from an inertial frame moving relative…

### Fabry-Perot and Michelson Interferometry: A Fundamental Approach

/
Fabry-Perot Effect: The Fabry-Perot effect is usually treated in most optics textbooks as the interference that results from multiple reflections of the…

### The Schwarzschild Geometry: Physically Reasonable?

/
In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres…

### The Schwarzschild Geometry: Coordinates

/
At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates: ds^2…

### The Schwarzschild Geometry: Key Properties

/
Not long after Einstein published his Field Equation, the first exact solution was found by Karl Schwarzschild. This solution is one of the…

### Explaining How Rolling Motion Works

/
Although rolling wheels are everywhere, when most people are asked "what is the axis of rotation of a wheel that rolls without slipping?", they will answer…

### Grandpa Chets Entropy Recipe - Determining the Change in Entropy

/
How do you determine the change in entropy for a closed system that is subjected to an irreversible process?Here are some typical questions we get…

### Orbital Precession in the Schwarzschild and Kerr Metrics

/
The Schwarzschild Metric A Lagrangian that can be used to describe geodesics is $F = g_{\mu\nu}v^\mu v^\mu$, where $v^\mu = dx^\mu/ds$…

/
A spacetime is often described in terms of a tetrad field, that is, by giving a set of basis vectors at each point. Let the vectors of the tetrad be denoted…

### Learn About Relativity on Rotated Graph Paper

/
This Insight is a follow up to my earlier tutorial Insight (Spacetime Diagrams of Light Clocks). I gave it a different name because I am placing more…

### Basic Kinematics in Classical Mechanics

/
There is an interesting thing in teaching of Classical Mechanics. Several theorems which presented below form a core part of kinematics for all Russian…

### Presenting a Rare Kinematic Formula

/
Here we present some useful kinematic fact which is uncommon for textbooks in mechanics. Consider a convex rigid body (RB) rolling without slipping…

### Elementary Construction of the Angular Velocity

/
Physics books  seldom contain accurate definition of the angular velocity of a rigid body. I believe that the following construction is as simple as possible…

### Frequently Made Errors in Vectors - Elementary Use

/
A vector has magnitude and direction. Pictorially, a vector can be imagined as a location in n-dimensional space relative to some fixed origin. …

### LightCone7 Tutorial Part III - How Things are Computed

/
In Part I and Part II of this mini-series, we have briefly discussed the basic user interface and the use of charts to depict the LCDM cosmological model.…

### LightCone7Combo Tutorial Part II - Charts

/
Part I dealt with the basic user interface of LightCone7Combo. This part of the tutorial is about potentially useful cosmological insights to be gained…

### LightCone7Combo Tutorial Part I

/
LightCone 7 Combo is a versatile tabulating/charting cosmological calculator, useful for understanding the expansion history of the universe (and even…

/

### A Short Proof of Birkhoff's Theorem

/
Birkhoff's theorem is a very useful result in General Relativity, and pretty much any textbook has a proof of it. The one I first read was in Misner, Thorne,…

### Frequently Made Errors in Heat: Elementary Level

/
1. Heat, Work, Internal Energy and Kinetic Energy "If heat is the motion of molecules, why isn't it Kinetic Energy?"In everyday use, we may think…

### Frequently Made Errors: Pseudo and Resultant Forces

/
1. Real versus Fictitious Pseudo, or "fictitious", forces can arise when a non-inertial frame of reference is used. Using a non-inertial frame…

### Why Renormalisation Needs a Cutoff

/
Introduction This is a follow on from my paper explaining renormalisation. A question was raised - why exactly do we need a cut-off. There is a deep reason…

### Frequently Made Errors in Mechanics: Springs

/
1. Springs in Series "A spring of constant ##k_1## is connected in series with a spring of constant ##k_2##.  What is the spring constant…

### Frequently Made Errors in Mechanics: Momentum and Impacts

/
An impact is an impulse (change of momentum) that involves arbitrarily large forces acting very briefly. These result in near-instantaneous…

### Frequently Made Errors in Mechanics: Hydrostatics

/
1. Archimedes' Principle X "When a body is placed in a liquid, the weight of the body equals the weight of the liquid displaced"That will…

### A Partial "Derivation" of Gauss's Law

/
Gauss's law was formulated by Carl Friedrich Gauss in 1835. It is one of the four Maxwell's equations that form the basis of classical electrodynamics.…

### Frequently Made Errors in Equation Handling

/
1. Algebra versus Arithmetic When numerical values are provided as inputs in a question, it is tempting to plug these into the equations…

### Frequently Made Errors in Mechanics: Kinematics

/
Kinematics is the subset of dynamics that only concerns itself with time, displacement, velocity and acceleration. A problem is…

### Frequently Made Errors in Mechanics: Moments

/
The term "moment" is used in various ways in Physics and Mathematics:Given a force and a reference point, the force has a moment (or…

/
What Is The Issue With Renormalisation If you have an interest in physics you have likely come across renormalisation before, although what it really…

### Misconceiving Mutual Inductance Coefficients

/
A commonly used formula for mutual inductance M between two nearby coils L1 and L2 is M = k√(L1*L2). This formula however assumes equal percentage…

### Frequently Made Errors in Mechanics - Friction

/
1. Direction of the normal Definition: The normal force that body A exerts on body B is that force of minimum magnitude which suffices to…

### Frequently Made Errors in Mechanics: Forces

/
Notation: In this page, a circumflex signifies an average. 1. Forces as vectors A force is a vector, i.e. has magnitude and direction.…

### Visualizing the 2-D Particle in a Box

/
Introduction The particle in a box is a staple of entry-level Quantum Mechanics classes because it provides a meaningful contrast between classical and quantum…

### Understanding Entropy and the 2nd Law of Thermodynamics

/
Introduction The second law of thermodynamics and the associated concept of entropy have been sources of confusion to thermodynamics students for centuries.…