# 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!

### Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

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Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief…

### Corrections to MIT Open Courseware: Systems of Varying Mass

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Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton's Laws to systems of Varying Mass PDF Course Link My concerns…

### Quaternions in Projectile Motion

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Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our…

### How to Solve Projectile Motion Problems in One or Two Lines

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Introduction We show how one can solve most, if not all, introductory-level projectile motion problems in one or maybe two lines. To this end, we forgo…

### Maximizing Horizontal Range of a Projectile

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Introduction A recent homework problem that appeared in the forums was concerned with maximizing the horizontal range of a projectile subject to the launch…

### A Numerical Electromagnetic Solver Using Duality

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In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein…

### The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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This Insight was inspired by the discussion in "electric field seen by an observer in motion", which tries to understand the relation between two expressions:…

### Valentine’s Reflections: Mathematical Matters of the Heart

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Introduction Being a somewhat geeky Maths 'nerd', I spent days leading up to Valentine's day trying to find a Maths function appropriate to the day. In…

### How to Use Duality in Computational Electromagnetic Problems

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Some weeks ago I happened across a post that caught my eye. Dale asked a question about the number of photons in an electromagnetic field. His question…

### Equations of Motion Revisited

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Introduction In any school Physics course, the Newtonian equations of motion are very much a 'stock' item. Students learn the equations and are given…

### How to Master Projectile Motion Without Quadratics

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Introduction In a homework thread a while back a PF member expressed dismay along the lines of "oh no, not another boring projectile motion problem."…

### Intro to Physically Reasonable Waves on a String

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Introduction Physics teachers who are either writing physics questions that deal with waves on a string or setting up equipment for a class lab or demo…

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

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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

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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. …

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### An Alternative Approach to Solving Collision Problems

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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

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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

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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

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

### Understanding Precession in Special and General Relativity

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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

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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…

### Exploring Fermi-Walker Transport in Kerr Spacetime

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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…

### Learning Fermi-Walker Transport in Schwarzschild Spacetime

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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

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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

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

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

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In this Insight, 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

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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…

### How to Calculate the Spin of Black Hole Sagittarius A*

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

### How to Solve Einstein’s Field Equations in Maxima

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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

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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

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

### Learn Statistical Mechanics: Equilibrium Systems

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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…

### Learn an Integral Result from Parseval’s Theorem

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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…

### Learn Relativity Using the Bondi K-calculus

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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

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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

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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

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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?

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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

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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

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

### Learn The Basics of Rolling Motion

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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…

### How to Determine the Change in Entropy

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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…

### Learn Orbital Precession in the Schwarzschild and Kerr Metrics

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

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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

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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…

### Learn Basic Kinematics in Classical Mechanics

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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

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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

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

### Frequently Made Errors in Vectors – Elementary Use

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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

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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

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Part I dealt with the basic user interface of LightCone7. This part of the tutorial is about potentially useful cosmological insights to be gained from…

### LightCone7Combo Tutorial Part I

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

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### Learn A Short Proof of Birkhoff’s Theorem

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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

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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

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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 in Quantum Theory Needs a Cutoff

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

### Frequently Made Errors in Mechanics: Springs

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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

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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

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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…

### Learn a Partial “Derivation” of Gauss’s Law

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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

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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

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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

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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…

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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

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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

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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

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

### How to Visualize the 2-D Particle in a Box

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

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

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