Tag Archive for: Undergraduate

partialfractions

Introduction to Partial Fractions Decomposition

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3 Comments
Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum…
micro3

Set-Theoretic Foundations of Numbers and Functions

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Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets.…
earthgravity

Learn All About Earth’s Gravity

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18 Comments
Introduction Earth's gravitational field at the surface is approximately 9.8 Newtons/kilogram or 9.8 meters/second/second. But how does that change due…
earlyuniverse

Was the Early Universe in a Disordered State?

No. The second law of thermodynamics says that entropy can only increase, so if the early universe had been in a state of maximum entropy, then the cosmos…
blackholeexists

Do Black Holes Really Exist?

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68 Comments
Introduction The purpose of this article is to discuss the title question from several different viewpoints, to show that it isn't as simple as it looks.…
speedoflight1

Learn About the Speed of Light and Galilean Relativity

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39 Comments
Introduction In this article, I will discuss some experiments in the 19th and early 20th centuries that looked at how the velocity of a light source affects…
buoyancy

Understanding Buoyancy: The Limits of Archimedes

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18 Comments
It’s a principle as old as science itself, and remarkably, remains essentially unchanged in the light of revolution after revolution in our understanding…
observableuniverse

Why the Observable Universe Radius Exceeds Its Age

The radius of the observable universe is about 46 billion light-years, which is considerably greater than its age of about 14 billion years. The radius…
heat

What Is Heat? Definition in Thermodynamics Explained

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13 Comments
One of the most frustrating misconceptions in thermodynamics centers on the actual definition of heat. Many science and engineering students refer to a…
ODE2

How to Solve Nonhomogeneous Linear ODEs using Annihilators

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3 Comments
My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations…
steadystate

What Is the Steady-state Model and Why It’s No Longer Viable

Introduction to the Steady State Model Back in 2011, I wrote a FAQ entry for Physics Forums on the steady state model and why it is no longer viable.…
ODE1

Solving Homogeneous Linear ODEs using Annihilators

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8 Comments
In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although…
lineartransformations

Learn About Matrix Representations of Linear Transformations

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19 Comments
Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…
electronatom

Why Electrons Don’t Crash Into the Nucleus — Quantum View

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12 Comments
Atoms and Coulomb forces If atoms were described only by Coulomb attraction, the electron and nucleus would attract each other and, according to classical…
blackholesuniverse

Why the Early Big Bang Universe Didn’t Form a Black Hole

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7 Comments
Why the Big Bang Is Not a Black Hole In the early universe, the matter was gathered together at very high density — so why wasn't it a black hole?Big…
electrostaticfield1

Learn About Energy Gained by Charge in an Electrostatic Field

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2 Comments
Energy gain of an electron in a potential V I have seen this question being asked frequently on here. A lot of students have a bit of an issue in understanding…
studymath2

How to Self-Study Calculus: Topics, Order & Book Guide

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25 Comments
Click For Complete SeriesHow to Study Mathematics How to Self-Study Basic High School Mathematics How to Self-Study CalculusWe often get questions…
peterblockuniverse

The Block Universe – Refuting a Common Argument

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153 Comments
The "block universe" interpretation of SR has come up repeatedly in threads here on PF. Rather than link to them, I want to summarize a common argument…
twinparadox

A Geometrical View of Time Dilation and the Twin Paradox

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90 Comments
Based on the number of questions we receive on the topic at Physics Forums, there is a lot of confusion in the general public about how time dilation works…
universespinning

Is the Universe Rotating? Evidence, Tests & Limits

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3 Comments
Can we tell whether the universe is rotating? If you believe wholeheartedly in Mach's principle, there is no way to test empirically for the rotation…
blockuniverse4

Blockworld and its Foundational Implications: General Relativity and Closed Timelike Curves

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2 Comments
 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). In part 3, I introduced…
blockuniverse3

Blockworld and its Foundational Implications: General Relativity and the Big Bang

 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). Geroch sums up the…
uncertaintyenergy

Does Heisenberg Uncertainty Violate Energy Conservation?

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The short answer is: No — there is no violation. The longer answer Given the probabilistic aspect of quantum theory, what do we mean now by "conservation…
animationwalk

Fun with Self-Avoiding Walks Simulations

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16 Comments
This post is about some simulations I did of self-avoiding random walks. These are what they sound like with each step, the position of the walk moves…
photonsmedium

Do Photons Move Slower in a Solid Medium?

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5 Comments
Scope and context This question appears often because it has been shown that in a normal, dispersive solid such as glass, the speed of light is slower…
blackbody

A Poor Man’s CMB Primer: Bumps on a Blackbody

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 Astronomers Arno Penzias and Robert Wilson discovered the cosmic microwave background in 1965. They were not looking for it.They were…
redshiftblackholes

Time Dilation & Redshift of Schwarzschild Black Holes

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76 Comments
Time dilation near a Schwarzschild black hole The following is an overview of the time-dilation and gravitational-redshift effects of a static (Schwarzschild)…
lightspeedrod

Why Rigid Rods Can’t Send Faster-Than-Light Signals

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52 Comments
Why a rigid rod can't send faster-than-light signals One common proposal for achieving faster-than-light (FTL) communication is to use a long, perfectly…
proofs

Mathematical Proofs: How to Understand and Write Them

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28 Comments
Introduction This FAQ is about proofs. Proofs are central to mathematics, and writing proofs is a skill many people find hard to master. There are two…
birthcmb

A Poor Man’s CMB Primer: The Birth of a Cosmic Background Radiation

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3 Comments
 The early universe was hot. So hot that nuclei boiled. The great thermal energy of the universe overwhelmed the confining efforts of the…
modulation

Mixing vs Beating: Superheterodyne & Beat Frequency

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36 Comments
Mixing vs. Beating: a brief history A long time ago I read a paper in the IEEE Proceedings recounting the history of the superheterodyne receiver. Overall…
bubbles

Explosion-Generated Collapsing Vacuum Bubbles Reach 20,000 Kelvin

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5 Comments
Experiment overview In a recent paper published in Physical Review Letters (PRL), one of the top physics journals, Jérôme Duplat and Emmanuel Villermaux…
cardinalnumbers

Informal Introduction to Cardinal Numbers

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Cardinal numbers We will now give an informal introduction to cardinal numbers. We will later formalize this by using ordinal numbers. Informally, cardinal…
cmb

A Poor Man’s CMB Primer: Orientation of the Universe

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4 Comments
 This is a picture of the cosmic microwave background:Fig 1. The cosmic microwave background as seen by the European Space Agency's…
speedoflight

Why Does C Have a Particular Value, and Can It Change?

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53 Comments
Short answer: Because c (speed of light) has units, its value is what it is only because of our choice of units, and there is no meaningful way to test…
planck

What Planck Length Is and It’s Common Misconceptions

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30 Comments
The Planck length is an extremely small distance constructed from physical constants. Many misconceptions overstate its physical significance, claiming…
gravitywaves

How Fast Do Changes in the Gravitational Field Propagate?

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20 Comments
Overview General relativity predicts that disturbances in the gravitational field propagate as gravitational waves, and that low-amplitude gravitational…
animal speeds

Animal Speed Scaling: Body-Lengths per Second Across Sizes

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19 Comments
Introduction In a recent American Journal of Physics issue, I read an interesting paper by Nicole Meyer-Vernet and Jean-Pierre Rospars examining the top…
infinity_faq

Infinity in Mathematics: Limits and Cardinality FAQ

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17 Comments
Introduction Understanding the behavior of infinity is one of the major accomplishments of mathematics. However, the infinite is often misunderstood and…
photoelectric

“Violating” Einstein’s Photoelectric Effect Model

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Einstein's Photoelectric Insight One of the most spectacular theoretical descriptions that Einstein had ever produced is the corpuscular nature of light…
999

Rigorous Proof: Why 0.999… Equals 1 (Geometric Series)

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101 Comments
Yes. What 0.999... Means First, we have not addressed what 0.999... means. So it is best to first describe what the notation [tex]b_0.b_1b_2b_3...[/tex]…
CosmologicalHorizons

Inflationary Misconceptions and the Basics of Cosmological Horizons

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Introduction It is a common saying that during inflation "space expanded faster than the speed of light."  This statement is meant to articulate the…
relativitytherorm

Learn the Relativistic Work-Kinetic Energy Theorem

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19 Comments
I was bothered for a long time by the reasons for the relativistic validity of the work-kinetic energy relation ##\Delta E=Fd##, which holds without any…
spaceship

Bell Spaceship Paradox: Why the String Breaks in SR

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33 Comments
Bell spaceship paradox — setup Bell describes two spaceships that start out at rest relative to each other with an elastic string between them, one…
hupmisconceptions1

Learn a Misconception of the Heisenberg Uncertainty Principle

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26 Comments
One of the common misconceptions about the Heisenberg Uncertainty Principle (HUP) is that it is the fault of our measurement accuracy.A description…
quantum_tunneling

Why You Can’t Quantum Tunnel Through a Wall

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64 Comments
Can You Quantum Tunnel Through a Wall? Short answer The short and sweet answer if a tennis ball, a bowling ball, or any other kind of ordinary macroscopic…
relmass

What is relativistic mass and why it is not used much?

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It happens that the term relativistic mass is used, in particular in the introductory text on special relativity. It should be noted that whether or not…
errorbars

Plus/minus What? How to Interpret Error Bars

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People sometimes find themselves staring at a number with a ± in it when a new physics result is presented. But what does it mean? This Insight aims to…
LCDM4

Approximate LCDM Expansion in Simplified Math (Part 4)

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  Part 4: Cosmic Recession Rates An astronomer, accompanied by his amateur relativist friend, aimed a telescope at a distant galaxy and measured…
LCDM3

The LCDM Cosmological Model in Simplified Math (Part 3)

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5 Comments
  Part 3: Important Cosmological Horizons and Distances A question that often comes up is: "how big is the observable universe?"The question…
lcdmjorrie

Approximate LCDM Expansion in Simplified Math (Part 2)

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14 Comments
  Part 2: Time and Distance In Part 1 we have seen how one of the simplest ways of expressing the fractional expansion rate of the universe,…
neutrino

Learn About Neutrino Masses and Speed

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28 Comments
It was hard to miss the 2011 OPERA neutrino speed measurement that indicated superluminal neutrino speeds (and turned out to be a measurement error), but…
gravity

Understanding the General Relativity View of Gravity on Earth

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102 Comments
Often students have difficulty reconciling the General Relativity (GR) view of gravity versus their own experience with gravity on the surface of Earth.…
cosmo

Approximate LCDM Expansion in Simplified Math

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11 Comments
 If we restrict ourselves to a spatially flat LCDM universe model (and ignore the early radiation energy density), the first Friedmann equation…
garrett

Garrett Lisi’s Superparticle Bet With Frank Wilczek

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52 Comments
There is an interesting story unfolding in particle physics. The main motivation for building the Large Hadron Collider was to search for new particles.…
oral presentation

Presenting Physics Research: Conference Tips for Students

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1 Comment
Part I: Early Physics Education in High Schools Part II: Surviving the First Year of College Part III: Mathematical Preparations Part IV: The…
masterdegree

How to Write a Master Degree Thesis As a Physics Major

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Your First Scientific Paper Ought to be good! Structure and Contents of a Master's Degree Thesis This guide was written some years ago as a guideline for…
arrowoftime

Retrocausality: How Fundamental is the Arrow of Time?

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26 Comments
In this Insight, I'll discuss a recent experiment ([1]  [2] [3]) by some Australian physicists that has brought a thought experiment into the real world…
gas compression

Reversible vs Irreversible Gas Compression and Expansion Work

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26 Comments
Introduction In learning Thermodynamics, one of the difficult concepts that many students struggle with is the difference between reversible and irreversible…
Distributions

Errors in Probability: Continuous and Discrete Distributions

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1. Classifying as discrete, continuous, or mixed These statements (or equivalents) can be found in authoritative-seeming websites:X "A random variable…
negative temperature

Negative Absolute Temperatures: Explanation & Examples

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It's a famous result in thermodynamics that you cannot reach absolute zero no matter how hard you try. Given the definition of absolute zero, it is also…
Coriolis

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…
LHC

LHC Part 3: Protons as Large as a Barn

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Introduction See also part 1 and part 2 of the series.The LHC commissioning phase is coming to and end. On Wednesday May 20th 2015, the first collisions…
errors springs

Frequently Made Errors in Mechanics: Springs

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3 Comments
  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…
natural language errors

Frequently Made Errors in Probability: Conditionals in Natural Language

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2 Comments
  1. Turning a verbal condition into Algebra An actual thread..."A study of auto accidents has found that 40% of all fatal accidents are…
impact errors

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…
matlab errors

5 Common MATLAB Error Messages and How to Fix Them

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4 Comments
In this Insight, I'll go over 5 common MATLAB error messages, what they mean, and how to fix them. Hopefully, after reading this post you'll find yourself…
hydrostatics

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…
variables

Some Conceptual Difficulties in the Roles of Variables and Constants

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1. Variables and Constants When is a constant not a constant? When it varies.In the standard equation ##y = a x + b##, we are used to thinking of…
gausslaw

Derivation of Gauss’s Law: Coulomb to Flux Explained

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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.…
physics grad school

Physics Grad School: Senior-Year Application Guide

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Full Chapter List - So You Want To Be A Physicist... SeriesPart I: Early Physics Education in High schools Part II: Surviving the First Year of College …
subway

Brachistochrone Subway Tunnels: Speed vs. Depth Tradeoffs

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Introduction A subway tunnel could, in principle, operate without external energy (like electricity) if we assume zero friction everywhere. The train would…
error handling

Frequently Made Errors: Equation Handling in Physics

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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 straight away.…
physics major

Physics Major Guide: Succeed in Years 3–4 and Research

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Full Chapter List - So You Want To Be A Physicist... SeriesPart I: Early Physics Education in High schools Part II: Surviving the First Year of College …
kinematic errors

Frequently Made Errors: Kinematics Mistakes Explained

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10 Comments
Kinematics is the subset of dynamics that only concerns itself with time, displacement, velocity, and acceleration. A problem is a kinematics problem…
math preparations

Mathematical Prep for Physics Majors — Essential Guide

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1 Comment
Full Chapter List - So You Want To Be A Physicist... Series Part I: Early Physics Education in High schools Part II: Surviving the First Year…
mistakes moments

Frequently Made Errors: Mechanics — Moments and Torque

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7 Comments
Entire List for Frequently Made Errors Series MechanicsForces Friction Moments Kinematics Hydrostatics Momentum and Impacts Springs Pseudo…
electrical wire

Mutual Inductance: When k≠1 and Two Coupling Coefficients

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4 Comments
Mutual inductance and the coupling coefficient A commonly used formula for mutual inductance M between two nearby coils L1 and L2 is M = k√(L1*L2).…
mermin

The Quantum Liar Experiment: An Instantiation of the Mermin Device

In 1981, Mermin published a paper[1], "Bringing home the atomic world: Quantum mysteries for anybody" in which he explained the mystery of quantum nonlocality[2]…
friction

Frequently Made Errors in Mechanics: Friction

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8 Comments
1. Direction of the normalDefinition: The normal force that body A exerts on body B is that force of minimum magnitude which suffices to prevent penetration…
classroom

Photoelectric Effect: Debunking the Photon Myth in Physics

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203 Comments
Introduction There are many sins in physics didactics. Usually, they occur, because teachers, professors, textbook or popular-science-book writers, etc.…
differentiable function 2

Nowhere Differentiable Functions: Fejér Kernel & Cesàro Sums

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2 Comments
This is Part 2 of a series of articles in which the goal is to exhibit a continuous function that is nowhere differentiable and to explore some interesting…
forces mistakes

Frequently Made Errors in Mechanics: Forces

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35 Comments
Entire List for Frequently Made Errors Series Mechanics Forces Friction Moments Kinematics Hydrostatics Momentum and Impacts …
2D Animation

Particle in a Box: 1D & 2D Quantum Visualizations

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11 Comments
Introduction The particle in a box is a staple of entry-level quantum mechanics courses because it provides a clear contrast between classical and quantum…
differentiable function

Constructing a Continuous but Nowhere Differentiable Function

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11 Comments
When studying calculus, we learn that every differentiable function is continuous, but a continuous function need not be differentiable at every point.…
Thermodynamics

Understanding the Second Law of Thermodynamics Clearly

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

Peano Axioms Explained: Building the Natural Numbers

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Using: Anderson-Feil Chapter 1.1 Is zero a natural number? This is a pretty controversial question. Many mathematicians - especially those working in foundational…