# Buckingham π theorem

This is a method used in applied mathematics, physics and engineering.

Let me start by saying that, unlike the other articles in this blog, this post will be quite technical, and it might be challenging for the lay public. Anyway, I invite you to read and ask me questions in the comments if something is unclear.

The Buckingham π theorem can be used to find the “hidden” relation between physical quantities that describe a given phenomenon. It relies on the principle of dimensional homogeneity which basically tell us that we can’t sum apples with melons.

I will use an example to explain the theorem. Let’s consider a pendulum, and suppose that we aim to find its period (i.e. the time that the pendulum takes to complete an oscillation). Our purpose is to find an equation describing how the period depends on the physical quantities involved. # Questions & Suggestions Please feel free to add your comments and critics about the blog here. For example, you can give me suggestions, criticize the way the blog is managed, or even suggest subjects for future articles that I may write. In other words, use this space to say whatever you wish, as long as it is unrelated with other published articles in this blog, for which you can just comment the articles themselves.

If you prefer, you can contact me by email: m.lopes @ exeter.ac.uk.

Marinho Lopes

# World of Particles II If you read Part 1 of this article you may wonder: “so, the bosons are like this, fermions like that, so what? What are the consequences of that?” Actually, these particles differ in something very important: while fermions interact with the forces, the bosons mediate the forces!

Newton introduced the idea of “action at a distance” (force at a distance) to formulate gravity. It was an extremely important concept to understand gravity and later electromagnetism. Then, it was discovered that electromagnetism is mediated by the photons (which are bosons), and that the weak and strong forces also have their own mediators which are also bosons. Interestingly, we are still looking for a boson to mediate gravity (the graviton), which constitutes a very important open problem in particle physics (within the Standard Model). In fact, the Standard Model is not compatible with the graviton, because this particle brings some contractions into the model. The most famous theory to solve these (and other) problems is String Theory (which I will talk about in another article).

# World of Particles I Have you ever found some strange names which supposedly refer to particles? No idea what do these strange names mean? Sometimes you may even wonder if such small particles actually exist, or if they are only the product of the imagination of some scientists.

# Black Holes

First of all, do black holes exist?

Yes, they do! (However, it is true that there are still some mysteries concerning black holes. Some of them are related to the origin of the universe! Solving these problems may expand our knowledge about the Big Bang towards its first instants, the Planck time, but that’s another “story”.)

A brief definition:

# The Forces of Nature

Everyone knows what a force is, however, the definition you might be thinking of may not coincide with the definition used in Physics, which is a convention proposed by Newton.

What is the convention? $F = m\times a$

F: force, m: mass, a: acceleration.

This is the Newton’s Second Law.

# Quantum Mechanics

In the late 19th century, many physicists thought that their job was almost finished, for well-understood mathematical laws could explain most of the observed physical phenomena. There were only a few things lacking explanation, but physicists thought that they were just details. They believed that soon it would be possible to describe the behavior of any system. (There is the story of a student who asked Lord Kelvin what he would advise him to choose to research in Physics, and Kelvin replied that he should choose another area, for Physics was almost “complete”.)

# The dual nature of light and matter

Dual nature? What are the two?

Physics studies two fundamental “objects”: waves and particles. These are the two “natures”.

• What is a wave?

A wave is a perturbation which propagates in time and across space. An ocean wave is an example. A wave can be divided, but cannot be localized with precision. A wave can be generated by applying an oscillatory force to one end of a string (by shaking your hand, for example). This figure shows a representation of a wave. This form is called sinusoid.

# Einstein’s Invariance and Photoelectric Effect

This article complements the previous two articles about Einstein’s Theory of Relativity (Special Relativity and General Relativity), namely by making the following remark: Einstein did not actually like the name “relativity” for his theory. I will explain why.

I will also tell you about the most famous equation in physics: E=mc^2. Furthermore, I will briefly explain the photoelectric effect, which was the discovery that earned Einstein the Nobel Prize in Physics.