The formulae in physics are not just jumbled meaningless symbols, which have to be memorized when you want to learn physics. They are like films, because they have actors and stories behind their mathematical equations. Mathematics is a language in physics, it is chosen due to its high consistency in logical flow, and its compactness in appearance. The sole disadvantage of the usage of mathematics is that we cannot easily grasp the stories embedded in the background. The disadvantage leads to difficulty, fallacy, and misconception in physics especially for beginners.We can catagorize the formulae into 3 groups according to the sort of their stories :
- cause-and-effect stories
- proportionality stories
- definition
Cause-and-effectThis kind of story is very common in physics. Actually, all phenomena in this world happen in accordance with the cause-and-effect pattern. While time is running, an incidence will cause another event in the future, and it is in fact caused by other phenomena taking place in the past.
A good example can be made by considering the most popular equation in mechanics, i.e. the Newton's second law :
F = m . a (1) where F is a force applied on a body whose mass is m. The force will accelarate the mass, thus introducing motion to it. The accelaration is denoted by a. F and a are vectors, thus having both magnitude and direction.
What is the story about?
The force, F, introduces an accelaration, a, to the mass. Without F nothing can accelarate by itself. In other words, we will be aware of the presence of a force (or forces), though not identified yet, if we experience an accelaration. The force (or the resultant of forces) will be in the same direction as the accelaration.
Who are the actors?
The actors are the suffering mass and an external agent exerting the force on the mass.
ProportionalityThis kind of story reveals the facts that there are many interdependent relationship between physical quantities, it is mostly time-independent in nature. A quantity is said to be proportional to another, if its multiplication causes the same multiplication to the other one.
We can take a simple example. Mass and volume are proportional to each other, because doubling the volume will result mass twice as big as before.
m ~ V
where m and V are the mass and the volume, respectively. The proportionality sign can be changed to an equality by introducing a constant of proportionality :
m = r . V (2) The mass density, r, takes the role of the constant. It is a constant, so it does not depend on the other two quantities, but on the nature of the substance itself.
What is the story about?
The mass of a body depends proportionally to its volume. The ratio between mass and volume is then constant, the fact of which is considered to be the definition of mass density, i.e. a measure of how much mass contained in a unit volume for a specific substance.
Who are the actors?
They are apparently the mass and the volume, and the proportionality constant acting as the mediator of their relationship.
Can we say that the density is inversely proportional to the volume (recalling that equation 2 can be rearranged as : r = m/V)?
It is indeed an interesting phenomenon good for understanding proportionality. The answer can be either yes or no. If you vary the volume to see how the mass behaves after the change, the answer is no, because the changing mass cannot be the proportionality constant. Then in what situation the answer will be yes? Consider a heated body that experiences expansion. In this case the mass of the body remains constant during the expansion, in which the density will change, i.e. decrease, along with the expanding volume.
DefinitionAll physical entities have their own definitions. For example, the density is defined by how much mass is contained in a unit volume:
r = m/V (3) You may see here that Eq (2) and Eq (3) are mathematically the same, but physically not.
Can we take the Newton's second law as an example for proportionality?Yes, we can, as long as only one rigid body is concerned. In this case, the force, F, is said to be proportional to the resulting accelaration, a. The mass m acts as the proportionality constant, which does not depend on the other two quantities, F and a, but on the nature of the body in action.
Is that all for physics formulae?
No. There are still assumptions to be understood in describing the formulae. These background stories are so vital that one should not overlook.