To describe the properties of a system for which we have designed mathematical symbols and in relating these properties in an expression, we often introduce some proportionality constant. What is the origin of this constant? Are there any things we missed earlier(some other property that we don't know about)?

  • $\begingroup$ Have you got specific examples in mind? They will make it easier to know exactly what you mean and therefore will make the answers more relevant to you. $\endgroup$ – Emilio Pisanty Sep 28 '15 at 11:40
  • $\begingroup$ Gravitational constant G in Newton's Gravitation formula $\endgroup$ – Neeraj kumar Sep 28 '15 at 13:02
  • $\begingroup$ Are there any thing we missed earlier what do you mean by this? $\endgroup$ – Kyle Kanos Sep 28 '15 at 14:04
  • $\begingroup$ @kyle, means in calculating gravitational force between two bodies we say F=Gmm/r2 so what property of this two system does this G represent? What is this missing prooerty? We simply say this is universal constant why? $\endgroup$ – Neeraj kumar Oct 8 '15 at 18:28
  • $\begingroup$ One could say that G is a result of our choice of units, we could have chosen masses or distances such that $G=1$, so there really isn't a property is represents. And it's universal because, given our MKS unit system, it does not change whether the two masses are tiny (microgram-scale) or massive (yotta-gram-scale) or a mix of the two, G is the factor that puts the force in Newtons from the masses (in kg) and distances (in m). $\endgroup$ – Kyle Kanos Oct 8 '15 at 18:39

Consider you have a copper wire with a certain diameter and length, an adjustable power supply and an ammeter. You do a set of experiments and find that as you increase the voltage the current increases. Then, you conclude that voltage is proportional to current and write
$$ V \propto I $$

But how are you going to convert this proportionality sign to an equal sign?
In this case, yes, there is a property which is missing, namely, resistance R, that is, the constant of proportionality. At this point, with the available data from your experiment, you do not even know what is the nature of this constant; whether it depends on the material or the length or diameter of the wire you used, etc.
Same logic is applicable for the proportionality between the force and the spring constant or the gravitational force and gravitational constant.
Also keep in mind that the values of these constants depend on the system of units one uses.


This isn't really all that sophisticated. Take, for example, Newton's law of universal gravitation:

The gravitational force between two objects is proportional to the mass of each of the bodies and inversely proportional to the square of their distance.

That is, if you have two bodies of mass $M$ and $m$, separated by a distance $r$ then the force $F$ between them obeys $$F\propto\frac{Mm}{r^2}.$$ Put another way, if you have two pairs of bodies, 1 and 2 and 3 and 4, with masses $m_1$, $m_2$, $m_3$ and $m_4$ and separated by distances $r_{12}$ and $r_{34}$ then the mutual forces inside each pair, $F_{12}$ and $F_{34}$, will obey $$ F_{12} \frac{r_{12}^2}{m_1m_2}=F_{34} \frac{r_{34}^2}{m_3m_4} $$ You can then choose some representative bodies and set $G=F_{12} \frac{r_{12}^2}{m_1m_2}$, and you'll get the same value regardless of which bodies you chose. In other words, $$ F \frac{r^2}{Mm}=G $$ for any two bodies, or $$F=G\frac{Mm}{r^2}.$$


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