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Someone told me that units for $G$ and $\epsilon_0$ (gravitational constant and Coulomb's constant) are placed there simply to make equations work dimensionally and that there is no real physical interpretation for them. Specifically he said $G$ was developed as the number such that when multiplied by the remainder of the expression for gravitational force creates the proper force and that its units are given to balance with the units of force. The same goes for $\epsilon_0$. Is this true?

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+1 I think philosophically this is deep question because it begs, when is a coefficient just a dumb interpolation, and when does it actually carry physical meaning. If Einstein said $E=K\,m$ then $K$ would be just a dumb constant, but he said $K=c^2$ and assigned physical meaning to it. –  ja72 Apr 12 '13 at 19:27
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Related: physics.stackexchange.com/questions/10709/… –  DJBunk Apr 12 '13 at 19:58
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Also related, perhaps a duplicate of: physics.stackexchange.com/questions/8373/… –  DJBunk Apr 12 '13 at 19:59

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Basically all dimensional constants (i.e. those that carry units) are just conversion factors between conventional units and physical laws. The units of mass and distance (e.g. kilograms and meters) were decided upon for historical and logistical reasons, just like the unit of force (e.g. Newton) - regardless of how gravity acts. None-the-less, there is a physical relation between those three quantities, namely $F_g \propto \frac{m_1 m_2}{r^2}$. To make the equation work out exactly, one needs to measure the constant which relates them---in this case $G$ the gravitational constant.

In this way, dimensional physical constants are often considered arbitrary in their value.

Dimensionless physical constants on the other hand, are much more puzzling in their nature.

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Nice answer, +1. A good way of looking at this kind of thing is that when you measure a dimensionful constant, you're really just measuring the standards used to construct the system of units. For example, the second was originally defined in terms of the rotational period of the earth, and the meter was originally defined in terms of the circumference of the earth. So when we say c equals so many meters per second, what we're really doing is describing our planet, not describing the speed of light. This paper by Duff is helpful: arxiv.org/abs/hep-th/0208093 –  Ben Crowell Apr 12 '13 at 21:30

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