Why do we need constants?

units physical-constants

6,465

Solution 1

Physical constants arise from the way we define units. Let's take the gravitational constant $G$ as an example. According to Newton's law of universal gravitation: $$F_{g} = G \frac{m_1 \times m_2}{r^2}$$ If you were to take two spheres, both with mass 1 kilogram, 1 meter apart, it turns out the gravitational attraction between them is not 1 newton: it would be $6.674 \times 10^{-11}\ N$. Thus, the gravitation equation needs a conversion factor with value $6.674 \times 10^{-11}$, which we call $G$. We need it just because of the way our unit of force, the newton, is defined: 1 newton is defined as the force needed to give a mass of 1 kilogram an acceleration of 1 meter per second squared.

The kilogram, meter and second are rather arbitrary quantities, and as such, the newton is an arbitary unit. We could have equally well chosen to use yards instead of meters, and pounds instead of kilograms, and define the newton as the force needed to accelerate a mass of 1 pound by 1 yard per second squared. This would give us other values for the physical constants. Nature, of course, couldn't care less how we humans define our units. Nature has it's own units.

We need the physical constants to convert the effects of nature into the units of our choice. The gravitational constant $G$ converts the gravitational force between masses (in kg) seperated by some distance (in meters) into Newtons. Planck's constant $h$ converts the energy of a photon with some wavelength (in meters) into Joules.

Solution 2

Actually, we do not really need constants in the sense that we choose to use them, but it’s just the way the universe works – or more precisely: very well seems to work.

For example, various experiments confirmed that the quotient of the energy of a photon and its frequency (when measured in the same units) is always the same within the accuracy of what can be measured. We did not chose the universe to be this way; it just is. In the above case, we agreed to call the quotient $h$ and we usually also agree on some units to communicate the value of this quotient. This in turn allows us to make accurate predictions about reality – in this case, what energy a certain photon has.

Now, the whole unit concept is again based on constants and, once more, they are brought upon us by the universe, which happens to be very well described by numbers. For a blatant example, suppose we take two blocks of wood and cut two pieces of string such that they just begin and end where one of the blocks begins and ends. If we then tie those strings together, we find that the composite string begins and ends where the two blocks of wood begin and end, when put together. This description is awfully complicated, because I tried to avoid the word length, which denotes the fundamental property of objects derived from such observations and which eventually lead to the invention of mathematics to better describe such properties.

However, for being able to talk mathematically about specific lengths without resorting to actual pieces of string, we need to agree on a unit length, which is nothing but an arbitrarily chosen constant to facilitate communication. While we chose what to use as units, we did not really chose to use units in general: The universe just happens to be such that they are damn useful and without them, we would not have the time to think about such questions.

6,465

Snerd

Updated on April 18, 2020

Comments

Snerd over 2 years

This question is driving me crazy because I cannot find a straightforward answer. I want to know what a physical constant exactly is. I know that it’s a value that doesn’t change, but what is it? Why do we need them in our equations?

For example what is the gravitational constant? Why do we need it in the universal gravitation formula? What is Planck’s constant? I don’t want to know the value, I want to what its use is. What does it allow us to do or find? What exactly does Planck’s constant tell us?
- John Rennie over 6 years
  
  Your question is hard to answer because it doesn't make sense. A constant is just a constant. It would be worth haveing a look through the existing questions related to this area.
- Thorsten S. over 6 years
  
  To only add an example to Jeff's already good answer: If you want to convert foot to meter, you also use constants: 1 foot is 0.3048 m. You can also use for own system with your own units (like foot-pound-second for imperial units) where most equation constants change because in many equations they really only have conversion meanings. But there are also constants like the fine-structure constant, the ratio of eletromagnetic interaction which are given by nature and have a deeper meaning. Physicists are also not completely sure that these values are really constant over time !
MPeti over 6 years

+1 Great answer, just one thing - I think dimensionless constants ought to be mentioned as well when going in this direction!
Kyle Oman over 6 years

In my world, $G\approx 4.301\times10^{-6}\,({\rm km}\,{\rm s})^{-1}{\rm kpc}\,{\rm M}_\odot^{-1}$ :)