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60

first of all, the question you are asking is very important and you may master it completely. Dimensionful constants are those that have units - like $c, \hbar, G$, or even $k_{\rm Boltzmann}$ or $\epsilon_0$ in SI. The units - such as meter; kilogram; second; Ampere; kelvin - have been chosen partially arbitrarily. They're results of random cultural ...


46

Something I posted on reddit answers this question quite well, I think: "Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number. By choosing the unit you use to express a ...


41

To formalize dushya's comment as an answer: Since the kilogram is an arbitrary, man-made unit, the actual numerical value of the proton mass in kilograms is meaningless (i.e. it's as good as its value in pounds, ounces, stones, solar masses, $\textrm{MT}/c^2$, etc.). The true fundamental constants of nature are dimensionless: they have the same value in ...


23

Special Relativity is based on the invariance of a quantity called the proper time, $\tau$, which is the time measured by a freely moving (i.e. not accelerated) observer. The proper time is defined by: $$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$ This is similar to Pythagoras' theorem as learned by generations of schoolchildren, except that it includes ...


23

The view of most physicists is that asking "How can it be that the speed of light is constant?" is similar to asking "How can it be that things don't always go in the direction of the force on them?" or "How can it be that quantum-mechanical predictions involve probability?" The usual answer is that these things simply are. There is no deeper, more ...


19

Yes, Avogadro's constant is a redundant artifact from the era in the history of chemistry in which people didn't know how many atoms there were in a macroscopic amount of a material and it is indeed legitimate to set Avogadro's constant equal to one and abandon the awkward obsolete unit "mole" along the way. This $N_A=1$ is equivalent to $$ 1\,\,{\rm mole} = ...


19

Nothing. From Nature's perspective speed of light is entirely artificial number. Imagine that you've discovered an alien culture that measured horizontal length $\ell$ and height $h$ in different units. They live on a planet with very strong gravitational force, and for them it is very difficult to rotate stuff in vertical plane. Such kind of rotations are ...


15

The infinitesimal length interval between two events in spacetime $ds$ is defined by $$ds^2=c^2 dt^2 - dx^2 - dy^2 - dz^2$$ The creature is dimensionally consistent, because time is multiplied with a speed. You can think of $(t,x,y,z)$ as the four coordinates of spacetime $(x^0,x^1,x^2,x^3)$ and $c$ appears naturally in the equations. However, the usual ...


15

No, the value $9.8\frac{\mathrm{m}}{\mathrm{s}^2}$ is an approximation that is only valid at or near the Earth's surface. You can go a few miles up or down and it'll still be good enough, but once you get any significant distance away from the surface of Earth, you would need to use a different value for gravitational acceleration. You can calculate the ...


15

$G$ is just a constant of proportionality to get the units right (so that when $m_1$ and $m_2$ are in kilograms and $r$ is in meters you get a force in Newtons rather than wingdingalings or something really weird). Indeed cosmologists like to work in a system of units where $G = c = 1 \text{ (dimensionless)}$, and particle physicists like to work in units ...


15

There was a proposal in 2006 trying to define NA as an exact number[1,2]: $$ N_A^* = 84\;446\;888^3 = 6.022\;141\;410\;704\;090\;840\;990\;72 \times 10^{23} $$ the problem? This value is incorrect, as the currently most accurate result is[3] $$ N_A = 6.022\;140\;84(18) \times 10^{23} $$ i.e. $N_A^*$ is now 3 s.d. away from $N_A$. As I have commented, if ...


14

To expand a little on David's point assume we move from the nominal "surface" where $g$ is $9.8\text{ m}/\text{s}^2$ to another point at radius $r + \Delta r$. How much does the acceleration of gravity change? $$ g = \frac{GM}{(r+\Delta r)^2} = \frac{GM}{r^2(1 + \Delta r/r)^2} $$ and as long as $\Delta r$ is small compared to $r$ we can reasonably ...


13

The expression $(\hbar G/c^3)^{1/2}$ is the unique product of powers of $\hbar, G,c$, three most universal dimensionful constants, that has the unit of length. Because the constants $\hbar, G,c$ describe the fundamental processes of quantum mechanics, gravity, and special relativity, respectively, the length scale obtained in this way expresses the typical ...


13

This does allow you to make a prediction--- the significance of the match tells you approximately the number of attempts you have made to get such a coincidence. The ratio of the mass of a proton to a mass of a steel cylinder in Paris was determined by the psychology of some French revolutionaries. But from the accuracy you get, one can be 99.99% sure that ...


13

You've seen the speed of light quoted as roughly $3*10^8\, \text{m/s}$, so the speed of light is very fast compared to one meter and one second. This is roughly a human walking speed, so your question could be interpreted as asking why light is few hundred million times faster than a walking speed. The speed people walk is rather anthropocentric, though. ...


13

$G$ is not exactly larger than $h$ by a factor of $10^{23}$ in SI units, as you are probably aware (just making sure). There is also no expected numerical relationship between the two that has a physical interpretation. You have to understand that these constants are mostly just due to our (to some extent) arbitrary choices of units. These are, of course, ...


12

Since in the limit of weak gravitational fields, Newtonian gravitation should be recovered, it is not surprising that the constant $G$ appears also in Einstein's equations. Using only the tools of differential geometry we can only determine Einstein's field equations up to an unknown constant $\kappa$: $$G_{\mu\nu} = \kappa T_{\mu\nu}.$$ That this equation ...


12

Your guess is correct. After electroweak symmetry breaking, the coupling constant for the residual $U(1)_\textrm{EM}$ gauge group can be written as a function of the couplings of the broken $SU(2)_L \times U(1)_\textrm{Y}$ gauge groups: $$ \alpha = \frac{1}{4\pi}\frac{g^2 g\prime^2}{g^2+g\prime^2} = \frac{e^2}{4\pi} $$ These couplings, however, are running ...


12

The problem is that you want your unit definitions to be realizable - so specifying "1 mol is long number molecules, 1 gram is 1/12 of the mass of one mol of $C_{12}$" is nice for your thought process, but as long as there is no practical way to count molecules at such scales to a precision of better than $10^{-9}$ (which I think is the precision of the ...


12

One should separate the question into two parts, the first of which is philosophical, and the second physics. The philosophical question is resolved by understanding that there are "constants" which are just those that set the system of units, and these are constant for the simple reason that they define our conventional units. The unit-defining constants ...


10

SPEED OF LIGHT: This is a very interesting question. Going through the foundations of electromagnetism and the theory that led to Maxwell’s equations, there is an interesting element that can grab your attention. You can see that the speed of light is not as abstract and mysterious as it appears to be, but only if you look from a different perspective. I ...


10

You can't calculate the numerical value of Newton's constant from the first principle because it is a dimensionful constant – it has units – so the numerical value depends on the magnitude of the units. And because e.g. the kilogram is defined as the mass of a platinum prototype hosted by a French chateau (the kilogram has the "least objective" definition so ...


10

No, as commented on above. Worse, we don't know its value very well. There are efforts underway to measure $G$ more accurately, as reported in Nature earlier this month: It is one of nature’s most fundamental numbers, but humanity still doesn’t have an accurate value for the gravitational constant. And, bafflingly, scientists’ ability to pinpoint G ...


9

It's all in how you define the units. $E = mc^2$ in nice MKSA units; but then change energy into BTUs and you'll need the ever-lovable "fudge factor" in there. People spent a lot (well, some) of time developing self-consistent sets of units largely to keep equations simple, tho' as Rijul pointed out, assigning ugly numbers to known constants hides a ...


9

It's a side effect of the unreasonable effectiveness of mathematics. You are in good company thinking it is a little strange. Many quantities in physics can be related to each other by a few lines of algebra. These tend to be the models that we think of as "pretty." Terms manipulated by pure algebra tend to pick up integer factors, or factors that are ...


9

Only dimensionless quantities are important. They are just pure numbers and there can't be any ambiguity about their value. This is not so with dimensionful quantities. E.g. if I tell you my speed $v$ relative to you is $0.5\, \rm speedons$ that doesn't give you much information as I have a freedom to define my $\rm speedon$ units any way I want. Only way I ...


9

In actual fact, the relative speed rule does not apply, ever. The relativistically correct speed addition rule is the following: $$s=\frac{v+u}{1+\frac{vu}{c^2}}$$ When $\frac{vu}{c^2}$ is close to zero (in other words when the velocities invloved are much less than the speed of light), then the correct formula is approximated to the Galilean version ...


8

Using fundamental physical constants try to construct expression which unit is legth. So using dimensional analysis, we have a data of: $G = m^3 \cdot kg^{-1} \cdot s^{-2}$, $c = m \cdot s^{-1}$ and $\hbar = J \cdot s = kg \cdot m^2 \cdot s^{-1}$. Than we are to construct length $l = m$ in the following way: $$l = G^a c^b \hbar^d = m^{3a + b+d} \cdot ...


8

Pure convention. There is no reason alternative conventions couldn't be used, apart from the need to avoid confusion. Newton introduced the constant to make the force law simple, whereas the electrostatic definition with the $4\pi$ is designed to make Poisson's equation (one of the equations for the electric field) look simple. You can write a Poisson ...


8

It depends on the unit you want to express it. If you choose c/100 as the speed unit, c will be expressed with a rational number. If you choose c/π, you'll have an irrational one. That depends on measure, not on nature.



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