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30

One reason you might think $T$ should be measured in Joules is the idea that temperature is the average energy per degree of freedom in a system. However, this is only an approximation. That definition would correspond to something proportional to $\frac{U}{S}$ rather than $\frac{\partial U}{\partial S}$, which is the real definition. The approximation holds ...


26

Another way to look at it $$ e^x = \sum_n \frac{x^{n}}{n!} = 1 + x +\frac{x^2}{2} + \dots $$ which comes down to adding quantities with different dimension, which you have already accepted makes no sense. This is why you can't exponentiate values with units. And we can do a similar thing with most transcendental functions.


25

Straight from the horse's mouth: Source: Bureau International des Poids et Mesures (Search for "dimensionless" for all guidelines.) The International Bureau of Weights and Measures (French: Bureau international des poids et mesures), is an international standards organisation, one of three such organisations established to maintain the ...


24

The units for torque, as you stated, are Newton-meters. Although this is algebraically the same units as Joules, Joules are generally not appropriate units for torque. Why not? The simple answer is because $$W = \vec F \cdot \vec d$$ where $W$ is the work done, $\vec F$ is the force, $\vec d$ is the displacement, and $\cdot$ indicates the dot product. ...


20

The length of one second in meters is the distance traveled by light in One second. $1$ sec $=c\times1$ sec $= 299,792,458$ m The reason we use the same units for time and distance is special relativity, whose foundation rests on the speed of light (in vacuum) being constant in all inertial frames of reference. Its universality allows us to use the same ...


19

Yes, logarithms always give dimensionless numbers, but no, it's not physical to take the logarithm of anything with units. Instead, there is always some standard unit. For your example, the standard is the kilometer. Then 20 km, under the log transformation, becomes $\ln(20\;\textrm{km}\;/\;\textrm{km}\;)$. Similarly, the log of 10 cm, with this scale is ...


19

Using the distance between the Sun and the Earth, at least for distances within the Solar system, just gives a better feel for the scales involved. You can't really imagine a distance of, say, 1000000000 kilometers -- or at least I can't. (I deliberately didn't include commas in that number, to illustrate the point.) But using a concrete physical distance ...


18

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} = ...


16

You're not wrong. However, there used to be an object exactly $1$ meter long until 1960, because a meter was defined to be the length of a certain platinum-iridium rod at certain conditions. Since then, the meter is defined in terms of interferometry, and now it is specifically the distance traversed by light in vacuum within a certain period of time. ...


15

Here's one "mathematical" but highly unphysical answer. Using that $km\cdot km = (km)^2$ etc, we can formally define arithmetic of numbers with units over a graded algebra $A = \oplus_{k\in \mathbb{N}} V_k$ where $V_k = \otimes^k V$ where $V$ is treated as a one-dimensional real vector space ($V_0$ is the scalar $\mathbb{R}$). The choice of unit is the ...


15

One Celsius (or Kelvin) degree as a temperature difference was defined as 1/100 of the temperature difference between the freezing point of water and boiling point of water. We call these points 0 °C and 100 °C, respectively. The number 100 arose because we're used to numbers that are powers of ten because we use the base-ten system. The Celsius degree is ...


14

A mole is nothing more than a countable number of things. Specifically, it is approximated by 6.02 × 1023 number of things. Hence, you can have a mole of argon atoms, a mole of electrons, or a mole of house keys all of which contain approximately 6.02 × 1023 of their respective items. Note that each of these groups of things will have a different mass. ...


14

Lubos Motl's answer is completely right, but I'll add my perspective anyway. For many compound units, you shouldn't try to "visualize" the meaning of the unit, but you should think of it as reminding you about relationships between that quantity and others. Why are the units of Newton's constant $G$ ${\rm N\ m^2/kg^2}$? It's because $G$'s "purpose in life" ...


12

By your question you could see divided units as rate, an amount of one quantity would be changed based on the amount of another. When looking it at that way you could think about multiplied units as conserved quantities, if you would double one you should half the other to have the same effect. Some interesting examples could be: Power $P = U I$ thus in ...


12

I suppose you mean $k_e=\frac1{4\pi\epsilon_0}$. That comes from the fact that Coulomb's law can be stated as : $$F= \frac1{\epsilon_0}\frac1{4\pi r^2}q_1q_2 $$ Now, $\epsilon_0$ is the electric constant, or the permittivity of free space, and it essentially scales the force. The $4\pi r^2$ comes from the surface ...


11

When the electrostatic force was originally being studied, force, mass, distance and time were all fairly well understood, but the electrostatic force and electric charge were new and exotic. In the cgs system, the charge was defined in relation to the resulting electrostatic force (it's called a Franklin (Fr) an "electrostatic unit" (esu or) sometimes a ...


10

There is no reason why you should be "imagining" a squared second. Most quantities in physics don't have any canonical "geometric" visualization and there is no reason why they should have. What matters is that you should be able to calculate with it. For example, the gravitational acceleration on Earth is $9.81\,\,{\rm m/s}^2$. This simply means that the ...


10

This is a fun question. I have a hard time getting a good grip on the transformation that is $ln$ so I'll write things in terms of exponents. $$value = \ln(10\ \mathrm{ km})$$ $$e^{value} = 10\ \mathrm{ km}$$ The number $e$ is, of course, unit-less. If I raise a number to a power, what are the permissible units of the power? If I write $x^2$, I have an ...


10

The metric tensor is unitless. That can be seen from the fact that $g_{\mu\nu}v^\mu v^\nu$ gives the square of the four-vector length of $v$, and thus has the unit of $v^2$. The scalar curvature is a contraction of the Ricci tensor. A contraction doesn't change the units. Also the Ricci tensor is a contraction of the Riemann tensor. The Riemann tensor is ...


10

All we're doing is using a set of units where certain quantities happen to take convenient numerical values. For example, in the SI system we might measure lengths in meters and time intervals in seconds. In those units we have $c = 3 \times 10^8 \text{m}/\text{s}$. But you could just as well measure all your distances in terms of some new unit, let's call ...


10

Temperature cannot be measured in units reserved for energy because, for instance, a grain of sand heated to the temperature as the Sun does not contain the same amount of energy as the Sun. Temperature is the property that, when two bodies in thermal contact have the same value of it, no net heat flows from one body to the other: they are in thermal ...


10

If you want to avoid factors of $\pi$ in the more fundamental equations like $\nabla . E = \rho / \epsilon_0$, you have to accept them where they belong, for instance in: $E = \frac{1}{\epsilon_0} \frac{Q}{4 \pi r^2}$. As remarked by others, Newton failed to put a factor $4 \pi$ into his gravitation equation (he stipulated $g = G \frac{M}{r^2}$, instead of ...


9

The reason $c$ is important is not because it is the speed of light. It's important because it is a universal conversion factor between time and distance. If you have a certain amount of time $t$, you can calculate the corresponding amount of distance by multiplying it by $c$. Note: I'm not talking about the distance any particular object travels in the ...


9

(Attn: non-seriousness ahead.) Since Helen, whose face could launch a thousand ships is the unit of beauty (as in a millihelen is a face that could launch one ship), perhaps Edison could be the unit of jerkishness. I base this, of course, on an entirely unbiased source.


9

Regardless of the context and the meaning of the symbols, both sides of the equation have perfectly the same units: they are dimensionless. The integral has units $js$ as you write, using your notation, but the functional derivative has the compensating units $1/(js)$ so the units cancel. To see that dimension of the functional derivative is $1/(js)$, one ...


9

I had an extensive look around, and I turned up four conventions. This included a short poll of google, other questions on this and other sites, and multiple standards documents. (I make no claim of exhaustiveness or infallibility, by the way.) Using $[q]$ to denote commensurability as an equivalence relation. That is, if $q$ and $p$ have the same ...


8

A mole is like saying "a dozen," as Luksen points out in a comment. But why pick $(6.02)(10^{23})$ instead of, say, $(6.07)(10^{24})$? The reason is that the number was strategically picked such that the weight, in grams, of a mole would be numerically exactly the same as its molecular weight. It is a number that was formulated for the convenience of the ...


8

The best way to think about it is that a number like 1 km consists of a dimensionless 1 multiplied by a unit, km. When you take the log of a product, you get the sum of the logs, so log(1 km) is the same as log(1)+log(km). This shows that the log of 1 km is neither a dimensionless quantity nor a dimensionful one. If it was dimensionless, then it would be ...


8

Ah, good question. The radian is actually a "fake unit." What I mean by that is that the radian is defined as the ratio of distance around a circle (arclength) to the radius of a circle - in other words, it's a ratio of one distance to another distance. For an angle of one radian specifically, the arclength $s$ is equal to the radius $r$, so you get ...


8

Radians are kind of a funny unit from the dimensional analysis perspective: radians are dimensionless. That means that rad/s and 1/s are equivalent from the point of view of dimensional analysis. One way to think about this is that angular measures in radians are really just ratios of like quantities: $\theta$ in radians is, by definition, the ratio of the ...



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