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56

Time, in which case each system's zero point is often called its epoch: http://en.wikipedia.org/wiki/Epoch_%28reference_date%29


53

Sound can be measured in deciBels ($\mathrm{dB}$) but also as an intensity measured in $\mathrm{W/m^2}$. $0\,\mathrm{dB}$ on this scale is equal to $1\times10^{-12}\,\mathrm{W/m^2}$.


34

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 ...


30

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. ...


29

Addendum: It has been brought to my attention that the argument herein is specious. See DOI: 10.1021/ed1000476 "Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions" Another way to look at it $$ e^x = \sum_n \frac{x^{n}}{n!} = 1 + x +\frac{x^2}{2} + \dots $$ which comes ...


29

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 ...


28

Luminosity. The Magnitude of a star is a logarithmic scale with an arbitrary zero point. In SI unit of brightness is the Candela or or there is luminosity if direction is not accounted for. From Wikipedia: In SI units luminosity is measured in joules per second or watts. Values for luminosity are often given in the terms of the luminosity of the ...


28

As you already indicated, physical units need to be considered. When working in SI units, the ratio of electric field strength over magnetic field strength in EM radiation equals 299 792 458 m/s, the speed of light $c$. However, the numerical value for $c$ depends on the units used. When working in units in which the speed of light $c=1$, one would ...


27

The thing is that $\mathrm{dm}$ is a single symbol, not a combination of two symbols. Yes, it can be understood in terms of a prefix and a base indicator, but it is still a single symbol. An analogy to the concatenation of variable is inappropriate. Reference to an authoritative statement: The grouping formed by a prefix symbol attached to a unit ...


24

Another example of an arbitrarily selected zero point is longitude. This was not always measured from the Greenwich meridian - Paris has been used, and the ancient Greeks (Ptolemy, specifically) used an island believed to exist off the west coast of Africa* in order to avoid dealing with negative numbers. Really, though, none of the examples anyone have ...


21

It is an ångström, a unit of length commonly used in chemistry to measure things like atomic radii and bond lengths. Although not an official SI unit, it has a simple relationship to the metric units of length: $$1\:\mathrm{ångström} = 1\:\mathrm{Å} = 10^{−10}\:\mathrm{m} = 0.1\:\mathrm{nm} = 100\:\mathrm{pm}.$$


21

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 ...


21

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 ...


20

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 ...


20

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} = ...


20

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 ...


19

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" ...


19

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. ...


17

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 ...


16

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 ...


15

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 ...


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

Seems valid to me... I think the interpretation of your example could be that you are doing something with some joules, for some period of time, and that the product of the number of joules and the number of seconds has a certain value, and the units would be... joule-seconds! I can picture the operator of an energy storage facility quoting a price for ...


14

Height / altitude. From Wikipedia: Indicated altitude – the altimeter reading Absolute altitude – altitude in terms of the distance above the ground directly below True altitude – altitude in terms of elevation above sea level Height – altitude in terms of the distance above a certain point Pressure altitude – the air pressure in terms of ...


13

it turns out this is exactly 273.15°C less the melting temperature of water. Actually, "Kelvin" and "degrees Celsius" are defined such that there are 273.16 degrees between absolute zero and the triple point temperature of water. Degrees Celsius are defined as $K - 273.15$. The freezing point of water is a measured quantity and is not exactly 273.15K ...


13

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 ...


13

If I saw the word "amp" written as such in a paper in my field (astrophysics) it would strike me as a bit informal. I would expect to see the full "ampere" written. That said, it is rare to actually write out the full name of a unit; usually it follows a number and is given its standard abbreviation. When abbreviated to e.g. "$5\ \mathrm{A}$", I would ...


13

Well, that equation for the force due to electric charges is only true for a very special choice for the unit of the electric charge. Typically, you would write down Coulomb's law as $k\frac{q_{1}q_{2}}{r^{2}}$, where $k$ is a constant of proportionality chosen to make the units work out. IN the SI system, the unit of charge is the Coulomb (C) and the ...


13

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 ...


13

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 ...



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