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31

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 ...

29

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. ...

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 ...

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.

26

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 ...

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: $$1\:\mathrm{ångström} = 1\:\mathrm{Å} = 10^{−10}\:\mathrm{m} = 0.1\:\mathrm{nm} = 100\:\mathrm{pm}.$$

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

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. ...

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, 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 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 ... 17 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 ... 17 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" ... 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 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 ... 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. ... 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 ... 12 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 ... 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 ... 11 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 ... 11 I've seen "(1)" used. Radians (and steradians) are also "unitless" but they're clearly not appropriate here. 11 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 ... 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 ... 11 Technically, apparently, your teacher is correct. BIPM and NIST In the official brochure from the Bureau international des poids et mesures (BIPM, the keepers of SI units) in §5.1 Unit symbols we find: It is not permissible to use abbreviations for unit symbols or unit names, such as sec (for either s or second), sq. mm (for either mm2 or ... 10 The only sensible rule when working with units is, that you can only add together terms which carry the same unit. Say  [x]=[y] , then x+y is unit-wise a valid statement. You may also multiply arbitrary units together. Whether that is physically sensible is another question. Obviously you cannot add, e.g meters and seconds, but multiplying to form m/s ... 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

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

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 ...

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