# Tag Info

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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 ...

23

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 ...

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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.

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 $$1\,\,{\rm mole} = ... 18 The length of one second in meters is the distance traveled by light in One second. 1 sec =c\times1 sec = 299,792,458 m The reason we use the same units for time and distance is special relativity, whose foundation rests on the speed of light (in vacuum) being constant in all inertial frames of reference. Its universality allows us to use the same ... 16 You're not wrong. However, there used to be an object exactly 1 meter long until 1960, because a meter was defined to be the length of a certain platinum-iridium rod at certain conditions. Since then, the meter is defined in terms of interferometry, and now it is specifically the distance traversed by light in vacuum within a certain period of time. ... 15 Yes, logarithms always give dimensionless numbers, but no, it's not physical to take the logarithm of anything with units. Instead, there is always some standard unit. For your example, the standard is the kilometer. Then 20 km, under the log transformation, becomes \ln(20\;\textrm{km}\;/\;\textrm{km}\;). Similarly, the log of 10 cm, with this scale is ... 15 One Celsius (or Kelvin) degree as a temperature difference was defined as 1/100 of the temperature difference between the freezing point of water and boiling point of water. We call these points 0 °C and 100 °C, respectively. The number 100 arose because we're used to numbers that are powers of ten because we use the base-ten system. The Celsius degree is ... 15 The units for torque, as you stated, are Newton-meters. Although this is algebraically the same units as Joules, Joules are generally not appropriate units for torque. Why not? The simple answer is because$$W = F \cdot d$$where W is the work done, F is the force, d is the displacement, and \cdot indicates the dot product. However, torque on ... 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" ... 13 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. ... 12 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 ... 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 ... 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 ... 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 One further point to note, is that strictly one is just saying that the exponent is dimensionless, not that it does not contain expressions with dimension. So for example we could have some expression like X=a^{(E/E_0)} where the exponent for a is a ratio of energies. There are several restrictions on the space (sometimes viewed as a vector space) of ... 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 ... 8 Multiplied units often turn up in "double proportion" situations, where a particular quantity is proportional to multiple dependent variables. To take a deliberately non-physics example, a difficult task might need either many people to work on it or people to work on it for a long time, so its difficulty is proportional to both the number of workers and the ... 8 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 ... 8 (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. 8 As the other answers (and dmckee's comments) note, yes, if you take the square root of a dimensional quantity then you need to take the square root of the units too:$$ \sqrt{4\;{\rm kg}} = 2\;{\rm kg}^{\frac12}  And no, I can't think of any meaningful physical interpretation for the unit ${\rm kg}^{\frac12}$ either. However, in the comments you say ...

8

Many questions here. Well, units have a formal definition in the sense that every physical quantity has two parts: A quantity and a unit. Take "distance". How would you describe distance using just a number? That's impossible. Stating just a number would immediately lead to the question "... of what?". Think about it like this: Everything you measure is ...

7

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 ...

7

1 kg is defined as the mass of a particular reference mass in France. So there is at least one thing that weighs exactly 1 kg. Furthermore, it may in general be possible to construct an arbitrary number of objects which weigh 1kg exactly. Assuming all Si atoms have the same mass, one could simply define the kilogram to be the mass of a certain number of Si ...

7

Trigonometric functions don't "preserve" units. The expression under a trigonometric function must be dimensionless and so is the value of a trigonometric function. Thus, C2 in your equations is in units of frequency: Hz or 1/s. There is an error in one of the equations, perhaps a missing constant.

7

Yes. The delta function always has the same dimensions as the inverse of its argument. You can read this from its definition, your first equation. So in one dimension $\delta (x)$ has dimensions of inverse of length, in three spatial dimensions $\delta^{3}(\vec x)$ or simply $\delta(\vec x)$ has dimension of inverse of volume, and in n spatial dimension ...

7

Any consistent system will do. That's the entire point of systems of units--if you stick to one, you don't need to worry about the units too much. And it never happens that a certain equation only works in a certain system*. In this case, you would use joules ($\:\mathrm{J}\equiv\:\mathrm{kg\:m^2\:s^{-2}}$), the metric unit of energy. If you were using the ...

7

A nautical mile is the length of one minute of arc (1/60 deg) along any meridian If you are navigating by measuring the angles of the sun and stars then it's a simple and obvious unit to use since it avoids a lot of calculation and it's close enough to a normal mile to be understood. It's also been an internation standard for quite a long time - unlike ...

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