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If you're setting $\hbar=1$, then you don't - you can't - distinguish between energy and angular frequency. They are, in fact, the same quantity, since $E=\hbar\omega=\omega$. Similarly, if $\hbar=1$ you can no longer draw dimensional distinctions between wavevectors and linear momenta, or between angular momenta and pure numbers. In general, you only set ...

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In the case of energy, I find it more useful to use the reciprocal of time rather than frequency. Time is a much more fundamental (and common) dimension. In other words $E=1/T$, and of course, $T=1/E$.

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When you work "fairly hard", your body can produce about 200 W of power - enough for two incandescent bulbs. Top athletes can produce more - in short bursts. Your body is roughly 25% efficient in converting "calories" (which are actually kilo calories) to Joules - meaning that if you work out hard enough to burn 600 kcal per hour, then you actually produced ...

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A (kilo)calorie is a unit of energy, while a watt is a unit of power, which describes the rate at which energy is expended. So a 100W bulb is using 100 joules a second. A kcal is about 4184 joules, so a 100W bulb takes about 42 seconds to consume (really: convert into light and heat) a kcal. The joule is the SI (derived) unit of energy. Units of energy ...

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An average person uses approx. 1500-2500kcal/day. Since one kcal equals 4148J in SI units, that's between 6.2-10.4MJ per day. A day has 86400 seconds, which brings us to an average power consumption of 72-120W... about as much as a light bulb. :-) Physical exercise varies between light (300kcal/h) at an additional 350W to very strenuous at probably six ...

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The Dimensions of Angle depend on one's viewpoint and purpose (of using dimensions). Likewies the Units (and implicitly scale) of angle also depend on the local customs and practices that support those viewpoints and purposes. Personally, I want Angles to be a dimension, particularly for error detection and correction in scientific and engineering ...

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(Thanks to Drebin J.) Let's suppose there's a building with many floors, labelled G, 1-10, and after that continues with A1, A2... A10, B1,B2 etc. A1 is floor 1 but actually the 11th, while A2 is 2nd but actually 12th. (Similar to Celsius in Kelvin) Supposing I jump from A2 to A1 (from the staircase), it would obviously hurt. The damage incurred ...

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In an exam, Alice scored $50$, Bob scored $40$, Eve scored $10$. Now to raise the class mean, the teacher decided to add $20$ points to every students. So Alice's score becomes $70$, Bob's score becomes $60$, and Eve's score becomes $30$. Now Alice complained to the teacher, "my score was higher than Bob's by $10$ marks, and you see, according to the ...

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This seems to me like the general "Problem", that differences of Observables behave in a different way, than the observables itself do: One other example (that has nothing to do with additivity or multiplicativity, as you stated) are differences of frequencies: You know that you can convert a frequency to a wavelength by dividing c by the frequency: ...

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Celsius and Kelvin are two scales that differs only for an additive factor, but the single increment corresponds to the same temperature difference. In other words, an object become "hotter" in the same way if you rise its temperature by 1K or 1°C. You can use conversion formula in differences, just make sure you use it for both terms and keep in mind that ...

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A temperature change of 1$^\circ$C is the same as a temperature change of 1K. So if you start at 30$^\circ$C (= 303.15K ) and increase the temperature by 5$^\circ$C (5K) the new temperature is 30 + 5 = 35$^\circ$C (303.15 + 5 = 308.15K).

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temperature difference is 5°C, which is convertible to 278.15K. This is where you go wrong. A difference of 5°C is a difference. Express it as you will in Celcius, Kelvin, Rankine or even in Fahrenheit. But it is a difference. Saying 5°C, which is convertible to 278.15K mean that you are not looking at a difference. Instead you are calculating how to ...

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No. Beside the already mentioned examples here some real problems with different dimensionless quantities and why they cannot be mixed. :( Starting with angle velocity units: Hertz (Hz) for frequency $f$ measured in periods per seconds (which is equivalent to $\frac{2 \cdot \pi}{s}$) or Angular frequency $\omega$ measured in $\frac{rad}{s}$ You can ...

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Of course, you can do it with Matlab, Mupad, Maple, Mathematica or even the Smart Math Calculator. Use this method: First define your variables with your units of choice, then tell the programm what the conversion factors from the given units to the target units are, for example, if you have km/h and need m/sec define 1km as 1000m and 1sec as h as 60² ...

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Here is an entertaining mathematical answer. (Or at least, I find it entertaining, anyway.) Let us take seriously the idea that we can treat radians as a unit, and proceed from there. This means that when we write an expression like $\sin \theta$, the argument $\theta$ must have units of radians, whereas the result (I'll assume) is just a number without any ...

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In your example, a wavelength and a wavenumber represent in a different way the same information as there is a direct conversion between the two quantities. You can use the same reasoning to answer your question about the m/s and s/m: You have a speed $v$. Let's define a new quantity based on that speed, $$\rho = \frac{1}{v}$$ which we conveniently call ...

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Thing is that the very concept of "dimensions" is unphysical in general, it's a human construct that was invented to allow people to do computations when not being able to compare different quantities for whatever reason (insufficient knowledge, wanting to use incompatible units in the same equation etc.). In reality, everything really is dimensionless, ...

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I personally think that one should not confuse an angle, say $\alpha$, and the ratio between $\ell$ the arc length of a circle and its radius $r$, at least from the outset. As far as fundamental concepts are concerned, angles need a new type of "thing" to be talked about; they are neither a length nor a time interval for example. Moreover, from a ...

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Think in terms of coordinate transforms as a generalization of unit conversions. When converting between units, you are doing a very simple coordinate transform on the, single, corresponding physical dimension: Multiplication*. When adding two angles, you are really dealing with, for example, a polar coordinate system. The underlying territory ...

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There are two mistakes. As AccidentalFourierTransform pointed out, the coefficient $7.181\times 10^{-16}$, when converted from MeV to eV, should give $7.181\times 10^{-46}$. Mega means a million, and it to the fifth power gives $10^{30}$, not just $10^{15}$. In this way, the OP has to add a $10^{-15}$ factor to his result. That makes his result $10^{-3}$ ...

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From the paper, which states fiber Bragg gratings (FBG) have been demonstrated to exhibit temperature dependent shifts in resonant wavelength of 10 pm/K it is fairly clear that the unit is picometer per kelvin. That is, you have some device with a resonance wavelength $\lambda_\mathrm{R}$ which depends on temperature, ...

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Think of it this way: is dozen dimensionless? radians (1), degrees (0.017), and gradians (0.0157) are all like dozen (12). Convention says that degree is for angles, and dozen is for eggs. No one goes around saying 562 degrees m/s/s, just like no one says 0.82 dozen m/s/s. They say 9.8 m/s/s. But they totally could. There's no fundamental mathematical or ...

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I answered another unit based question very much related to this. In it, I pointed out that units are not a fundamental concept in the underpinnings of the universe. They are a concept which people have found helpful for relating the real world to mathematical equations we use to describe the world. Thus, their primary purpose is to be useful. In ...

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Few of the answers thus far have directly addressed empirical equations, such as the following: Vapor pressure of isobutane (source) $$\log_{10}{P_\mathrm{mmHg}}=6.74808-{882.80\over 240.0+T_\mathrm{^\circ C}}$$ Seeton model for kinematic viscosity of various liquids (source; $K_0$ is the zero-order modified Bessel function of the second kind) $$... 1 As mentioned by the other answers, it is the dimension which essentially needs to be the same on both sides of an equation and not the unit. This has already been spoken with the example of 1 \,\mathrm{hour} = 60 \,\mathrm{minutes}. Let me give you one example and illustrate to you that: The thing called unit has been merely developed for our own ... 7 You can't add dimensionless quantities willy-nilly for the simple fact that a particular dimensionless quantity represents a particular physical thing. Using the examples you gave, you can't add m/m to kg/kg because they represent different quantities; one is an angle and one is a partial mass content. This can even go for dimensional quantities though. So ... 74 Whenever I think about this problem I go back to one of Joel Spolsky's articles, "Making Wrong Code Look Wrong", which talks about Hungarian notation. Not only the useless kind of Hungarian notation, where variables are named in a way that describes their types (f_pos for a float, d_pos for a double, etc.) - this is "Systems Hungarian" in the article - but ... 79 The answers are no and no. Being dimensionless or having the same dimension is a necessary condition for quantities to be "compatible", it is not a sufficient one. What one is trying to avoid is called category error. There is analogous situation in computer programming: one wishes to avoid putting values of some data type into places reserved for a ... 4 If I've understood your question correctly, you're looking for a case in physics where angles are added to any dimensionless but non-angular quantity. I don't think this happens too often, but it's possible. For example, consider a gauge transformation of QED. The electron field transforms as$$\psi \to e^{ie\theta(x)} \psi$$so \theta(x) is an angle. But ... 10 It's worth noting natural unit systems, which may appear to violate this rule. Since certain physical constants (e.g., G and c) simply reflect an arbitrary choice of units, it can be convenient to change units so that they are identically 1. For example, in Planck units, where G=c=1, we can write the Schwarzschild radius as r_s = 2m. While it ... 5 There is a simple argument to see why dimensions must agree on both sides. To use innisfree's example, consider the (obviously wrong) equation$$m_e = c\tag{*}$$m_e being the mass of the electron and c the speed of light. I assume I have written this equation in the International System units (kilograms, meters, seconds). Now if I want to write this ... 15 No. All equations have the same dimension on both sides. Dimensions are mass, distance, time, speed, acceleration, force, power, electric current, electric charge etc. As long as you work with symbolic relations, you only care about dimensions. The equation$$v = \frac{s}{t}$$(velocity = distance / time) works with any units as long as they are units for ... 33 It depends what you mean by "unit". If you mean something like "seconds", then no. Counterexample: 1 minute = 60 seconds has different units on both sides, but they're both representing a duration, so they can still be equal. If you mean something like "time", then yes. An equation means two things are equal, i.e. the same. For that to be true, they have ... 8 They have to be equal, because if the units are not identical, we will add fudge factors to make them identical. What you are looking at is called dimensional analysis. Dimensional analysis is a tool that lets us turn equations like x(t) = \frac{1}{2}at^2 + vt + x_0 into something meaningful in the real world. The real truth is that there are no "units" ... 1 Hint: what do you get if you add two apples to three oranges? You can only add like things to like things. In this sense it is technically correct to add 1 m to 3 inches and quote the result as 1 m 3 inches (both are measurements of lengths), but it would not be very useful or good practice. There are seven fundamental units: kilogram, metre, candela, ... 17 The dimensional units in an equation must balance. Sometimes a dimensionless "unit" may appear on one side and not be obvious (or even present) on the other side. For example, consider the kinetic energy of a spinning object:$$K_s = \frac{1}{2}\mathcal{I}\omega^2.$$A comparison of SI units yields the following:$$[J]=[kg\cdot m^2]\frac{[rad]^2}{[s]^2}$$... 3 Yes for sure. After all you cannot say 5 chickens = 2 buffaloes. Here is an excerpt from NCERT physics for class 12 Chapter 2. I hope this helps. The recognition of concepts of dimensions, which guide the description of physical behaviour is of basic importance as only those physical quantities can be added or subtracted which have the same dimensions. ... 82 It doesn't matter where the equation came from - a fit to experimental data or a deep string theoretic construction - or who made the equation - Albert Einstein or your next-door neighbour - if the dimensions don't agree on the left- and right-hand sides, it's nonsense. Consider e.g. my new theory that the mass of an electron equals the speed of light. It's ... 7 Every equation should have corresponding dimension. Either by the natural dimensions of the equation$$\text{Average Speed} = \frac{\text{Distance}}{\text{Time}}$$or by some constant which gives the correct dimension$$F = \frac{Gm_1 m_2}{r^2}$$Where G has dimension [M^{-1}] [L^3] [T^{-2}] to ensure that the dimensions are equal on both sides. 2 Under the suggestion of L. Levrel, I'll expand upon my comment. We can look at the constants included within an equation to get an idea of whether it was theoretically derived, or an empirical result. If we have fundamental constants, such as \hbar, \epsilon_0, e, c, etc, then it was probably theoretically derived. An empirical law would have an arbitrary ... 4 First of all, there is no real or observable lines. Even the magnetic and electric fields are nice and abstract fields which describe observable forces. The term "line" you read is an old unit of magnetic flux. One line is the flux of a uniform magnetic field of one gauss across a surface of one square centimeter perpendicular to the field,$$1\ line = 1 ...

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Field lines are a good concept for imagining things, but it does not reach too far. Imagine for example the field of two distinct sources -- the field lines would cross if you just draw them both. But this does not represent the sum field. Field lines are drawn by convention so, that their density is approximately proportional to the field strength. This is ...

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Are you confused about how you get into this? \begin{align} v & = \omega\,r \\ ({\rm m/s}) & = ({\rm rad/s})\,({\rm m}) = ({\rm m\,rad/s}) \end{align} Radians are not units with dimensions. They can be seen as $({\rm rad}) = ({\rm m/m})$ like arc length to radius. This makes to above right hand side equivalent to the left hand side.

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This is all true because the ratio of a circle's diameter to circumference is constant (pi). The unit of radian is actually chosen so that l = r.theta. So if you go theta radians around a circle you travel r.theta. For the angular to linear velocities, think of a disc rotating at an angular speed omaga. then the further out from the centre of the disc you ...

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Many of the units related to physical phenomena were defined before the phenomena in question were particularly well understood. For example, electric current was measured based upon its magnetic effects before it was understood that the amount of current called "1 ampere" represented the flow of some number of electrons per second, and "2 amperes" ...

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Actually, we do not really need constants in the sense that we choose to use them, but it’s just the way the universe works – or more precisely: very well seems to work. For example, various experiments confirmed that the quotient of the energy of a photon and its frequency (when measured in the same units) is always the same within the accuracy of what can ...

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Physical constants arise from the way we define units. Let's take the gravitational constant $G$ as an example. According to Newton's law of universal gravitation: $$F_{g} = G \frac{m_1 \times m_2}{r^2}$$ If you were to take two spheres, both with mass 1 kilogram, 1 meter apart, it turns out the gravitational attraction between them is not 1 newton: it would ...

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Just ignore the dashes. They should just be written as 7'11.5", etc., as seen in this standard reference. But some times people like to put dashes just because they think it looks ugly without them. Here's one example of someone explicitly saying he wants to do it the way seen here, mentioning that it's what "you would see on a set of plans here in ...

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Suppose you take a circle of radius $\ell$ and take an arc of length $\ell$ along the circumference, then 1 radian is the angle subtended by the arc: More generally, if the length of the arc is $\ell$ and the radius of the circle is $r$ then the angle in radians subtended by the arc is $\ell/r$. So the radian is a derived unit because it is the ratio of ...

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On dot product you get magnitude, in units of product of operands. On cross product you get vectors with direction , in units of product of operands.

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