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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 ...


81

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 ...


76

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 ...


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 ...


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: $$[J]=[kg\cdot m^2]\frac{[rad]^2}{[s]^2}$$ ...


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 ...


14

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 ...


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 ...


8

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 ...


8

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 ...


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" ...


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.


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 ...


6

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 ...


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 ...


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 ...


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 ...


3

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 ...


3

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 ...


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. ...


2

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 ...


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 ...


2

The speed of light was first measured by Ole Christensen Rømer (Danish pronunciation: [ˈo(ː)lə ˈʁœːˀmɐ]; 25 September 1644 – 19 September 1710) was a Danish astronomer who in 1676 made the first quantitative measurements of the speed of light. When Maxwell formed what is the classical electromagnetic theory it was evident that the speed of light would ...


2

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, ...


2

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, ...


2

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 ...


2

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 ...


2

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 ...


2

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 ...


2

I think your problem is that you didn't change the units in the constant g. It has a value of approximately $9.8ms^{-2}$. Notice that it depends on meters. To obtain the correct result, you should use $980cms^{-2}$. Notice that this constant is off by a factor of 100, so that the result (after the square root) is off by a factor of $\sqrt{100}=10$.



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