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11

It's a unit conversion: $$1\,{\rm yr}=\frac{365\,{\rm days}}{\rm year}\times\frac{24\,{\rm hrs}}{1\,{\rm day}}\times\frac{3600\,{\rm sec}}{1\,{\rm hour}}=3.1556926\times10^7\,{\rm sec}$$ Since $3.1557$ is (somewhat) close to $\pi\sim3.1416$, we use the approximation you cite. Technically, the year is actually 365.25 days long, so using that gives a ...

7

The kinetic energy is given by $E_c=\frac12mv^2$, as the factor $1/2$ is dimensionless, you can see that $\mathrm{[m^2.s^{-2}]=[J.kg^{-1}]}$. Dimension analysis remains correct if the velocity $v$ takes the value $c$, because $c$ is also a velocity.

7

The answer to your question lies in simple dimensional analysis. Joules are the units of energy, so also the units of work. Work, as we all know, satisfies the relationship $$W=\vec F \cdot \vec s$$ Meanwhile, $$\vec F= m \vec a$$ Substituting in this relationship gives us $$W=m\vec a \cdot\vec s$$ Now, let's look at the units of this equation: ...

9

This is a good question - as in the example with $L,\lambda$ you provide, not every rescaling and not every set of constants is valid. The recipe for the set of good natural units is the following: take all the units that appear in your theory and create a space with one dimension for every one of them. Say we have a theory with time, length and energy - ...

-1

"Noah!", said the Lord, "Build me an ark 300 cubits long (137.16 m, 450 ft), 50 cubits wide (22.86 m, 75 ft), and 30 cubits high (13.716 m, 45 ft). How many cubiccubits of volume does your ark measure? How many squarecubits of timber does your ark need on the outside? Do your math wisely before you order wood from the local lumbar yard and make sure you do ...

2

You seem to be doing dimensional analysis in SI units. The paper seems to using Gaussian units. The magnetic field differs between these units by a factor $c$. In SI units we have $$\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B \tag{SI})$$ but in Gaussian units $$\mathbf F = q(\mathbf E + \frac{\mathbf{v}}{c} \times \mathbf B \tag{G}).$$ The latter ...

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