etotheipi
I'm only really just learning how natural units work so forgive me if this seems like a silly question.
I was just wondering if someone could verify whether the following line of reasoning is valid (I will use joules instead of electron volts just so we can ignore the e conversion factor for now). I'm just going to convert say 5J into a mass.
I first set c to 1 light second per second. Then noting that $$m = \frac{E}{c^{2}}$$ I say that $$m = \frac{5J}{(1 ls s^{-1})^{2}} = 5 \frac{J}{(ls s^{-1})^{2}}$$ Though this is an ugly unit so I'll just write it as $$5 \frac{J}{c^{2}}$$ When I feel like converting back into SI units, this is equivalent to $$5 \frac{J}{(3*10^{8} m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} \frac{J}{(m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} \frac{kg m^{2} s^{-2}}{(m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} kg$$
The logic seems ok to me when I consider the unit to be just as important a part of the overall quantity as the numerical value preceding it and so I treat it just like a normal algebraic variable.
I guess the main point of my question is whether we can move values between the units and their numerical 'coefficients' so to speak. So something like $$300 MeV / c^{2}$$ can be rearranged to $$\frac{300}{(3*10^{8})^2} \frac{MeV}{(m s^{-1})^{2}}$$ and so on.
I was just wondering if someone could verify whether the following line of reasoning is valid (I will use joules instead of electron volts just so we can ignore the e conversion factor for now). I'm just going to convert say 5J into a mass.
I first set c to 1 light second per second. Then noting that $$m = \frac{E}{c^{2}}$$ I say that $$m = \frac{5J}{(1 ls s^{-1})^{2}} = 5 \frac{J}{(ls s^{-1})^{2}}$$ Though this is an ugly unit so I'll just write it as $$5 \frac{J}{c^{2}}$$ When I feel like converting back into SI units, this is equivalent to $$5 \frac{J}{(3*10^{8} m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} \frac{J}{(m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} \frac{kg m^{2} s^{-2}}{(m s^{-1})^{2}}=\frac{5}{(3*10^{8})^{2}} kg$$
The logic seems ok to me when I consider the unit to be just as important a part of the overall quantity as the numerical value preceding it and so I treat it just like a normal algebraic variable.
I guess the main point of my question is whether we can move values between the units and their numerical 'coefficients' so to speak. So something like $$300 MeV / c^{2}$$ can be rearranged to $$\frac{300}{(3*10^{8})^2} \frac{MeV}{(m s^{-1})^{2}}$$ and so on.