 # E=mc^2 Applied

I was asked today what exactly E=mc^2 meant, and I replied to the best of my knowledge that m was Mass c was the speed of light and E was energy. Einstein came up with the theory that the speed of light was the only constant in the universe and proved it by showing you can fine the Energy of any object given the mass and vise versa. And thats where my knowledge crapped out.
So, I’m looking for a practical applied and hopefully interesting way of explaining this to my pals. I thought I could take the mass of one of my friend, do the calculation and show them how much energy was stored in them. Followed by comparing this amount to something else so they could understand if it was a lot of a little amount.
But alas, I have no idea how to convert the (metric) units, and nothing to compare the results to. So I come to the Dope.
OK, you take mass (in grams I assume?) time c^2 (c in m/s, or something else?) and end up with energy (joules? Kj?). How can I figure this out given lets say a 70kg person? What can I compared the amount of energy to?

I’d always assumed mass was in kilograms. The energy is in kg x m^2/s^2, but I forget if that’s a joule.

AFAIK, it’s the conversion if you managed to convert all of the mass into energy… how… I don’t know.

I don’t think the speed of light is the only constant in the universe but Einstein discovered that it was one of them.

E=MC[sup]2[/sup] simply says that mass and energy are related and convertable from one to the other.

As it turns out I asked awhile ago on this message board how big a boom I’d make if you converted my mass to 100% energy in a fashion that made a big boom (ala tossing me in an equivalent amount of anti-matter). Here’s the answer (among a few) that I got (assuming I weigh 75 kilos).

Joules is the correct unit for your answer and the thread linked above gives the answer at around 10[sup]19[/sup] joules for 75kg. However, I am unsure if the 10[sup]19[/sup] includes the energy from the antimatter I was speculating that was being used to convert me to energy. If it was (and I think it was) then take hlaf that number for just your mass. Still a LOT of energy!

Should be standard IS units; J, m/s and kg.

There are several standards of metric units. Most scientists use the MKS system which use the meter, kilogram second and ampere as the base unit. Joule is a derived MKS unit, defined as
Joule = Newton * meter = kg(meter/sec)[sup]2[/sup]*
So if you take the mass in kg and c in m/s you get the energy in Joule. It’s the same for simple kinetic energy E=1/2 m v[sup]2[/sup] - use kg for mass and m/s for velocity and you get the kinetic energy in Joule.

The other, less commonly used metric system is the CGS system based on the centimeter, gram and second. The CGS unit of energy is
erg = dyne * cm = gram * (cm/s)[sup]2[/sup]

One way to explain it is by using the earth’s sun. Our sun uses fusion to fuse hydrogens into a helium. But there is a difference in weight between 4 hydrogens and one helium, and this weight is converted into engery. If you take that difference in weight and plug it into E=mc^2 this very small amount of difference in mass is converted into a ton of energy. Hence you can see why the sun is so powerful.

This site illustrates it a little better than I can.

E = mc[sup]2[/sup] relates mass-energy to mass, it’s useful, for example, in working out the kinetic energy of beta particle (a type of nuclear radiation):

K.E.[sub]β[/sub] = [m[sub]nuclide[/sub] - (m[sub]decay product[/sub] + m[sub]β[/sub])]c[sup]2[/sup]

There are many constants in physics and c is one of the fundamental physical constants (which includes Dirac’s constant and Boltzmann’s constant). c itself was not discovered by Einstein, it actually comes out of the Maxwell’s equations

You can choose any unit system as long as you are consistent. Scientits prefer the SI system which means that you would take the mass in kg and the speed in m/s which would mean that E would be in joules. For a 70kg person, taking c as 3.0 x 10[sup]8[/sup] ms[sup]-1[/sup] this wold give you a mass-energy of 6.3 x 10[sup]18[/sup] J.

The average electrical power consumption in the republic of ireland is roughly 3000 MW… sorry very rough figure but it will do for this calculation.
so a 70kg person
Energy = 70 * (3 x 10^8)^2
Electrical energy consumed per year in ireland
E= 3000 * 10^6 * 60 * 60 * 24 * 365.25 (seconds in a year)

Divide the first figure by the second figure = ~66

So 1 person contains enough energy to power ireland for 66 years.

I googled for the US, but the figure i got from here
http://www.corrosioncost.com/utilities/electrical/
of 3,240 billion GWh seems way too high. Assuming that they meant 3,240 TWh, that would mean you or I, converted (without loss) to electrical energy could power the US for roughly 6 months.

Mass is energy is what it is saying. Whenever energy is released from a systen the mass of the system has to decrease. This does not just apply to matter-antimatter, theromnuclear, and atomic reactions but chemical and mechanical ones as well, but the amount of mass lost is very small to be almost unmeasurable in the last 2.

The reverse is also true.

Or at least this is my understanding of it.

So what is a joule then? I see scr4 defines it as:

``````
Joule = Newton * meter = kg*(meter/sec)2

``````

but what’s the concept behind it? It looks like a joule is basically the amount of energy needed to accellerate a mass to a certain speed. Am I close?

A joule is defined as theamount of work done when a force of 1 N moves though 1 m.

What “the reverse” implies is that, in order to produce 1 kg of matter, you’d have to run Ireland for 66 years, or that 100-watt lightbulb for 100 million years. And that’s assuming 100% efficiency and no losses, which is impossible by thermodynamics.

croakdale, did you mean 3000 MWh or MW for Ireland? There’s a big difference.

bnorton, since a Joule is defined as Newton * meter, and a Newton is the force to accelerate 1 kg at 1 ms[sup]-2[/sup], in exerting that force to move 1 m, you have expended 1 J of energy. If you do it in one second, that is 1 W of power. Ain’t SI great? Relativity is time reversable, infact you get reverse beta decay in which an electrons kinetic ends up forming part of the mass of a neutron, which is the process that forms neutron stars.

av8rmike, I did mean 3000 MW, instantaneous demand. I worked in the Irish Electricity co (ESB) when i was a student, not too many years ago, so I was just pulling that out of my head.

http://www.eirgrid.ie/EirGridPortal/DesktopDefault.aspx?tabid=Power%20System%20Operations
total electrical consumption in '99 was 21,080 GWh. 6% annual consumption growth, my estimate was close enough.

Thanks croakdale, thats just what I’m looking for. I’m not looking to give a complete and perfect example of mass/energy conversion, just a rough sketch to give them an idea.
btw, how much energy is in a 1 megaton nuclear bomb? I think telling them how big of a bomb they’d make would catch their attention.

Well, google calculator fails me for this, but it seems 1 megaton = 4.2x10^15 Joules.

Just FYI, that equation is unitless; you can shove in whatever you want. If your guy is 200lbs, and since the speed of light in feet per minute is 5.90142634 × 10^10, E = 1.2 x 10^13 pound-feet per minute. The pound-foot per minute being… uh… the amount of energy exerted over 1 minute by a force of one pound-foot. Nanoda, you mean E=1.2x10[sup]13[/sup] pound ft[sup]2[/sup] min[sup]-2[/sup].

Good point though. The question works for any unit. If you plug in mass in kg and speed in m/s you get energy in kg m[sup]2[/sup] s[sup]-2[/sup]. Now you use the definitition

``````
Joule = kg m[sup]2[/sup] s[sup]-2[/sup]

``````

to convert the units. The conversion factor happens to be 1 so it’s a very easy conversion. C’mon…did you read my post or the link I provided? My guess is no as the answer is there.

A one megaton bomb is around 10[sup]15[/sup] joules. A 75 kilogram person would equate to a roughly 6,750 megaton bomb by themselves. The only way to make somone explode in such a fashion (that I’m aware of) would require 75 kilograms of antimatter getting you a 13,500 megaton explosion. As mentioned in my quote and link above the largets bomb ever tested was around 100 megatons by the former USSR (although some cites suspect the yield was much lower than that). So, by yourself (assuming you weigh 75 kilos) you represent over 67x the strength of the biggest bomb ever tested.

Whoopsie, forgot the ^2. What’s a few thousand exploding-person units between posters anyhow?

Just to set the record straight the equation isn’t really

E = mc[sup]2[/sup]

Its

E[sup]2[/sup] = m[sup]2[/sup]c[sup]4[/sup] + p[sup]2[/sup]c[sup]2[/sup]

Or setting c = 1, E[sup]2[/sup] = m[sup]2[/sup] + p[sup]2[/sup]

So mass is therefore equal to the energy of the system that cannot be transformed away. IOW mass cannot really be converted to energy. Mass and energy are not things they’re properties of a system and in SR they are both universally conserved.