E=MC^2

[Kebsis]

When calculating with this equation, what are the measurements used? For instance if you use 1gram for M, then E will equal however many what?

[Bob]

The cgs system (centimeter-gram-second) unit of energy is the erg. The preferred mks system (meter-kilogram-second) unit of energy is the joule.

[joema]

Note there's an easy conversion tool built into Google.

While at the Google.com search box, just type in your conversion request. It outputs the answer. It accepts a wide range of inputs. Examples:

convert 10000 ergs to watt seconds
convert 2 parsecs to furlongs
convert 2e4 cubic km to cubic miles
convert 10000 pascals to psi

[01101001]

While at the Google.com search box, just type in your conversion request. It outputs the answer. It accepts a wide range of inputs.

Hrmph. I was going to chuckle at Google by typing in "convert 1 kg to joules", thinking it could only translate units and wouldn't understand converting matter to energy.

The laugh is on me.

1 kilogram = 8.98755179 × 10^16 Joules

Edit: PS: I think the 'convert' is superfluous. "1 kg to joules" yields abot the same. In fact, I think 'convert' and 'to' are not recognized as keywords by the calculator, because besides a numerical result, Google also shows search results. The form I learned for conversions, "1 kg in joules", where 'in' is a keyword, only shows the numerical result.

[frogesque]

Remember that all units have to be consistent so if you are using the cgs system c must also be the speed of light in cm/sec and if using SI units then c must be in m/s. A small point that makes a big difference and one that can be easily overlooked.

[Normandy6644]

Remember that all units have to be consistent so if you are using the cgs system c must also be the speed of light in cm/sec and if using SI units then c must be in m/s. A small point that makes a big difference and one that can be easily overlooked.

I did that in an E&M class that used cgs units all the time. The speed of light is not 3x10^8 anymore. :lol:

[Evan]

Be careful. The Google calculator does not always give correct results. I don't have the example handy but was using it for some simple trig and it was consistently wrong in the .001 significance range. I e-mailed their tech support and they said they would look into it. I haven't checked it lately. It might be wise to do a few spot checks on sin/cos to see what it returns.

[Saluki]

Remember that all units have to be consistent so if you are using the cgs system c must also be the speed of light in cm/sec and if using SI units then c must be in m/s. A small point that makes a big difference and one that can be easily overlooked.

This is critical. One of the things that I learned well in engineering school was always include units in your calculations when you start replacing variables with actual numbers. It takes slightly more time, but the payoff is massive in terms of being able to catch many common errors. Your units should cancel out to a final unit that makes sense. For example, in Einstein's mass-energy equation if:

m = 1 kg
c = 3 x 10^8 m/sec

E = (1 kg)(3 x 10^8 m/sec)²

E = 9 x 10^16 kg-m²/sec²

but a kg-m/sec² is a Newton (the SI unit of force). See how Newton's second law leaps out here? We have a mass (kg) times an acceleration (m/sec²), which gives us a force. Beautiful!

so, E = 9 x 10^16 N-m

Here, we see a force times a distance, which tells us we have Work, which is equivilant to Energy, and in fact, 1 N-m = 1 J.

This may seem like a complicated way to analyze it, but IMO, it is important to understand these relationships between the defined units (distance, time, mass, etc.), and the derrived units (force, energy, velocity, pressure, etc.). If we don't, we do not even know if we used the right units.

[frogesque]

Saluki I had that dinned into me during my Engineering course too. In both Fluids and Thermodynamics if the units didn't pan out then it was a big fat zero for fundamental error. My early tech college was based on Foot Pounds Seconds and then higher education was SI

[Saluki]

I recall doing physics and chemistry homework with friends who were non-engineers. They would just start plugging in unitless numbers, and then apply the units they thought should be there at the end. They laughed at how "tedious" I was, but stopped laughing when they noticed that I seldom had to re-work a problem to get the correct answer.

Edit: Another nice thing about it is when it comes to unit conversions. If you dont recall how many joules there are in a foot-pound force, you can just break the derrived units down into their component units and convert in multiple steps.

[Ricimer]

Definetly remember the units. I've done quick back of the envelope proofs for my physics homework using "only" the units. No values, I didn't even know what gets multiplied, what gets divided etc. All I know, is I had to end up with something (like energy) and I knew what I started with.




Anyway, for units: when using kg for mass, use meters for distance. When using grams for mass, use cm for distance. using pounds for mass...you're nuts.

[Saluki]

What if I use Slugs for mass?

[Bad jcsd]

What if I use Slugs for mass?

It would depned which units you used for speed (snails perhaps?)

The thing to remember about dimensional analysis is that it can never tell you if you are correct it can only tell you when you are wrong (i.e. having the incorrect units guarentees you are wrong, having the correct units gurantees nothing).

[Saluki]

What if I use Slugs for mass?

It would depned which units you used for speed (snails perhaps?)

The thing to remember about dimensional analysis is that it can never tell you if you are correct it can only tell you when you are wrong (i.e. having the incorrect units guarentees you are wrong, having the correct units gurantees nothing).

Slugs are actually a USCS unit. It is the mass that would be accelerated 1 foot per second² by a force of one pound. The USCS system is a little backward in that it uses force as a defined unit and mass as a derived unit. A slug is equal to about 32.2 pounds mass (because g≈32.2 ft/s²).

You are right that it does not guarantee you the correct answer, but it seems to eliminate 70%+ of the common errors made in a complex problem.

[A Thousand Pardons]

having the incorrect units guarentees you are wrong

Not quite. :) Notice the equation that 01101001 found above (http://www.badastronomy.com/phpBB/viewtopic.php?p=395903#395903):


1 kilogram = 8.98755179 × 10^16 Joules

That's certainly right, in the context of this thread.

[01101001]

having the incorrect units guarentees you are wrong

Not quite. :) Notice the equation that 01101001 found above (http://www.badastronomy.com/phpBB/viewtopic.php?p=395903#395903):


1 kilogram = 8.98755179 × 10^16 Joules

Yeah, the numbers don't matter quite so much, as long as you include the units, with a smart calculator:

Google query: 1km / 2 hours in furlongs/fortnight

Result: (1 km) / (2 hours) = 835.122882 furlongs / fortnight

[Bad jcsd]

having the incorrect units guarentees you are wrong

Not quite. :) Notice the equation that 01101001 found above (http://www.badastronomy.com/phpBB/viewtopic.php?p=395903#395903):


1 kilogram = 8.98755179 × 10^16 Joules

That's certainly right, in the context of this thread.

ahh, but that's not quite true.

if you take kg and J as units of different quantites

1 kg = 8.98755179e+16 J

iis incorrect and must be amended to:

1 kg x c^2 m^2/s^2 = 8.98755179e+16 J

Howvere you can take them as units of the same quantity (though in the SI system they are units of different quantities) then the firts statemnt is true, howvere diemnsional analysis will not tell you that it is incorrect as it iis simply a conversion from one set of units to another.

[Evan]

I found the example of where Google calculator makes a mistake. When calculating the arctan of a slope to find the angle:

Enter .45 inch per 10.18 inch in degrees and it gives 2.5327211 degrees. This is incorrect.

The correct answer is 2.531073365

This of course makes Google calculator suspect on all other calculations.

[01101001]

I found the example of where Google calculator makes a mistake. When calculating the arctan of a slope to find the angle:

Enter .45 inch per 10.18 inch in degrees and it gives 2.5327211 degrees. This is incorrect.

The correct answer is 2.531073365

This of course makes Google calculator suspect on all other calculations.

Don't you want arctan (.45 inch per 10.18 inch) in degrees ?

It gets that right, no?

Edit: arctan (x) ~ x for small x. You're just giving it a ratio close to zero and it yields a dimensionless quantity (which is then interpreted as radians and converted to degrees). Don't you want the angle that corresponds to a rise of .45 per run of 10.18 (arctan) and then convert that to degrees?

Reedit: An example less close to 0:

Query: (1/1) in degrees

Result: 1 / 1 = 57.2957795 degrees

Query: arctan(1/1) in degrees

Result: arctan(1 / 1) = 45 degrees

[Saluki]

I guess it depends on what you are doing with the information. Three sig figs is good enough for the vast majority of applications, but is woefully inadequate in some cases.

They probably are using some estimating technique to do these calculations, and you are finding the limitations of the technique.

If you want to get picky, the correct answer is:


atan (0.45/10.18) ≈ 2.5310733653778412722809963115323º

[Evan]

I don't consider 0.044204322 close to zero. Google is calulating to ten sig. There is something wrong with the algorithm that is doing the interpreting. I entered it as someone might that doesn't know trig.

[01101001]

I don't consider 0.044204322 close to zero. Google is calulating to ten sig.
Edit 2: I'm not sure you got my point there. I mentioned "close to 0" because the angle you are dealing with maybe is making you think there is just a small defect in the trigonomy functions that is yielding a result off by a tiny amount. That is not the case. See the 100/1 example below for a result that is right, but "off", for you, by a factor of about 64.


There is something wrong with the algorithm that is doing the interpreting.
I disagree. It did as you requested.


I entered it as someone might that doesn't know trig.
So... 1 inch per 1 inch in degrees should not be 180/pi degrees, and should be 45 degrees?? I wouldn't use it if it did that.

Edit, after thinking of a more extreme case:

I really really really want the calculator to give the same result for 100 in degrees, 100/1 in degrees, 100 per 1 in degrees, 100 inches per 1 inch in degrees and 100 seconds per 1 second in degrees to all give the same result: 5729.57795 degrees

I do not want one of those equivalent expressions to give the incorrect result 89.4270613 degrees. Too bad for those who doesn't know trig.

About all I'd be willing to budge is for it to give the correct answer and maybe ask: Did you mean arctan (100 inches per 1 inch)? in one case. But only maybe.

[pghnative]

I don't consider 0.044204322 close to zero. Google is calulating to ten sig. There is something wrong with the algorithm that is doing the interpreting. I entered it as someone might that doesn't know trig.

If I interpret 01101001 correctly, Google is interpreting your expression differently than you think. It is taking your expresion (0.45 inch per 10.28 inch) and presumes that that value (0.044204322) is in radians. It then converts radians to degrees.

So the algorithm is fine. It is the conversion of Evan-speak to math that is the problem. I doubt that google will ever be able to guess the different ways all users may try to write something.

[tlbs101]

What if I use Slugs for mass?

You would crash into Mars.

:lol:

[Evan]

Yes, I see what Google is doing. Part of the problem though is that the Google calculator allows "natural language" queries. This can easily lead to someone assuming they have phrased a query correctly and then produce a subtle error that looks reasonable as in the example I gave.

[01101001]

Yes, I see what Google is doing.

Ah, good. Thanks. I hope I wasn't too strident. I blame it on Huygens-induced sleep deprivation.


Part of the problem though is that the Google calculator allows "natural language" queries. This can easily lead to someone assuming they have phrased a query correctly and then produce a subtle error that looks reasonable as in the example I gave.
Well, yeah, it seems like it processes natural language, but I think that is just because they did a really good job of handling units and common natural-language phrasing for mathematical operations (like per). I don't think they have any intention, nor claim, to be able to parse meaning out of ambiguity.

The only way I can see them make your example work is by breaking what for me is a consistent language for calculating, one that I can rely on now.

[A Thousand Pardons]

Yes, I see what Google is doing. Part of the problem though is that the Google calculator allows "natural language" queries. This can easily lead to someone assuming they have phrased a query correctly and then produce a subtle error that looks reasonable as in the example I gave.
Yahbut you have the same problem with English (or any other language). When you said

The correct answer is 2.531073365

This of course makes Google calculator suspect on all other calculations.
I assumed that you were claiming that their calulations were inaccurate, but your later post

There is something wrong with the algorithm that is doing the interpreting. I entered it as someone might that doesn't know trig.
seems to say that you only have a problem with their interpretation algorithm.

That's unfair, then, to say that their calculator is suspect on all other calculations.