Reading science books is scary ! All those formulas ! To tell you the true such books intimidate me too.
Still ... a science book without a formula does not provide any insights on the topic.
As a layman myself I can appreciate the warm, fuzzy feeling people get reading the popular books on Universe and everything, but then comes the realization..
Now WHAT ? I got the general gist, but how this works in real life, what does these ideas really mean, what is the deeper true.
The only way to grok what the ideas mean is to have a formula or two deciphered.
It seems to me nowdays scientists shy away from using formulas when they write popular books, either because they think that the books would become unsellable or
are just too lazy to spend some more time and "wear the shoes" of us, the mere-mortals. What a waste :(
Pick some book from the 20's, 30's, 50's, 60's .... Scientists back then had the knack of explaining ideas via formulas, diagrams and words.
"Picture is worth a thousand words, formula is worth a thousand pictures".
So I would take upon a task in this article to try to provide basic information for layman's like me how to read formulas without being afraid.
I still have just a vague idea what exactly I will come up with, but I hope some brainwave will command those agile fingers to type some
non-gibberish. Wish me luck ;)
So here goes.. We should start with the algebraic formulas first, those would be easily decipherable from anyone who has attended high school.
Once we move from pure mathematic to the realm of Physics all
symbols in the formulas take on a new life from abstract quantities to describing real measurable "entities" or patterns of reality.
Symbols representing discoveries after myriad of experiments, logical deduction and intuition from science legends and geniuses, which we lovely 'engraved' for posterity in the Unit names we use in our everyday life.
Almost any physical law contains some sort of constants and variable parts.
The constants normally allow us to convert a possible deduction or pattern between properties from
relation(~) to
equation(=).
For example it is one thing to say :
`F ~ (M*m)/r^2`
it is another to state :
`F = G * ((M*m)/r^2)`
Do you see the difference in the second case we can exactly calculate the quantity of the force or any of the masses.
(I hope you recognized that this is the Universal law of gravitation).
As you probably expect those Constants most of the time are 'bolted' in the equation initially and later their value is experimentally calculated.
Constants normally are not derived from first principles they are just given.
The second part, the most important one is our primary goal here, the relation i.e. the pattern of nature we want to capture, it involves the measurable variables.
These variables are axiomatic things like volume, pressure, weight, temperature, speed... Things of which we have some intuitive idea of what they mean.
Some of those have been refined over time as we learnt more about how nature behaves... for example we know now that weight is not an intrinsic property of matter
but mass is the important one. Or expanded others to acquire new meaning .. for example we use speed only sporadically, but use velocity instead.
Again the major theme in the whole exercise is to provide a quantitative predictability, which underpins today's technological advancements.
Now that we know WHY, lets see HOW to interpret it.
For this we will use Plank law (black body radiation):
If you look carefully you will see that the only variable thing in this formula is the
frequency(f) and
temperature(T) the rest are just constants.
One more thing if we strip the equation from those constants we will get something along the lines of `B ~ f^3/e^(f/T)`.
What can we deduce from this shorthand :
- first if we look at the nominator, if the frequency increases we expect the radiation to increase thrice-exponentially `f^3`
- second if we look at denominator if the frequency increases the denominator grows i.e. the radiation drops i.e. we could expect this to
balance in some way the nominator. Then if the temperature grows the denominator will drop i.e. radiation will increase.
So what we see here is a subtle interplay of two variables. To be sure exactly what "curve" of radiation we will get we could try to plot
a graph and see how it really looks like. We can see that here :
wolframalpha.com
If we go back to the Universal gravitation formula things are even easier to decipher.. the bigger the masses of any of the object the bigger the
gravitational force... the further away they are the gravitation falters even faster (`r^2`).
So this in essence is how we can read simple algebraic physical formulas. First thing first mentaly divide contant and virable part of the equation i.e. watch which variable part change the result in what direction, try to figure out the curve that will
be drawn if we decided to plot it. Even if still don't know how to solve specific physical problems we have more intuition of how the underlying reality plays out in gross terms.
So far so good but what if any variable in our formulas is not a single value like temperature, frequency, mass ? OR !
What are we to do if we have to apply the gravitational law on multiple bodies which can be anywhere in 3D space ?
We can probably use the same law but use some geometry to account that the object does not lay on the same plane or we can do the calculations
for the bodies two by two and find some other way to correlate the results. But the more we think the more it seems like a waste
of time. There should be a better way to handle these cases.
Yeah, there is such idiom, it is called
VECTORS
More to come ...