Why is stuff squared?

[CoolATIGuy]

Well, the subject line was a bit blatant. Basically, I'm wondering why many physics equations are squared, e.g. E=MC2, kg*m/s2=Newton, etc. Maybe it's just coincidence that it happens to solve the equation, but I've got a hunch that maybe there is some common physics factor that requires things to be squared for a common purpose. Does it have anything to do with the inverse square law ( http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html )? If I'm totally off-track and it's not, then why are things like the afore mentioned squared, i.e. how does it complete the equation for those two examples?

 

Cheers!

 

 

CoolATIGuy

[Severian]

For E=mc² the 'c' needs to be squared to make the dimensions balance. If you remember from your Newtonian dynamics that kinetic energy is 1/2 m v² (for non-relativistic objects), since velocity (v) is measured in m s^-1 (where here 'm' is metres) then energy must be kg m²s^-2. You can see that if we had E=mc, we would not have an energy on the right-hand-side of the equation - c has to be squared.

 

The inverse square law is a little different, since in principle there is somthing to stop us having a 1/r or 1/r³ law (or something completely different). It is interesting to note though that the 1/r² force produces stable orbits (or to be more exact, orbits which are stable to a small perturbation) - the 1/r, 1/r³ (or any other power) do not, so if gravity had been one of them there would be no formation of planets and no life as we know it.

[swansont]

For E=mc^{2[/sup'] the 'c' needs to be squared to make the dimensions balance.}

 

^{Well, there's more to it than that, of course. Just making the units match doesn't mean an equation is valid; it's a necessary but insufficient condition.}

[Skye]

Look at how the relationships are derived mathematically, it will differ from equation to equation.

[Severian]

Well, there's more to it than that, of course. Just making the units match doesn't mean an equation is valid; it's a necessary but insufficient condition.

 

Obviously! But a mass-energy equavilance equation in relativity would have to be of this form since c is the only fundamental constant you have got to play with.

[Martin]

ATI,

 

cant answer yr question (beyond what severian, swansont et al said already)

but I can comment.

 

Suggest you browse around at Motion Mountain

http://www.motionmountain.net/

 

look back at the roots of physics---archimedes, kepler,....

they were looking for ALGEBRAIC and GEOMETRIC relations between

things in nature

 

the most simple relation is straight proportionality (straight line relation)

the next most simple is proportional to square (parabola relation, quadratic)

 

Archimedes found some relations between things that involved squaring and cubing (volumes and areas of cones and spheres and cylinders, plus lots of greeks were interested in conic section curves like parabolas and stuff)

 

Kepler 1618 found a relation between the period of a planet and its distance to the sun----it involved squaring and cubing (or the one-and-a-half power)

 

If something is just not PROPORTIONAL to something else (pretty damn simple) then the NEXT SIMPLEST THING is for it to be related by some square law or inverse-square law, or cube law.

 

For 2300 years physicists have been looking to find the simplest relations they can between things. the way they think, the algebraically simpler it is the more beautiful. It is not their fault that not all things are straight-line proportional, they are after finding the simplest formula that will work.

 

but nature is not all straightline-----the area of a square is not simply proportional to the diameter, the area of a sphere is not simply proportional to the diameter, the volume of a sphere is not simply proportional to the area.

 

when you drop something it doesnt fall at a constant speed----at least at first it accelerates-----feet per second per second is the same algebraically as feet per second-squared.

 

Yeah, as you suspected, there are several reasons that square occurs a lot in formulas. Severian and swansont told some reasons. I am not telling you more reasons. I am telling you to be glad so much is so simple. Be glad it isnt worse. Be glad that even tho some things are not just straight proportional that at least they are related by the square, the next simplest.

It could be worse.

[revprez]

The inverse square law is a little different, since in principle there is somthing to stop us having a 1/r or 1/r^{3[/sup'] law (or something completely different).}

 

^{It's called area. For forces emanating from a point source, you would expect intensity to diminish as an inverse function of area. Since area is a squared function of distance, you'd expect an inverse square relationship between a force field's strength at any given point and distance.}

[CoolATIGuy]

Wow! Thanks for the posts, guys!

 

However, I am a bit confused...maybe it would help to start by asking for clarification on this:

 

For E=mc2 the 'c' needs to be squared to make the dimensions balance.

 

Why in (Energy) = (Mass) x (Speed of Light, squared), is the SoL squared at all? I'm trying to grasp the reason, but all I can come up with is the fact it is going through 3d space, but in that case it would be cubed. I'm just very lost as to *why* it completes the equation...

 

Thanks, and I look forward to great enlightenment!

 

 

CoolATIGuy

[Martin]

ATI, maybe what you should be asking right now is why, in the metric system, a joule of energy is defined like this:

 

[math]Joule = kilogram\frac{meter^2}{second^2}[/math]

 

If energy is meaured in such units then if a things energy is going to be proportional to its mass then the only constant of proportionality possible is going to be the square of some speed.

 

So it is almost a foregone conclusion!

 

So suppose you ask that.

 

Well you already said in an earlier post how the FORCE unit is defined:

 

[math]Newton = kilogram\frac{meter}{second^2}[/math]

 

and the root concept of energy is WORK which is force x distance---a unit of work is performed by pushing with unit force for unit distance---like raising a weight one meter by pulling with Newton force on a pulley rope.

(ultimately physics boils down to childishly simple situations which are then elaborated to intricacy by a maze-like web of equivalences)

 

[math]Joule = Newton \times meter = kilogram\frac{meter}{second^2} \times meter [/math]

[math]= kilogram\frac{meter^2}{second^2}[/math]

[Severian]

Or to put it another way, we know that momentum is proportional to mass times speed (well velocity...). And speed is measured in metres per second.

 

So momentum must be kilogram x metre / second.

 

Now in relativity, energy and momentum are related in the same way as time and distance. Clearly time and distance are related by a speed (a distance per unit of time), so energy must be momentum times a speed, ie. E = p c.

 

But since momentum is energy times speed we must have E = mc².

 

[There was some very vigorous hand-waving there. For those who are happier with things a little more complex, the correct mass energy equation is [math]E^2 = m^2 c^4 + p^2 c^2[/math]. When m=0, this reduces to E=pc, while with p=0 it reduces to E=mc².]

[granDODyssey]

square means a rectangular shape right?...so it is a 2D geomatry shape...

i would look back to the pythagoras' theoram.... if you have a right triangle(having one of the angle 90 degree)

the relationship of it's sides can be expressed as a*a + b*b = c*c ( being a , b and c are the 3 sides of the triangle. c is the one facing the right angle of course)

let's say the length to a to b is 3 ( so you can represent it a line of 3 smiles

but it's a just a line

 

if you make the square of it

effectively you got a total of 9 smiles which is 3 * 3 ( that's why we call it "squaring". Whenever you multiply a number(which can be represented by a line), you can get the square shape of that line)

 

so Pythagoras could see that if one can make a square shape cut-outs for each side of the right trangle. the sum of the area of the smaller two cut-outs( e.g a*a + b*b ) will be the same as the area of the largest one..

 

[CPL.Luke]

well the original idea of 1/2mv^2 was probably found by making objects fall with different speeds and weights, from this you find the relationship

 

from there when einstein came to the conclusion that E=mc^2 he new that the energy (momentum essentially) has to be equal to mv^2 except that the speed of light is v

 

so you replace v with c

 

Momentum with e because you are measuring the total amount of energy in an object not just the momentum

 

you have e=mc^2

 

the whole squareing thing comes mostly off of peole along time finding relationships between thigs such as velocity being heavily weighted over mass

and a number of other fields that I just can't think of now