What's the significance of the squaring of symbols in many laws and equations eg E=MC2, the inverse square law

Question

I know it means the multiplaction of the factor by itself but am interested to know why the function crops up so often in famous expressions and what physical phenomena it represents.

Answer 1

Squaring shows up in many equations involving energy.

Interestingly, Isaac Newton started out thinking that kinetic energy was proportional to velocity (and not the square of the velocity) until in the 1730's Emilie Du Châtelet, a French woman and friend of Voltaire, evaluated some experimental data taken from a fairly simply test--dropping weights into soft clay and measuring the distance they pressed into the clay. She found, for example, when a weight hit the clay at twice the speed, it buried itself 4x as far into the clay.

I'm sure Newton was annoyed at having one of his ideas dressed down a bit, but that's why they run the experiments!

Answer 2

OK, you have a lot of very good answers here. I think some of them, like Prof. Zikzak and others, are probably right. a while back, I was reading an article on this very question, why c was squared in Einstein's equation, E=MC^2. I don't recall the exact details now, but the article stated that c was squared as an artifact of the technique that Einstein used to derive his famous formula. It said that the C^2 had to do with quantum effects and the fact that time itself is the 4th dimension. You may want to do some additional research on the question.

Answer 3

The real reason it pops up so often is that many physical phenomena are independent of the direction of the vector being squared; eg, in Ek = ½mV², the direction of the velocity can be in the + or - direction and give the same result.

By the same token, if you see a factor squared in a formula, you can ask yourself "should this give the same answer if I reverse the sign of (X)?" ------as a check on whether the formula is a valid one.....

Answer 4

Change in energy = dE = pdv; where p is momentum = mv and dv is change in the velocity of the body with mass m. Thus kinetic energy is E = Int(pdv) = Int(mv dv) = m Int(v dv) = (1/2) mv^2. Thus the square term comes from integrating (Int) the change in energy that is related to the change in velocity of a mass with momentum p.

Similar kinds of development for relativistic energy can be made to show the c^2 factor in E = mc^2. In other words, energy is directly related to the square of velocity for a given rest mass (and not inversely related as you state).

You might be interested to note that energy = fs = Mas, which is force f acting to move a mass M s distance. So the units of the RHS are (kg-m/sec^2)m or on rewriting kg(m/sec)^2 where m/sec are the units for velocity (meters per second) and kg is the unit for mass. Thus the units for energy include (m/sec)^2 which are the units for v^2. And there we have the square factor from a different approach.

Answer 5

I guess a lot of the time the variable being squared is a vector and a lot of things are scalar. And so the vector is squared. Really I like to think of it like this the universe is simple, so simple relationships exist normally a maximum of cubic ones.
Inverse square laws usually mean something is being spread over an area, inverse cube laws, over a volume.

Answer 6

Its just the way the universe is, a better explanation will be possible when a theory of everything uniting the forces but until then we are in the dark as it were.

Answer 7

Well, in this particular case, c is the universe's natural conversion factor from meters to seconds, and you need two of these to properly convert units from kilograms to Joules.

Answer 8

reflects the proportionality of the equation. as a mathmatical principle it doesn't have to be a part of a larger statistical group. (it can stand by itself.)