Why do we square 'things' ?

[steveb]

Originally Posted by amilton542

The video is called 'Einstein's Big Idea' which is a breakdown of scientific breakthroughs leading upto his famous formula ...

However, the chapter '2 is for squared' and the work Émilie du Châtelet did makes my problem more clear.

Steveb I would be grateful if you could view this ten minute chapter of the video and explain in more detail.

The chapter '2 is for squared' begins at 54:36 and goes on for approximately ten minutes.

Thanks for posting the video. I watched the whole thing and found it educational, interesting and entertaining. I especially enjoyed seeing my heros (Faraday, Maxwell and Einstein) come to life.

I'm not sure of what to say about that section of the video, but I can make an attempt as follows.

My best interpretation is that we need to realize that Newton and the scientists of his day did not have our perspective on Newton's laws of mechanics. There is no doubt that Newton fully understood his own laws, but how these laws interface with all other physics that we know today is something that he could not know about. In a sense, he invented the first reasonably complete theory of mechanics, and it was difficult to link this theory with other areas of science. He had no knowledge of theories of relativity, quantum mechanics and electrodynamics, nor did he live to see the Hamiltonian and Lagrangian formulations of his own laws. Hence, the fact that Newton and Leibniz might not see eye-to-eye on the importance of kinetic energy versus momentum is perhaps not too surprising.

We now readily think about and understand the concept of energy. We know that for the many experiments energy is conserved and mass is conserved independently. We also know that other experiments can show that mass and energy can be converted and are in a sense different forms of the same thing.

Apparently, Newton did not think from the same point of view, even though his laws of mechanics had the concept of energy embedded within them without doubt. Is seems Newton viewed momentum (mv) as the more important quantity of interest, while Leibniz recognized kinetic energy (mv^2/2 or mv^2 or anything proportional to v^2) as an important characteristic of a body. In some sense this comes down to semantics. Newton was not really saying that mv is energy, but most likely was just saying that mv is important to consider when looking at interactions. In reality both are equally important, so maybe the video is giving a biased view of what the debate is about. Fundamentally, momentum and energy are both conserved and are both important quantities to consider. Indeed, modern relativity theory links both of these into one 4-vector indicating that our perception of these quantities depends on our frame of reference.

[amilton542]

@ Steveb

Thank you for your generous interpretation. Me, myself, also enjoyed the video and I now fancy the actor who played Émilie du Châtelet.

As a reward for my gratitude, you will recieve new bedsheets of your named 'heros' through the post in the next few days.

[panic mode]

Originally Posted by K7GUH

OK, guys, now that you're squared away, how do you explain the formula for the volume of a sphere?

it is actually very simple, in fact you get formula for any shape - using calculus.
the key idea is to divide object into many small pieces and 'count them up'.
the smaller the pieces (use limit), the more accurate sum. this is what integrals are - sums when 'pieces' are infinitely small.

area is square function, volume is cubic... cube of side a has volume a^3 etc. there may be some other scaling factors involved (like 4pi/3) but ultimately volume is cubic (in volume equation for sphere it is r^3).