Boy of Age 16 Asks Me about Relativity, etc. 11. How Do We Prove a=F/m?

Newton's first and second laws, in Latin, from the original 1687 Principia Mathematica. Source: http://www.loc.gov/exhibits/world/images/s123.jpg
[Public domain], via Wikimedia Commons.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: How do we prove a=F/m?

Ted: Your question is considered to be equivalent to this: Can we derive Newton's second law of motion F=ma from a more fundamental principle? (Here, F is the net external force acting on a body, m is the mass of the body and a is the acceleration the body obtains.) We cannot derive this equation from any other principle. Newton assumed it and presented it as a law. The validity of the equation can experimentally be confirmed. For example, we measure accelerations of the bodies of different masses by applying the same force to find that the result is inversely proportional to the known masses. Another confirmation comes from the application of the law to planetary motion. The law combined with Newton's law of universal gravitation neatly explains Kepler's laws of planetary motion empirically found. Thus, the equation is certainly a law. This law can be considered equivalently as a fundamental postulate in the (classical) mechanics of a particle* or as a definition of force and mass [1].

* The mechanics of a particle treats the motion of bodies (objects) which are considered or idealized as a particle, in the sense that the extent of the body is neglected in the evaluation of its motion, i.e., the object is small compared to the distances involved in the analysis, or the deformation and rotation of the body is of no importance in the analysis [2].

By the way, April 18 was the anniversary of Albert Einstein's death, so that I made a short blog post about it. Please enjoy the article here.

References

1. H. Goldstein, Classical Mechanics (Addison-Wesley,1950) p. 1.
2. "Newton's laws of motion," Wikipedia: The Free Encyclopedia (19 February 2012 at 09:36).

(Originally written on April 20 and 22, 2011)