Topic 2-02: Mass, force and Newton's Second Law: The view of "newtonians"
Some readers probably find the path that leads to the concept of "mass" and then
to the second law of Newton as outlined in the previous topic rather twisted and
unsatisfactory. To shed more light on this topic we will reintroduce here the
concept of "mass" and "force" through the eyes of a species of intelligent creatures
in a fictitious world.
The "newton" is a strange astronomical object. It is not a planet, since it does not
orbit any star. It sits in the middle of nowhere, far from any stars and even from
nearby galaxies. It is not a star, since it does not radiate vast amounts
of energy, and its surface is solid rather than gaseous as stars are. However, it does radiate
a certain amount of heat as infrared radiation. The
newton is perfectly spherical, with a uniform crust several thousand kilometers thick. Inside this
uniform crust is a perfectly spherical cavity filled with some kind of "air". Within
the crust, some kind of thermal reaction occurs that maintains the
the cavity at room temperature. The inner shell of the crust
periodically lights up and darkens, creating day and night inside the cavity. Life
has evolved within the cavity. The intellectual creatures living inside the cavity is
called the "newtonian". Unlike the world of earthlings, the world of newtonians
is weightless; the reason why it is weightless inside the cavity of the newton will become
clear in later chapters.
Like earthlings, newtonians have an intuition of distance and time. However,
unlike earthlings, newtonians do not have legs, since legs are useless in a weightless
environment. They have a mouth to speak to one another, and a nose to suck in
“air”. They absorb a chemical compound in their “air” as fuel to warm up
the sucked-in air. The expanded air is expelled through their tube-like body for
forward momentum. The body of the newtonian is a flexible tube that allows them to change direction easily. If they want to
slow down and then stop, they only need to bend their body tube to point in the opposite
direction. They are experts at regulating the airflow through their body-tube to control the
speed of their motion. In other words, they are like super-sophisticated jet planes. If
readers want to know more about newtonian air, we will say that the magical chemical compound
that serves as fuel is constantly replenished by the inner surface of the crust of the astronomical
object called the “newton”. No matter how to look at it, the newton is probably a lab of
genesis created by some super intelligent beings. Actually, this basic lifestyle, including the
mode of propulsion, is universal for all the creatures in the world of newtonians; only newtonians
have evolved a superb brain and very high intelligence. Since all the creatures in this
world derive energy from their “air” and are all unisex, there is no need
to devour each other as food or to fight one another for “companions”. Thus,
the world of newtonians is perfectly peaceful, and newtonians can devote a lot of time to, you
guess it, studying physics.
Among many things that newtonians have created are springs. They anchor one end of a
spring to the wall of the cavity, and grab (yes, they have hands) the other end of the spring.
When they try to fly away by expelling a fixed amount of air at a fixed speed per unit time,
the spring stretches by a certain amount. If they expel more air or expel the same amount of air
but at a higher speed, the spring stretches more. Thus they learn to use the spring to measure
their air thrust; they call the application of air thrust "force". They also know that by
applying a constant air thrust, or force, and moving in a straight line, their flying speed will
increase steadily, but the acceleration is constant. If they increase the "force", the
acceleration increases accordingly. Thus they have discovered the relation
f = ca,
(1)
where f is the force measured by the stretch of a standard spring, a is the acceleration
and c is just a constant. They also discovered that the proportionality constant c is not universal
but depends on who is doing the experiment. In general, a larger newtonian will result in a larger c.
To understand more about the proportionality constant c in Eq. (1), newtonians performed the following
experiments. They made many identical balls. A newtonian, carrying no ball, flies with a fixed force f,
or thrust, and achieved a constant acceleration a0. The same newtonian, carrying one ball and
using the same thrust, flies with an acceleration a1. Carrying two balls, it achieved an acceleration
a2, and so on. Then the results are plotted on an x-y graph as shown in Fig. 1, where the
vertical axis represents the acceleration and the horizontal axis represents the number of balls. It
turns out that the curve of Fig. 1 can be fitted by the equation
where f is the thrust, or the force, n is the number of balls, and m and d are constants. If the size
of the balls are changed, but the material used remains the same, then m is proportional
to the volume of the ball. The constant d depends on the size of the newtonian, larger the size larger
the constant d becomes. At this point newtonians conclude that every object has an intrinsic quantity
called "mass". The mass of a ball is m, and the mass of the newtonian who is carrying out the experiment
is d. Thus nm+d is the mass of the system that consists of the newtonian and n balls. With a system of
mass m, Eq. (2) becomes
f = ma .
(3)
Newtonians call Eq. (3) the "golden rule", but earthlings call it Newton's Second Law. With their golden
rule, newtonians have developed a classical mechanics similar to the classical mechanics of earthlings.