Force, gravity, inertia and mass
By Tom Brown
Wed, 29 Oct 2014
Classical mechanics is a branch of physics that deals with physical force and gravitational attraction, inertia and mass of particles and rigid bodies. This then in the geometrical framework of kinematics– time, space, distance, speed and acceleration. It concerns as well the concepts of momentum and energy. The foundations were established by Sir Isaac Newton.
The terminology has changed with time for example Newton spoke of Force in terms of “Action”. The notation of Calculus has also become quite different and favours Wilhelm Leibniz who developed Calculus independently, concurrently. The principles and laws as set out by Newton are the starting point and all of classical mechanics may be derived from them.
In fact it is the same axiomatic approach as in any formal mathematical theory, for instance our familiar school geometry. A handful of axioms and definitions are set out and the whole theory of Euclidean geometry is systematically derived from there using only the rules of logic.
The development of theoretical mechanics coincides exactly with that of Calculus which must the most powerful tool the human mind has ever invented (admittedly “discovered” is probably more accurate). In my opinion mechanics in turn has to be the most successful scientific theory to date, no doubt partly due to its amazing simplicity.
Statics comprises the study of mechanics without consideration of time for instance in structural analysis of buildings. Mostly it involves torque, forces such as weight, and geometry i.e. lengths and angles. It is applied in things such architectural planning of buildings, bridges, furniture and machines such as workshop hand tools and pulleys. The aim is usually for strong and optimal design.
As well as space and geometry, and force and torque, dynamics is concerned with time, movement, or change in a physical system. Time manifests as speed, rotation and acceleration. Dynamics is the basic modelling tool for analysis and design for such as in racing cars and aircraft, ballistics planetary motion and astrophysics, of space vehicles, weapons rockets and many machines and inventions, almost any conceivable physical thing involving time and motion, mass and force.
Weight and Mass
When one wants to understand the principles of mechanics there are a few basic things that you must know. Most people are aware weight and mass are not the same thing and this should be common knowledge. Mass is a measure of “amount of matter” whereas weight is the force of attraction on an object due to gravity.
For example although your mass is exactly the same everywhere, you would weigh zero in far outer space, or say you landed on the surface of Mars, with a gravitational acceleration of 3.8m/s² you will be 39% of your weight on Earth. The SI unit for mass is kilogram. Weight is a force, and is rightly measured in Newton. Standing on the ground, on the Earth surface if you are 75kg your weight is 75kg x 9.8m/s² = 735 N.
Galileo Galilei observed that if a mass is in linear motion at constant speed and there is no resultant external force then it will just carry on in the same manner, straight ahead indefinitely holding the same velocity and direction. This is usually called Newton’s First Law but had already been demonstrated by Galileo, whose very simple experiments yielded some unexpected and surprising results.
To explain, a racing car for example experiences an external force that is the effect of friction such as air resistance, which slows an object down. The engine in turn could cause increase in speed, that is acceleration. The external forces experienced by the vehicle on a level road would be that because air drag and those from the wheels, the contact surface of the tyres of a car. When the resultant external force is not zero speed increases. If the resultant is zero then all the forces balance and the car would ride at a constant speed.
Mass as inertia in the sense of sluggishness, is the ability of a body to resist, to withstand force and keep momentum. The relation between force and acceleration known as Newton’s Second Law can be formulated in the equation
F = ma
It states that the external force needed to move an object is the product of its mass and its acceleration. This law was originally formulated in terms of momentum.
The Principle of Action-Reaction
As an interesting example of how force works in practice, I have a stone in my hand and I squeeze it gently– the stone pushes back and just as lightly. I grip the stone in my hand as hard as I can but it doesn’t collapse it just pushes back harder, just as hard as my grip. Then let me put it in a bench vice. The force now is terrific but even so the stone pushes back the same. If it is a diamond it will never break a diamond is too hard the vice will give in first. This is how a human’s will to live also works. As hard as you force it, as strong it becomes.
The principle which Newton called Action-Reaction does not have the same meaning as the familiar idiom. Newton called force “action” so that “reaction” would mean “counter-force”.
The idea may be explained easily. If you stand on a bathroom scale your feet act on the scale giving your weight, and the scale pushes back with the same force. The scale experiences a force and you feel the same force, but in the opposite direction- upwards. When thinking of a falling brick the Earth pulls the brick downwards falling all the faster, whereas the brick pulls the Earth up with the same force, the brick’s weight.
This principle applies to every force. It is independent of any considerations of acceleration and with no regard to inertial systems or any other circumstances.
Newton’s Universal Law of Gravitation can be formulated in the equation
F = GMm/R²
F is the force of attraction between two bodies due to gravity, G is a (universal) constant, M and m are the respective masses, and R is the distance between the bodies.
Surprisingly (in fact!) in Newton’s 2nd Law ( F = ma ) and in the law of gravitation the mass m is equivalent and may be taken as equal then.
In the given two equations the mass m has different and distinct meanings. In the second law m is inertia as “sluggishness”, and in the law of gravitation m is mass in the sense of “amount of matter”. Thus Newton had in fact made an assumption– that gravitational mass is equivalent to inertia. Galileo as a predecessor of Newton had discovered in experiments that all bodies fall at the same rate, in absence of other influence irrespective of their weight. The motivation for equivalence of inertia and gravitational mass is exactly Galileo’s experiment.
The dependence of gravitational force on distance R is something totally apart. Newton made the discovery by thorough study and analysis of the orbits of planets and specifically the geometrical laws of Johannes Kepler. His inverse square law of gravitation could explain these so that in fact the observations of Kepler was the evidence and motivation.
I am sure we have all engineers and professional scientists in smiling and confident smug superiority by now. Thus may I offer a few very elementary challenges.
We were told the story of the great Archimedes and his buoyancy law in high-school, and everyone threw up their arms in great joy and ran out of the classroom shouting Eureka Eureka!
Great! “We have it!!” My question simply is, what have we? I mean, why is it so? Why does a floating object displace its own weight with water? If you don’t know, then you can’t run around shouting Eureka Eureka now can you?
The following very successful basic “rule” is usually used in modelling of engineering mechanics problems and comes back even from high-school days. F represents a frictional force, N is the normal force acting between two surfaces and μ is a constant:
F = μN
An example would be when a block of wood is pushed sliding on a smooth level cement floor at constant speed. The resisting force F would now equal the applied force, and N the block’s weight. In practice the coefficient of friction μ is measured experimentally and depends only on the surfaces. No other considerations are directly involved.
The question then is, what is the motivation behind this (heuristic) rule?
The well-known formula for kinetic energy is E = ½ mv² whereas Albert Einstein’s equation for equivalence of mass and energy is E = mc² with c as light-speed. Why?
Bob Dylan– Did Einstein really play the electric violin?
As I learnt long ago from an old jailbird and expert con-artist, “It is not the complicated thing
that confuses simple man, it is the simple thing that confuses complicated man”.
The Visionary of Isaac Newton
“The changing of Bodies into Light, and Light into Bodies, is very conformable
to the Course of Nature, which seems delighted with Transmutations”.
Opticks, 2nd edition (1718), Book 3, Query 30, 349.