*Notes to a video lecture on http://www.unizor.com*

__Newton's Three Laws__

Newton's three laws of motion are axioms that we accept to be true in any

*inertial frame of reference*.

As a very good approximation of the

*inertial frame of reference*, we can consider the heliocentric system of coordinates with an origin at Sun and axes directed to stars.

In many cases we will consider geocentric system of coordinates with

origin at the center of the Earth and axes directed to stars or

ground-related system with origin and XY-plane fixed on the ground as

approximately

*inertial*.

Another important consideration is that in their classic formulation the

Newton's laws address the principles of motion of point objects, where

all the mass of an object and all the forces applied to it are

concentrated in one point with three coordinates.

So, the geometric size of the point object is zero, and in most cases we

will omit the word "point", implying it when talking about objects,

unless otherwise is explicitly specified.

**First Law**

The

**Newton's First Law**is the familiar

**Law of Inertia**that states that

**an object**(that is, no change in

*at rest*__stays__*at rest*and an object in*uniform motion*__stays__in this*uniform motion**velocity vector*)

**, unless acted upon by**.

*unbalanced forces***Second Law**

The

**Newton's Second Law**is the most fundamental law of classic Mechanics. It establishes a quantitative relationship between

*vector of force*,

**F***vector of acceleration*and

**a***mass*(scalar) of an object.

**m**These concepts were explained in the previous lecture "Force,

Acceleration, Inertial Mass". The Newton's Second Law brings

quantitative relationship to these concepts as follows.

*F = m·a*Notice, this is a vector equation, the force and the acceleration are

vectors. If in certain cases we omit the symbol of vector, the line

across the top, we just assume that the force and the motion, including

the velocity and acceleration, are all occurring along one straight

line.

The above formula is based on the process of establishing the units of measurement for

*force*and

*mass*, described in the previous lecture, and is experimentally confirmed.

Let us recall how we introduced the units of measurement of

*force*and

*mass*.

1. We have taken one particular object (a small cylinder of

platinum-iridium alloy) and said that this object has, by definition,

the

*inertial mass*(or simply

*mass*) of

*1 kilogram (kg)**unit mass*.

2. Then we defined any force acting on this cylinder of

*unit mass*and pushing it with an acceleration of

*1 m/sec²**unit magnitude*of

*1 newton (N)*3. Using the force of

*unit magnitude*of

*1 N**mass*of any object by the acceleration it gets, if the

*unit force*acts upon it. If the

*unit force*causes it to accelerate with a value

*a m/sec²**mass*, by definition, is

*1/a kg**mass*was defined as the measure of

*inertia*, hence higher acceleration of an object signifies proportionally smaller inertia and, therefore, proportionally smaller

*mass*.

4. Using the object of

*unit mass*of

*1 kg*

*a m/sec²**unit mass*, then, by definition, the magnitude of this

*force*is

*a (N)*So, we have defined the units of measurement of

*force*and

*mass*in such a way that the Newton's Second Law is true by definition in two simple cases.

*Case 1*. For the

*unit of force*

*1 N***the mass, by definition, is**

*a(m/sec²)***and the formula**

*1/a(kg)*

*F = m·a*

*1 (N) = 1/a (kg) · a (m/sec²)**Case 2*. For the

*unit of mass*and acceleration

**the force is**

*a(m/sec²)***and the formula**

*a(N)*

*F = m·a*

*a (N) = 1 (kg) · a (m/sec²)*It has been experimentally confirmed that

*mass*is additive. That is, if one object has mass

**(that is, the force of**

*m*_{1}

*1 N*

*1/m*_{1}m/sec²**(that is, the force of**

*m*_{2}

*1 N*

*1/m*_{2}m/sec²**(that is, the force of**

*m*_{1}+m_{2}

*1 N*

*1/(m*_{1}+m_{2}) m/sec²It has been also experimentally confirmed that

*force*(as a vector) is additive. That is, if one force is

**(that is, an object of**

*F*_{1}

*1 kg*

*|F*_{1}| m/sec²**(that is, an object of**

*F*_{2}

*1 kg*

*|F*_{2}| m/sec²**+**

*F*_{1}**, applied together to an object of**

*F*_{2}

*1 kg*

*|F*_{1}+F_{2}| m/sec²In other words, these additive properties of mass and force state that

**an increase in mass of an object causes proportional decrease in acceleration**, if acted by the same force, and

**an increase in force causes proportional increase in acceleration**, if acted on the same object.

Using this additive property of mass and force and the basic equation that defined our units of measurement

*1 (N) = 1 (kg) · 1 (m/sec²)*we increase the mass by a factor of

**without changing the force, thus decreasing the acceleration by the same factor of**

*m***, getting**

*m*

*1 (N) = m (kg) · 1/m (m/sec²)*Then we increase the force by a factor of

**without changing the object, thus increasing the acceleration by the same factor of**

*m***, getting**

*m*

*m (N)=m (kg) · m/m (m/sec²)=*

= m (kg) · 1 (m/sec²)= m (kg) · 1 (m/sec²)

Then we increase the force again by a factor of

**without changing the object, thus increasing the acceleration by the same factor of**

*a***, getting**

*a*

*m·a (N) = m (kg) · a (m/sec²)*which means that the force, applied to an object of mass

**and accelerating it with acceleration**

*m(kg)***equals to**

*a(m/sec²)***, which constitutes the Newton's Second Law.**

*m·a(N)*The above logic is not a rigorous proof but a reasonable foundation for

the Newton's Second Law, which was established experimentally and

accepted as an axiom.

Incidentally, the Newton's First Law (Law of Inertia) follows from the Second Law. Indeed, if

*force*is zero,

*acceleration*is zero as well for an object with non-zero

*mass*.

That means that in the absence of external forces there is no

acceleration and an object stays at rest or moves uniformly with

constant

*velocity vector*, exactly as the Law of Inertia states.

Two important side notes about the Newton's Second Law.

Firstly, more general form of this law, when force and, therefore, acceleration depend on time looks exactly the same:

*F(t) = m·a(t)*Secondly, this equation can be considered as a differential equation of the second order since

*vector of acceleration*

**is the second derivative of a**

*a(t)**position vector*

**:**

*P(t)*

*F(t) = m·**d²*

**P(t)/**d**t²**In coordinate form, where

**and**

*F(t) = {F*_{x}(t), F_{y}(t), F_{z}(t)}

*P(t) = {P*_{x}(t), P_{y}(t), P_{z}(t)}the vector equation is split into three coordinate differential equations of a second order:

*F*_{x}(t) = m·*d²*

**P**d_{x}(t)/**t²**

*F*_{y}(t) = m·*d²*

**P**d_{y}(t)/**t²**

*F*_{z}(t) = m·*d²*

**P**d_{z}(t)/**t²****Third Law**

The

**Newton's Third Law**states that for

**for every action there is an equal in magnitude and oppositely directed reaction**.

In other words, forces are always paired. When object A acts with a force

**upon object B, object B acts at the same time upon object A with a force**

*F***.**

*−F*Notice that these two forces are applied to different objects, the one

originated at A is applied to B, and the one originated at B is applied

to A, and, therefore, strictly speaking, they are not balancing each

other. However, their magnitude is the same and the direction is

opposite to each other.

So, why then a cup on the table is not moving, implying that all the forces acting upon it are balanced?

Consider the following example.

A cup is on a table. It pushes down on a table with its weight (more

precisely, the Earth pulls it down with its gravitation), but nothing

moves, there is no acceleration of either a cup or a table. Why? Because

opposite forces of reaction participate in balancing, counteracting the

force of gravity.

A cup has two balancing forces acting on it: the gravity (its weight,

the force of Earth's gravity that pulls it down) and the reaction of the

table, that is equal in magnitude to the weight and pushes a cup up.

So, the gravity and the reaction of a table, acting upon the same cup,

balance each other, a cup is at rest.

A table is at rest as well because, again, there are balancing forces:

the gravity (table's weight) plus the weight of a cup on a table push

the table down, but the reaction of the floor pushes a table up, thus

balancing the downward forces.

So, forces of

**action**and

**reaction**do not balance each

other, since they are applied to different object. But they do

participate in balancing by nullifying other forces, like gravity.

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