Tuesday, May 8, 2018

Unizor - Physics - Mechanics - Dynamics - Newton's Three Laws

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 at rest stays at rest and an object in uniform motion stays in this uniform motion (that is, no change in 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 a and mass m (scalar) of an object.

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

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) - an object of the 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 as being a force of unit magnitude of 1 newton (N) and direction coinciding with the direction of the acceleration.

3. Using the force of unit magnitude of 1 N and applying it to different objects, we defined the 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, its mass, by definition, is 1/a kg. This is because 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 and applying to it different forces, we defined the magnitude of these forces. If the force causes an acceleration of a m/sec² of an object of the 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 and acceleration a(m/sec²) the mass, by definition, is 1/a(kg) and the formula F = m·a is true:

1 (N) = 1/a (kg) · a (m/sec²)

Case 2. For the unit of mass and acceleration a(m/sec²) the force is a(N) and the formula F = m·a is also true:

a (N) = 1 (kg) · a (m/sec²)

It has been experimentally confirmed that mass is additive. That is, if one object has mass m1 (that is, the force of 1 N gives it an acceleration of 1/m1 m/sec²) and another object has mass m2 (that is, the force of 1 N gives it an acceleration of 1/m2 m/sec²), then a combined object has mass m1+m2 (that is, the force of 1 N gives it an acceleration of 1/(m1+m2) m/sec²).

It has been also experimentally confirmed that force (as a vector) is additive. That is, if one force is F1 (that is, an object of 1 kg accelerates by it at a rate |F1| m/sec² in the direction of this force) and another force is F2 (that is, an object of 1 kg accelerates by it at a rate |F2| m/sec² in the direction of this force), then a vector combination of these two forces F1 +F2 , applied together to an object of 1 kg will cause the acceleration of this object at the rate |F1+F2| m/sec² in the direction of a vector sum of these forces.

In 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 m without changing the force, thus decreasing the acceleration by the same factor of m, getting

1 (N) = m (kg) · 1/m (m/sec²)

Then we increase the force by a factor of m without changing the object, thus increasing the acceleration by the same factor of m, getting

m (N)=m (kg) · m/m (m/sec²)=

= m (kg) · 1 (m/sec²)

Then we increase the force again by a factor of a without changing the object, thus increasing the acceleration by the same factor of a, getting

m·a (N) = m (kg) · a (m/sec²)

which means that the force, applied to an object of mass m(kg) and accelerating it with acceleration a(m/sec²) equals to m·a(N), which constitutes the Newton's Second Law.

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 a(t) is the second derivative of a position vector P(t) :
F(t) = m·P(t)/d

In coordinate form, where

F(t) = {Fx(t), Fy(t), Fz(t)} and

P(t) = {Px(t), Py(t), Pz(t)}

the vector equation is split into three coordinate differential equations of a second order:

Fx(t) = m·Px(t)/d

Fy(t) = m·Py(t)/d

Fz(t) = m·Pz(t)/d

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 F upon object B, object B acts at the same time upon object A with a force −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.

No comments: