Unizor - Physics4Teens - Mechanics - Dynamics - Newton's Three Laws
Channel: Zor Shekhtman on UNIZOR Education
Date: 2018-05-08

Unizor - Physics through a Looking Glass of Mathematician

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.

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.

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.
F = m·a 

More general form of this law, when force and, therefore, acceleration depend on time looks exactly the same:
F(t) = m·a(t)
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·d²P(t)/dt²

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·d²Px(t)/dt²
Fy(t) = m·d²Py(t)/dt²
Fz(t) = m·d²Pz(t)/dt²

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.

Consider a cup on a table. 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.