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NEWTON’S LAWS OF MOTION
1.
Introduction
Newton by his three laws of motion changed our understanding of the Universe.
Newton’s laws of motion are three physical laws that together laid the foundation for classical
mechanics. They describe the relationship between a body and the forces acting upon it as they can
help in explaining and investigating the state of motion or of rest of many physical objects and
systems.
They can be stated as follows
First law
An object at rest will stay at rest and an object in motion will stay in uniform linear
motion (motion at constant velocity) unless acted upon by an external unbalanced
force.
Second law The net force on an object is equal to the product of its mass and acceleration. The
acceleration of a body is directly proportional to, and in the same direction as, the net
force acting on the body, and inversely proportional to its mass:


F = m⋅a


where F is the net force acting on the body, m is its mass and a is its acceleration.
Third law
To every action there is always an equal and opposite reaction: or the force of two
bodies on each other are always equal and are directed in opposite directions.
When two bodies interact, the forces exerted on the bodies by each other are always
equal in magnitude and opposite in direction


F12 = − F21
where F is the net force acting on the body, m is its mass and a is its acceleration.
In his “Principia” Newton defines the quantities:
Mass
in terms of density and magnitude;
Motion
in terms of velocity and quantity of matter;
Forces
inherent
as intrinsic property of a body
impressed
if originating from an external source.
Inherent force is the ability of matter to preserve its state of rest or uniform motion (inertia);
Impressed force is an action exerted upon a body, in order to change its state, either of rest, or of
uniform motion on a straight line.
In modern language we call force any action exerted upon a body, in order to change its state of rest
or motion.
2.
First law (Law of Inertia)
The first law states that if the net force (the vector sum of all forces acting on an object) is zero then
the velocity of the object is constant. Velocity is a vector quantity which expresses both the object’s
speed and the direction of its motion; therefore, the statement that the object’s velocity is constant is
a statement that both its speed and the direction of its motion are constant.
After Galileo we call Inertia the resistance of any physical object to a change in its state of motion
or of rest. The first law we recognize as essentially Galileo’s concept of inertia, and this is often
termed simply the “law of Inertia”.
This law can be stated mathematically as


∑ F = 0 ⇒ ∆v = 0
Consequently,
• an object that is at rest will stay at rest unless an external, unbalanced, force acts upon it;
• an object that is in motion will not change its velocity (uniform motion) unless an external,
unbalanced, force acts upon it.
An object continues doing whatever it happens to be doing unless a force (unbalanced) is exerted
upon it.
If it is at rest, it continues in a state of rest. On a book lying on a table act two balanced forces:
gravity and the normal force of the table: the book will stay in its initial state of rest forever unless
another force acts upon it.
If it is moving, it continues to move without turning or changing its speed. When a skydiver falls
from a hovering helicopter, as her speed increases, the air resistance on her also increases;
eventually, it is enough to balance her weight, and she gains no more speed. She is at her terminal
velocity and she keeps moving at uniform motion.
The terminal velocity is the constant value the falling velocity reaches when the air resistance
balances the weight of the falling body.
We call inertial reference frame a reference frame where Newton’s law of Inertia is valid; we say
a reference frame is a non-inertial one if it is accelerated and Newton’s law of Inertia is NOT valid
in it.
Any reference frame that is in uniform motion with respect to an inertial reference frame is also an
inertial frame (Galilean invariance or the principle of Newtonian relativity).
3.
Second law
The second law states that the net (unbalanced) force on an object is proportional to the rate of
change of its velocity that is to its acceleration. The net force applied on a body produces a
proportional acceleration: if a body is accelerating, then there is a force on it.
The law applies to the behavior of objects for which all existing forces are balanced.
The mathematical expression for the second law is


F = m⋅a
This is the most powerful of Newton’s three laws, because it allows quantitative calculations of
dynamics: how do velocities change when forces are applied.
The quantity m on the right hand side is the inertial mass: it is a scalar quantity and it provides a
measure of an object’s resistance to a change in its state of motion or rest when an unbalanced force
is applied. It is determined by applying a force to an object and measuring the acceleration that
results from that force. Its unit of measurement is the kg.
Newton’s second law enables us to compare the results of the same force exerted on objects of
different mass: the same force exerted on a larger mass produces a correspoinding smaller
acceleration.
The weight on the other hand is a vector quantity and it represents the gravitational force the Earth
exerts on an object


W = m⋅ g
It acts downwards, towards the center of the Earth.
Its unit of measurement is the N.
Then, (inertial) mass can be defined in the context of Newton’s first law as a quantitative measure
of the resistance an object has to change in its velocity and weight can be defined in terms of
Newton’s second law as the force on an object due to gravity.
A free body diagram (FBD) is a diagram the represents the object of interest and all the external
forces that act on it.
Procedure to draw it
1. isolate the object (body)of interest;
2. draw all external forces vectors for that body;
3. choose a convenient coordinate system that is common for all vectors.
4.
Third law

The third law states that all force exist in pairs: if an object A exerts a force FA (action) on a second

object B, then B simultaneously exerts a force FB (reaction) on A and


FB = − FA
Action and reaction forces:
• are opposite in direction;
• are equal in magnitude;
• act on different bodies.
Since they are simultaneous, it doesn’t matter which one is called the action and which one is called
reaction. All forces are interactions between different bodies and there is no such thing as a force
that acts only on one body. A single force is impossible.
This law is exemplified by what happens when we step off a boat onto the bank of a lake: as we
move in the direction of the shore, the boat tends to move in the opposite direction (leaving us
facedown in the water, if we aren’t careful).
5.
Range of validity
Newton’s laws are valid only in an inertial reference frame. In a non-inertial reference frame the
laws of physics depend upon the acceleration of that frame of reference, and the usual physical
forces must be supplemented by fictitious forces.
Newton’s laws are applied to bodies as long as they can be idealized as single point masses. This
can be done when both the object is small if compared with the distances involved in its motion and
the deformation and rotation of the body are of no importance.
Newton’s law of motion, together with his law of universal gravitation and the mathematical
technique of calculus (due to Newton as well) provided for the first time a unified quantitative
explanation for a wide range of physical problems. However Newton’s laws are inappropriate for
use in certain circumstances:
• very small scale (quantum mechanics);
• very high speed, not much lower than the speed of light (special relativity);
• very strong gravitational fields (general relativity).