Newtonian Mechanics
Single Particle, Chapter 2
• Classical Mechanics:
– The science of bodies at rest or in motion +
conditions of rest or motion, when the bodies
are under the influence of forces.
– HOW bodies move, not WHY
• Other than the fact that forces can cause motion.
– Sources of forces? Outside the scope of mechanics
(recall the introductory lecture on the structure of
physics!).
Galileo Galilei (1564-1642)
Note that G’s
death year is
N’s birth
year!
Sir Isaac Newton (1642-1727)
Newtonian Mechanics
• 2 Parts to Classical Mechanics:
– Kinematics: Math description of motion of
objects (trajectories). No mention of forces. Concepts of
position, velocity, acceleration & their inter-relations.
– Dynamics: Forces  Produce changes in motion
(& other properties). Concepts of force, mass, &
Newton’s Laws.
• Special case is Statics. Total force = 0. (Boring!)
• Newton’s Laws: Direct approach to dynamics: Force.
• Lagrange & Hamilton formulations: Energy (Ch. 7)
Introduction
• Mechanics: Seeks to provide a description of
dynamics of particle systems through a set of
Physical Laws
• Fundamental concepts:
– Distance, Time, Mass (needs discussion: Newton)
• Physical Laws: Based on experimental fact.
(Physics is an experimental science!). Not
derivable mathematically from other relations.
• Fundamental Laws of Classical Mechanics 
Newton’s Laws.
Newton’s Laws:
Based on experiment!
Sect. 2.2
• The text contains some discussion which is
philosophical & esoteric. We will not dwell
on this!
• Here, we will emphasize the practical
aspects.
• However, I find the many historical
footnotes interesting.
– I invite you to read them!
• 1st Law (Law of Inertia): A body remains at rest or in
uniform motion unless acted upon by a force.
F = 0  v = constant! First discussed by Galileo!
• 2nd Law (Law of Motion): A body acted on by a
force moves in such a manner that the time rate of change of
its momentum equals the net force acting on the body.
F = (dp/dt) (p=mv). Discussed by Galileo, but not
written mathematically.
• 3rd Law (Law of Action-Reaction): If two bodies
exert forces on each other, the forces are equal in magnitude
& opposite in direction.
– To every action, there is an equal and opposite reaction.
2 bodies, 1 & 2; F1 = -F2
(Acting on different bodies!)
First Law
• All of Newton’s Laws deal with Inertial
Systems (systems with no acceleration). The
frame of reference is always inertial.
– Discussed in more detail soon.
• First Law: Deals with an isolated object
 No forces  No acceleration
F = 0  v = constant!
First Law: Alternate statement: It is always
possible to find an inertial system for an
isolated object.
First Law: First Discussed by Galileo
Second Law
• Deals with what happens when forces act.
– Forces come from OTHER objects!
– Inertia: What is it? Relation to mass. (Mass is the
inertia of an object).
– Momentum p = mv if m = constant!

∑F = (dp/dt) = m (dv/dt) = ma
– This gives a means of calculation!
– If mass is defined, the 2nd Law is really a definition
of force, as we will see.
– If a  0, the system cannot be isolated!
Third Law
• Rigorously applies only when the force
exerted by one (point) object on another
(point) object is directed along the line
connecting them!
• Force on 1 due to 2  F1.
• Force on 2 due to 1  F2.
F1 = - F2
• Alternate form of the 3rd Law (& using the 2nd
Law!): If 2 bodies are an ideal, isolated system,
their accelerations are always in opposite
directions & the ratio of the magnitudes of the
accelerations is constant & equal to the inverse
ratio of the masses of the 2 bodies!
• 2nd & 3rd Laws together (leaving arrows off vectors!):
F1 = - F2
 dp1/dt = -dp2/dt
m1(dv1/dt) = m2(-dv2/dt)
m1a1 = m2(-a2)  m2/m1= -a1/a2
• More on the 3rd Law: For example, if we take
m1 = 1 kg (the standard of mass!). By comparing
the measured value of a1/a2 when m1 interacts with
any other mass m2, we can measure m2.
• To measure accelerations a1 & a2, we must
have appropriate clocks & measuring sticks.
– Physics is an experimental science!
– Recall, Physics I lab!
• We also must have a suitable reference frame
(discussed next).
More on Mass
• A common method to experimentally determine a
mass  “weighing” it.
– Balances, etc. use the fact that weight = gravitational force
on the body.
F = ma  W = mg (g = acceleration due to gravity)
– This rests on a fundamental assumption that Inertial
Mass (the mass determining acceleration in the 2nd Law) =
Gravitational Mass (the mass determining gravitational
forces between bodies).
– The Principle of Equivalence: These masses are
equivalent experimentally! Whether they are fundamentally
is a philosophical question (beyond scope of the course).
See text discussion on this! This is discussed in detail in
Einstein’s General Relativity Theory!
Third Law & Momentum Conservation
• Assume bodies 1 & 2 form an isolated system.
• 3rd Law: F1 = - F2
 dp1/dt = -dp2/dt
Or:
d(p1 + p2)/dt = 0
 p1 + p2 = constant
Momentum is conserved for an isolated
system!
Conservation of linear momentum.
Frames of Reference: Sect. 2.3
• For Newton’s Laws to have meaning, the motion of
bodies must be measured relative to a reference
frame.
• Newton’s Laws are valid only in an
Inertial Frame
• Inertial Frame: A reference frame where
Newton’s Laws hold!
• Inertial Frame: Non-accelerating reference frame.
 By the 2nd Law, a frame which has no
external force on it!
Newtonian/Galilean Relativity
• If Newton’s Laws are valid in one (inertial)
reference frame, they are also valid in any
other reference frame in uniform (not
accelerated) motion with respect to the first.
• This is a result of the fact that in Newton’s 2nd
Law:
F = ma = m (d2r/dt2) = mr
involves a 2nd time derivative of r.
 A change of coordinates involving constant
velocity will not change the 2nd Law.
• Newton’s Laws are the same in all inertial
frames  Newtonian / Galilean Relativity.
• Special Relativity  “Absolute rest” &
“Absolute inertial frame” are meaningless.
• Usually, we take the Newtonian “absolute” inertial
frame as the fixed stars.
• Rotating frames are non-inertial 
Newton’s Laws don’t hold in rotating
frames unless we introduce “fictitious”
forces. See Ch. 10. See example at the end of Sect.
2.3.