Newtonian MechanicsSingle 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)

Sir Isaac Newton (1642-1727)

Note that G’s death year is

N’s birth year!

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 accelerationF = 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

∑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!

if m = constant!

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) = mrinvolves 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.