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 ( ) Sir Isaac Newton ( ) 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!

1 st 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! 2 nd 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. 3 rd 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; F 1 = -F 2 (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 ∑F = (dp/dt) = m (dv/dt) = ma –This gives a means of calculation! –If mass is defined, the 2 nd 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  F 1. Force on 2 due to 1  F 2. F 1 = - F 2

Alternate form of the 3 rd Law (& using the 2 nd 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! 2 nd & 3 rd Laws together (leaving arrows off vectors!): F 1 = - F 2  dp 1 /dt = -dp 2 /dt m 1 (dv 1 /dt) = m 2 (-dv 2 /dt) m 1 a 1 = m 2 (-a 2 )  m 2 /m 1 = -a 1 /a 2

More on the 3 rd Law: For example, if we take m 1 = 1 kg (the standard of mass!). By comparing the measured value of a 1 /a 2 when m 1 interacts with any other mass m 2, we can measure m 2. To measure accelerations a 1 & a 2, 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 2 nd 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. 3 rd Law: F 1 = - F 2  dp 1 /dt = -dp 2 /dt Or: d(p 1 + p 2 )/dt = 0  p 1 + p 2 = 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 2 nd 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 2 nd Law: F = ma = m (d 2 r/dt 2 ) = mr involves a 2 nd time derivative of r.  A change of coordinates involving constant velocity will not change the 2 nd 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.