Transcript Lecture 5
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.