1Physics 325 Spring 2015Lecture 1 20 Jan 2015Course OverviewPhysics 325 is the first half of the two-semester classical mechanics sequence (Physics 326 is thesecond half). Physics 325 will probably require more work than your previous physics courses andis a serious departure from "the plug-n-chug from the formula sheet" mode of operation withwhich you might have become accustomed. You will be deriving analytic expressions, andlearning to critique and interpret and apply them. I expect you to know first year calculusthoroughly and be prepared to go beyond that level. Topics such as matrices, vector mathematics,multivariable calculus, and differential equations covered in PHYS 225 will be assumed.Concurrent registration (or prior completion) of Math 285 (ODE's) or equivalent is also expected.The web page has links to a couple of collections of math and physics formulas that you may finduseful during this and other courses, some of which you may recognize.Please read the course web pages.Let me draw your particular attention to a few of the main course-structure items:There will be two weekly HW assignments "A" and "B" (pink and green). They aretypically due in class on Thursdays ( though you may opt to submit in the yellow interpass HWmailboxes ) They will ( or so is the plan ) be returned to you graded 7 days later.There are weekly Monday evening Discussion classes. Attendance and participation thereis part of your grade.Most Lectures will conclude with an in-class miniquiz pertaining to the material of thatlecture. They are not graded – except for attendance - so feel free to consult with your neighbors.There will be two (in class?) midterms… times and places TBA.The text is the book by Taylor. Also recommended is the book by Thornton and Marion. Both ofthese are on reserve at Grainger. An additional recommendation is the "approximate lecture notes"( an edited photocopy of a previous year's lectures) available for purchase at FedEx on WrightStreet. You are strongly encouraged to read the appropriate section of the Approximate LectureNotes as a preview before each class. You may also enjoy being able to follow along in lecturesby referring to these notes. After each lecture, the actual this year's Lecture notes (including thequiz) will be posted at the course web site.Topics to be covered are listed at the course web site class schedule.Newton's Laws – (Reading: Taylor 1.4, 1.5)1) The momentum (in an inertial frame) of a particle is constant unless a force acts on it rp is a vector, rp = mrv,equal (non relativistically) to the particle's mass times its velocity2N.B. rv is a vector, with a number of components, and its expression in terms of how theparticle is moving is potentially problematic; e.g. Given its polar coordinates for example, and howthey are changing, what is rv?2) The time–rate of change of momentum (as observed from an inertial frame) is the net (vectorsum) of all forces on the particle ΣrF = drp / dtN.B. the time derivative of a vector may require careful definition.3) Particles act on each other by means of forces. These forces are equal and opposite -fromwhich we will later derive conservation of momentum. rf1− >2= −rf2 − >1Euler noted that the third law ought be supplemented: The forces must be co-linear, i.e, forces arecentral. (otherwise angular momentum is not conserved)Comment on the 3d law: The notion that forces are equal and opposite is often mis-interpreted.It is the force from particle 1 onto 2 that is equal and opposite to the force from 2 onto 1. Thecondition has nothing to do with static equilibrium. The distinction is illustrated by considering ablock of mass m sitting on a table. The forces on the block are mg down and some normal force NupThese forces N and mg may be equal and opposite – but that is not what Newton's 3d law isreferring to. N = mg only because the box is in static equilibrium and there are no other forces.3Newton's 3d law actually corresponds to the following picture, with one free body diagram for thebox and another for the earth+table We notice that N appears in both pictures as does mg. Just asthe table pushes up on the box with force N, the box pushes down on the table with force N. Justas the box is pulled towards the center of the earth with force mBox g, the earth is pulled up towardsthe box with force mbox g. (By the way g = G Mearth/Rearth2.)free body diagram for box free-body diagram for earth+tableNewton's laws contain some subtleties. Eddington quipped that the first law merely says that aparticle continues with constant velocity, except when it doesn't. The second law seems to merelystate that if it doesn't have constant velocity, then there must be a force (without ever definingforce, so the law by itself says precisely nothing.) The third law is a bit more substantive, sayingthat the forces (whatever they are) must occur in these pairs of opposites.There is a further subtlety: the laws hold only in inertial frames. And what is an inertial frame? Itis a frame of reference in which the laws hold. The definition is a bit circular, but Newtonidentified it with the frame of reference supplied by distant fixed stars.In practice these laws are usually not complete – one will also need extra information to solve realproblems, often some prescription that tells us what the forces are or how to calculate them.========Conservation of momentum (for a system of particles with no external forces) may be derivedfrom Newton's laws. Consider a set of N particles of masses mi and velocities rvi. The system hasa total momentum rP = Σrpiwhose time derivative is (assuming the system is isolated, i.e, has no external forces acting on it) drP / dt = Σidrpi/ dt = ΣiΣjrfj onto i= Σpairs{ i j }{rfj onto i+rfi onto j} = 0A similar procedure can be used to derive conservation of angular momentum.=======4Further comment on Newton's laws: The above quantity m appearing in F = m a is the particle'sinertial mass, minertial. Inertia is resistance to change in velocity.We are perhaps more familiar with weight – related to passive gravitational mass. rW = mpassive gravitationalrg( where g is given in terms of the mass of the earth by g = GMearth/R2 and M is the earth's activegravitational mass.)Recall G mparticle Mearth/R2 = mparticle apartticle and recognize that the two factors of mparticle logicallyneed not be the same! ( They are