Physics 351 — Wednesday, April 29, 2015

I Final exam is Wednesday, May 6, from 9am-11am, in DRLA4. You can bring a sheet of your own handwritten notes.

I Final exam includes Ch7 (Lagrangians), Ch9 (non-inertialframes), Ch10 (rigid-body rotation), Ch13 (Hamiltonians),and will (like midterm) be mainly based on HW problems.

I You can submit XC work as late as Tuesday, May 12.

I Do you want a review session? Maybe Monday afternoon orevening (May 4)?

Vorticity is defined as Ω = ∇× v. Feynman says, “If you put alittle piece of dirt—not an infinitessimal point—at any place in theliquid, it will rotate with angular velocity Ω/2. Try to see if youcan prove that. You can also check it out that for a bucket ofwater on a turntable, Ω equals twice the local angular velocity ofthe water.” Go ahead and try it now! (How do you relate v toangular velocity?)

It’s also easy to show that (if ωx = ωy = 0) Ωx = 0 and Ωy = 0.

Viscosity (η) describes the frictional drag force experienced e.g.between two parallel surfaces with area A, separation d, andrelative velocity v

Fx = −η Avxd

= −η A dvxdy

Picture a molasses sandwich . . .

For two large planes, vx of fluid varies linearly with y: fluid speedadjacent to each plane matches speed of that plane.

Note ∂vx/∂r < 0, which explains sign of 2nd term in force eqn.

Remarkably, the total flow (volume/time) through a cylindricaltube (assuming steady irrotational flow, no turbulence) scales asR4. It’s not surprising that it scales as 1/`, but one’s first guessmight have been that the flow would scale like R2 instead. But itdoesn’t, because the fluid speed is largest in the center of thecylinder (where the factor 2πr is smallest).

I happen to have here a container of water (η ≈ 0.001 Pa · s) andtwo tubes of length L = 0.30 m. One has R = 0.0010 m (1 mm),and one has R = 0.0005 m (0.5 mm). How can I estimateP1 − P2?

For R = 1 mm, h = 5.5 cm, I estimate

flow =πR4

8ηLρgh = 7× 10−7 m3/s

so (10 mL)/flow ≈ 14 s to reach 10 mL.

R = 0.5 mm should take 16× as long.

I estimate Reynolds number 2Rvρ/η ≈ 450 for the wide tube and≈ 60 for the narrow tube, so flow should be laminar.

A brick sinks when it is dropped into a bucket of water. Supposethat the same brick is supported by a string and slowly lowered (atconstant speed) into a bucket of water. How do the tensions in thestring compare at the four positions shown?

(A) T1 > T2 > T3 > T4

(B) T1 < T2 < T3 < T4

(C) T1 > T2 > T3 = T4

(D) T1 = T2 = T3 = T4

A boat carrying a large boulder is floating on a small lake. Theboulder is thrown overboard and sinks. As a result, the water level(with respect to the bottom of the lake)

(A) rises

(B) drops

(C) remains the same

Do you want a hint? (Hint on next page.)

A boat carrying a large boulder is floating on a small lake. Theboulder is thrown overboard and sinks. As a result, the water level(with respect to the bottom of the lake)

(A) rises

(B) drops

(C) remains the same

Hint: What volume of water is displaced by floating the boulderinside the boat? What volume of water is displaced by sinking theboulder into the lake?

Question:

I When is H conserved (i.e. a constant of the motion)?

I When does H equal the total energy?

I Notice that these are two different questions.

Morin 15.11. A bead is free to slide along africtionless hoop of radius R. The hoop is forcedto rotate with constant angular speed ω around avertical diameter. Find H in terms of θ and pθ,then write down Hamilton’s equations. Is H theenergy? Is H conserved?

I I’ve structured this course in a way that has let me get toknow most of you as individuals, and has let me work togetherwith many of you on solving homework problems.

I I’ve enjoyed immensely the time I’ve spent this semesterdiscussing physics with you — both in person and by email.That interaction with you is what I enjoy most about teaching.

I The more interactive format is a big help to me in trying toadjust the course to suit you better — for instance, whetherto work some computer problems into the course; how muchtime to spend on review at the outset; the number and style ofhomework problems; what to cover at the end of the course.

I I’d like your feedback on what to change or not to change if Iteach this course again two years from now. Here’s mycurrent thinking:

I While Taylor’s writing is a bit verbose, nearly all of you foundthe textbook very clear and easy to learn from. I’d keep it.

I Most of you enjoyed the brief review of Phys 150/170/230topics. I’d keep it, but keep it short, as we did this term.

I I wasn’t sure how required readings with online questionswould work out for a course at this level. In retrospect, I thinkthat with a clear and readable book like Taylor’s, requiredreading really facilitates your learning. Then, when we discussa topic in class, you’re not seeing it for the first time. It alsoleads to some fruitful email exchanges about the material,which in turn helps me to decide where to focus the class time.

I I really enjoyed our spending a good fraction of the “lecture”time on working through problems together. To me, that feelslike a more fruitful use of class time than derivations thatyou’ve seen in the book.

I But next time around, I would spend more time looking forreally good example problems that are suitable for in-class use(illustrating the key ideas, requiring conceptual insight, usuallywithout lengthy calculation). Many of Taylor’s problems areeither too simple or too laborious for in-class use.

I I like giving you a moment to think about a problem before Iwork it on the board. But I wonder how to do it moreeffectively: worksheets? better seating? And it’s not alwaysobvious how long we should wait before moving ahead.

I And for those times where a lengthy derivation or calculationis appropriate, I’d rehearse it in advance, so that I can moreconsistently get through it without falling back on my notes.

I I have mixed feelings about the projector. I wish we couldpush it off farther to the side, without removing it completely.

I I also have mixed feelings about the quizzes. On the onehand, they take up class time. On the other hand, I thinkthey’re probably more effective motivators than additionalmidterm exams would be.

I I think making exam problems closely resemble homeworkproblems had the desired effect of emphasizing the importanceof the homework, though it probably places too much weighton avoiding careless errors.

I You’ve given me excellent feedback on the length and qualityof the homework assignments. If I teach this course again, I’llkeep the mid-semester assignments from getting quite so long,toss out the problems you declared uninteresting, and addmore problems of the “insightful” type.

I Nevertheless, I’ve tried to pace out the workloadsystematically, so that you can plan your week, know whatyou’re responsible for, and learn steadily week-by-week.

I Many of you seem to have enjoyed the computer-basedproblems. This component seems worth keeping.

I I would certainly keep the extra-credit problems.I I’d like to find a better way to motivate the Hamiltonian

formalism.I One way would be to flesh out further the connection with

quantum mechanics — e.g. Poisson brackets vs. commutators.I Another way (related) would be to do a proper treatment of

canonical transformations, which basically re-map phase spacein such a way that Hamilton’s equations remain valid. Usuallythe goal is to transform the equations of motion into a formthat is more easily integrated; another goal is to identifyconserved quantities. The trouble with this topic is that it feelsvery abstract.

I I’d like to identify at least one good classical problem (maybefrom celestial mechanics) for which the Hamiltonian formalismhas some clear advantage over the Lagrangian formalism.

I Several of you asked for some systematic guidelines on the useof approximations, e.g. Taylor series.

I If you have opinions on any of these points (or others!), I’dlove to hear your thoughts either openly (by email) oranonymously (via course review). This course is here for yourbenefit. Your input can improve it for future students.

From Richard Feynman (Feynman Lectures):I “I think, however that there isn’t any solution to this problem of

education other than to realize that the best teaching can be doneonly when there is a direct individual relationship between a studentand a good teacher — a situation in which the student discussesthe ideas, thinks about the things, and talks about the things. It’simpossible to learn very much by simply sitting in a lecture . . . .”

From Mary Boas (Mathematical Methods):

I “One point about your study of this material cannot be emphasizedtoo strongly: To use mathematics effectively in applications, youneed not just knowledge but skill. Skill can be obtained onlythrough practice. You can obtain a certain superficial knowledge ofmathematics by listening to lectures, but you cannot obtain skill thisway. How many students have I heard say, ‘It looks so easy whenyou do it,’ or ‘I understand it but I can’t do the problems!’ Suchstatements show lack of practice . . . .”

I’ve tried to focus your time in this course on solving (worthwhile)problems, and I’ve tried to facilitate your “talking about thethings” with me, Tanner, and each other as much as possible.

I Since so much knowledge can be found online now, the role of thecollege teacher has shifted from being “the sage on the stage”(lecturing at a podium) to being “the guide on the side” (coachingyou to develop your own problem-solving skills).

I My role this term, as I see it, was to “coach” you as you workedyour way through Lagrangian mechanics, fictitious forces, rigid-bodyrotation, Hamiltonian mechanics, and so on.

I The right coach, trainer, piano teacher, etc. will help you to makeefficient use of the time that you invest. But it’s still up to you todo the work to build your own skills.

I You’ve learned a whole new way to solve mechanics problems, andsolved 145 homework problems (plus XC) along the way. I’m veryhappy with the hard work you’ve done this term. I hope this hasbeen a fun course for you, and generally not a source of stress.

I It’s hard to express how much fun I’ve had working with you,getting to know most of you, and being your “coach” this term forPhys 351. I’ve enjoyed every moment of it. Best wishes on yourfinal exams, summer jobs, and whatever you decide to pursue next!

Physics 351 — Wednesday, April 29, 2015

I Final exam is Wednesday, May 6, from 9am-11am, in DRLA4. You can bring a sheet of your own handwritten notes.

I Final exam includes Ch7 (Lagrangians), Ch9 (non-inertialframes), Ch10 (rigid-body rotation), Ch13 (Hamiltonians),and will (like midterm) be mainly based on HW problems.

I You can submit XC work as late as Tuesday, May 12.

I Review session Monday afternoon or evening (May 4)?