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Home , Analytical mechanics

Welcome! 1 300: Analytical 1 Some practical information 2 • Classes MWF 2-250 pm Faraday 237 • Taylor’s is the required textbook • Expect a basic knowledge of Newton’s laws, coordinate systems, concepts like conservation of (will touch on all these subjects) • Odd vs even problems • Odd (mostly) have answers in back of book - use this! • Plan on following the book quite closely front and • Useful math formulas: book cover back! Some practical information 3

• For those who are not fully comfortable with , highly, highly, highly recommend “Div, Grad, Curl, and All That” • A great reference for E&M, too What we will cover (following Taylor) 4 1. Coordinate systems and Newton’s laws 2. Projectiles and air resistance/drag, and charged 3. and 4. Energy and conservative 5. Oscillations and 6. 7. Lagrange’s Equations 8. Central- problems 9. Mechanics in non-inertial frames 10.Rotational motion of rigid bodies (if there is )

With 1 problem per topic above Why are we studying this? 5

A bunch of stuffy- looking old white dudes from a long time ago Why are we studying this? 6

• But... classical mechanics underlies all the newer, more • The class will teach you key tools necessary for advanced E&M, , relativity, ... • The material here also covers the more relevant physics for our every-day lives • We don’t have much daily, direct interaction with the quantum world or where is important And hopefully the material can still be interesting 7

You have 2 minutes - with your neighbor (if you want), write down the most interesting reasons/subjects to study classical mechanics Some of my favorite examples 8 New EMDrive for propulsion - many scientists (myself included) are very skeptical of this “technology” Sports 9

http://www.symmetrymagazine.org/article/august-2014/ lhc-physicist-takes-on-new-type-of-collisions Sports 10 An unfortunate example for the engineers 11

http://upload.wikimedia.org/wikipedia/commons/ 1/19/Tacoma_Narrows_Bridge_destruction.ogg And my favorite 12

Candidate Higgs boson event decaying to ZZ* decaying to 4 electrons Grading 13 • Problem sets every 1-2 weeks, each with the same weight: combined total, 50% of grade • All to be due 1 week, in class, after assignment • To be distributed after we finish a chapter/topic • The HW will include computer assignments • Start the HW early! If you get stuck and need help, please come by during office hours • Please ask for help if you don’t understand solutions after they are posted (we’ll briefly go over them in class, but not over everything) • Midterm: 25% of grade • Date TBD, but at halfway point in course. Will be A TAKE-HOME MIDTERM (DUE 1 WEEK LATER) • Final: 25% of grade at nominal time (Dec 8 2-350 pm) • Late HW/exams NOT accepted without valid Doc note or excuse Grading 14

• After weighting problem sets, midterms and final as on last slide, the grades will be: • A: 85-100% • A-: 78-85% • B+: 70-77% • B: 62-69% • B-: 56-61% • C+: 48-55% • C: 40-47% • D: 25-39% • F: 25% or less On homework 15

• I don’t want to keep you from working with others, but any that you hand in must be your own • Solutions found on the web are a form of plagiarism • “Can I copy your solutions” are also plagiarism • And “Tell me the Answer” is academically dishonest, as well • You will anyway not get credit for answers without showing your work • I do want you to help your classmates, however... and don’t forget that office hours are there for those who need help

• Remember to use units (and label graph axes) where appropriate or you will not receive full credit Office hours 16

• Office hours (Faraday 220): Monday and Wednesday 3-4 pm (right after class) or by appointment • I may spend significant time at ANL and also traveling, so if you want to meet at any time other than during the set office hours please e-mail me ([email protected]) to set up an appointment • You can always try and stop by, but you will have better luck if you set up an appointment • Can also try the phone (753-6468) but email is preferred The class 17 • Roughly once a week, I will try and post previous slides on the class website for you • http://nicadd.niu.edu/~jahreda/phys300.html • Should not be considered a substitute for note-taking, but can hopefully help you in preparations for exams and homework • Hope to split you into groups of a few people for brief (5-10 minutes, at most!) exercises somewhat regularly • To help you feel comfortable with material • We will also work out some problems together in presentations here or on blackboard • So you can see that I can also get stuck :) Attendance 18

• Please come to class (shouldn’t need to ask this of you, but I state it anyway) • You can’t hand in homework without being in here • The small group and other problems that we go over will be important to follow and understand • I am not taking attendance - but if you don’t come to class, you will not do well in this course The classroom 19

• Please avoid food in the classroom • Bottles and cans of liquid are OK (no straws!) so that we can all stay hydrated, but is otherwise disruptive to me and to others. And the class is only 1 hour long • Talk to me privately if this is a problem • Cell phones need to stay in your pocket and be turned off • If your phone rings, we will know it was you (this class is that small). This is unfair to your fellow students • If you are using your phone in class, I will see you and call you out on it About me ... and you 20 • For those who do not know me, I am new to the department, to the campus, to teaching at this level • I’m a particle working on searches for new physics with Higgs bosons using the ATLAS experiment at the LHC (at CERN) • Ask me after class or during office hours about my research. I like to talk about it :) About me ... and you 21

• I’ll try to update my teaching style as the semester goes on, based on my experience, observations and your feedback • If I am going too fast... or too slow, or if my style (or hand- writing) is incomprehensible, please speak up And now it’s your turn 22

• We are small enough to go around the room • What’s your name? • What’s your major? • Why are you in this class? • What do you like (and dislike) about physics? • What do you like (and dislike) about classical mechanics? Before we get to physics... 23

Any questions? Before you do anything else in classical mechanics... 24

SET UP A (also known as the boring stuff to get through before the interesting physics) Ways to define our in space 25 Will try and use bold-face for vectors, though sometimes Unit arrow notation instead vectors (hats) r = ~r = xˆx + yˆy + zˆz r =(x, y, z) r = r1e1 + r2e2 + r3e3 Cartesian coordinates 3 r = riei are what we are most i=1 familiar with, but not always the easiest or P best choice Ways to define our position in space 26

r = ~r = xˆx + yˆy + zˆz r =(x, y, z) r = r1e1 + r2e2 + r3e3 3 r = i=1 riei Our unit vectors do not have P to be constant (they can be functions of time and/or position!) Playing with vectors 27

r =(r1,r2,r3), s =(s1,s2,s3) p = r + s =(p1,p2,p3)=(r1 + s1,r2 + s2,r3 + s3) s = cp =(c p ,c p ,c p ) · 1 · 2 · 3 Some vector operations 28

r =(r1,r2,r3), s =(s1,s2,s3) r s = rs cos ✓ = r1s1 + r2s2 + r3s3 · r s = 3 r s · i=1 i i P r = r = pr r | | ·

when ✓ =0, cos ✓ =1 r s = rs when vectors parallel when ✓ = ⇡/2, cos ✓ =0 !r s· = 0 when vectors perpendicular ! · Another way to think of the 29

r = ~r = xˆx + yˆy + zˆz r =(x, y, z) r = r1e1 + r2e2 + r3e3 If B has unit length, A·B3 = |A| cosθ is the length of Ar when= projectedi=1 r ontoiei the axis given by B P More vector operations (cross product) 30 r r ,r ,r , s s ,s ,s “p x y zq “p x y zq p r s rysz rzsy,rzsx rxsz,rxsy rysx “ “pp ´r s r ´s sin ✓ ´ q ë | |“| |“| || | ë

when ✓ 0, sin ✓ 0 r s 0 when vectors parallel when ✓ ⇡ 2“, sin ✓ 1“ rÑ|s r|“s when vectors perpendicular “ { “ Ñ ë“| || | ë Don’t forget the right-hand rule 31

Find B given current (we’ll have something similar for and rotating objects)

Z = X x Y Calculus with vectors? 32

r r1,r2,r3 “p dr dr q dr dr r9 1 , 2 , 3 dt “ “pdt dt dt q d r s r9 s9 dt p ` q“ ` d fr f9r fr9 dt p q“ ` Remember, base vectors can be functions of time p r e r e r e “ 1 1 ` 2 2 ` 3 3 d d d p9 r1e1 r2e2 r3e3 “ dtp q`dtp q`dtp q p9 r1e91 r91e1 r2e92 r92e2 r3e93 r93e3 “ ` ` ` ` ` What about polar coordinates? 33 x r cos ✓,y r sin ✓ x2 “r2 cos2 ✓,y2“ r2 sin2 ✓ x2 y“2 r2 cos2 ✓ “sin2 ✓ r2 y `x “r sinp✓ r cos` ✓ q“tan ✓ { “p✓ arctanq{p y xq“ “ p { q How to think about unit vectors in polar coordinates? 34

Vector pointing in direction of increasing u

r e B u “ u in any arbitrary Beu direction (u, for example) is a ˆe u normalized vector that points “ eu in the direction of increasing u | | e Whydu is that true? 35 u ¨ dr is some small r from eu B point r “ u Beu ˆe u “ eu r uu vv ww | | “ ` ` r r r dr B du B dv B dw “ u ` v ` w B B B e du u ¨ Start by thinking about unit vectors in rectangular coordinates 36 r = ~r = xˆx + yˆy + zˆz @r e = = ˆx x @x @r e = = ˆy y @y @r e = = ˆz This is simple because z @z xhat, yhat, zhat do not vary and are constant How to think about unit vectors in polar coordinates? 37

x = r cos ,y = r sin ,z = z r = rˆr + ˆ + zzˆ r = xxˆ + yyˆ + zzˆ r = r cos xˆ + r sin yˆ + zzˆ @r e = = cos xˆ +sinyˆ r @r @r e = = r sin xˆ + r cos yˆ @ @r e = = zˆ z @z How to think about unit vectors in polar coordinates? 38

@r e = = cos xˆ +sinyˆ r @r er cos xˆ +sinyˆ ˆr = ˆe r = = = cos xˆ +sinyˆ ˆe 2 2 | r| cos +sin @r e = p = r sin xˆ + r cos yˆ @ e r sin xˆ + r cos yˆ ˆ = ˆe = = = e/r = sin xˆ + cos yˆ ˆe 2 2 | | r cos +sin p How to think about time derivatives in polar coordinates? 39

Also see derivation in book!

ˆr = cos xˆ +sinyˆ ˆ = sin xˆ + cos yˆ dˆr d d = sin xˆ + cos yˆ = ˆ˙ dt dt dt dˆ d d = cos xˆ sin yˆ = ˆr ˙ dt dt dt Putting it all together 40

Let’s work on 1.1, 1.2, 1.6 and 1.14 in Taylor

We’ll do the first two together on the board, and then you work on 1.6 and 1.14 in small groups So .... 41

How will we use any of this? Why did we bother? Well, let’s go back to our favorite physical laws ... Our good ‘ol friend Newton and his laws 42 1. An object in motion stays in motion, and an object at rest stays at rest (law of ) 2. Our favorite equation, F=ma, remembering, though, that F and a are really vectors 3. The law of equal and opposite reaction, F12 = -F21, again remembering that forces are vectors Our good ‘ol friend Newton and his laws 43 1. An object in motion stays in motion, and an object at rest stays at rest (law of inertia) 2. Our favorite equation, F=ma, remembering, though, that F and a If no are really vectors other forces 3. The law of equal and opposite reaction, F12 = -F21, again remembering that forces are vectors Our good ‘ol friend Newton and his laws 44 1. An object in motion Lift force generated stays in motion, and by wings overcomes an object at rest stays , and plane at rest (law of inertia) accelerates up 2. Our favorite equation, F=ma, remembering, though, that F and a are really vectors 3. The law of equal and opposite reaction, F12 = -F21, again remembering that This law is the one that forces are vectors will help us to solve ... Our good ‘ol friend Newton and his laws 45 1. An object in motion Rocket exhaust given stays in motion, and a strong force an object at rest stays downward... which at rest (law of inertia) means a strong 2. Our favorite equation, opposite (ie upward) F=ma, remembering, force on the rocket though, that F and a itself are really vectors 3. The law of equal and opposite reaction, F12 = -F21, again remembering that forces are vectors The second law 46

dv d2r F = ma = m = mv˙ = m = m¨r dt dt2 p = mv p˙ = mv˙ = ma ! p˙ = F ! A reminder of the simplest use case 47 One-dimensional motion subject to constant force

r(t)=y(t), F(t)=F0 F y¨(t)= 0 m F F y˙(t)= y¨(t)dt = 0 dt = v + 0 t m 0 m Z Z F F y(t)= y˙(t)dt = (v + 0 t)dt = y + v t + 0 t2 0 m 0 0 2m Z Z Recall from into mechanics: 1 constants of integration d = d + v t + at2 0 o 2 (initial position and ) A reminder of the simplest use case 48

One-dimensional motion subject to constant force = gravity (with no air resistance or ... we’ll get to that soon) And a reminder on frames of reference 49

For now, consider only motion in inertial frames (will come back to noninertial frames at the end of the course). What do we mean by this? Newton’s second law for collections of 50

Let’s start with 3 particles (read Taylor Section 1.5 for expansion to N particles)

1 2

3 Newton’s second law for collections of particles 51

Each of 3 particles is subject to potential external forces, and also to forces on each other

1 2

3 Newton’s second law for collections of particles 52

ext F1 = F12 + F13 + F1 ext F2 = F23 + F21 + F2 ext F3 = F31 + F32 + F3

1 2

3 Newton’s second law for collections of particles 53

ext p˙ 1 = F1 = F12 + F13 + F1 p˙ = F = F + F + Fext Momentum of 2 2 23 21 2 ext total system p˙ 3 = F3 = F31 + F32 + F3 P = p1 + p2 + p3

P˙ = p˙ 1 + p˙ 2 + p˙ 3 ˙ ext ext ext P = F23 + F32 + F1 + F13 + F31 + F2 + F12 + F21 + F3

0 0 0

˙ ext ext ext P = F1 + F2 + F3 Newton’s second law in polar coordinates 54

A nice form to have, but only works if given Fr(t) and FΦ (t), which can be trivial ... or non-trivial

F = m¨r = Frˆr + Fˆ r = rˆr So maybe we need to use the second line and differentiate that... r˙ =˙rˆr + rˆr˙ dˆr Luckily, we derived this = ˆr˙ = ˆ˙ dt recently Newton’s second law in polar coordinates 55

r˙ =˙rˆr + rˆr˙ dˆr = ˆr˙ = ˆ˙ dt v = r˙ =˙rˆr + rˆ˙

vr =˙r, v = r˙ = r!

Radial is as expected, is what we learned in intro mechanics Newton’s second law in polar coordinates 56 v = r˙ =˙rˆr + rˆ˙ d a = ¨r = (˙rˆr + rˆ˙) Apply chain dt rule... ˙ ¨r =¨rˆr +˙rˆr˙ +˙rˆ˙ + rˆ˙ + rˆ¨ dˆ ˙ = ˆ = ˆr ˙ dt Luckily, we also derived this recently Working out the algebra ... 57 d a = ¨r = (˙rˆr + rˆ˙) dt ˙ ¨r =¨rˆr +˙rˆr˙ +˙rˆ˙ + rˆ˙ + rˆ¨ from class from class ¨r =¨rˆr +˙r˙ˆ +˙rˆ˙ rˆr˙2 + rˆ¨ ¨r = ˆr(¨r r˙2)+ˆ(2r ˙˙ + r¨)

Fr Fϕ Putting it all together 58

¨r = ˆr(¨r r˙2)+ˆ(2r ˙˙ + r¨)=F/m F = m(¨r r˙2),F = m(2r ˙˙ + r¨) r IF and only if r!2 r¨ r constant: F = mv2/r Let’s work out some problems 59

• Taylor Example 1.2 let’s work it out together • Taylor problems 1.36, 1.41 to be done together • On your own in small groups: Solve for R and h below! m

v0 h

R Homework due back in 1 week! 60

• Taylor 1.8,1.10,1.22,1.26,1.30,1.38,1.39