Introduction to Spacetime Geometry

Save this PDF as:

Size: px
Start display at page:

Download "Introduction to Spacetime Geometry"

Transcription

1 Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the crdinates f its endpints using the fllwing relatin. s x y Fr example, the length f the line segment shwn in Figure 1 is 10 m. s x y 7m1m 10m2m 6m 8m 10m 2 2 One f the prperties f the length f a line segment is that it s the same regardless f the crdinate system used t measure it, smething that wuld be difficult t demnstrate if we measured x and y in different units. In spacetime gemetry there s an analgus relatinship: t t x Fig 1: A line segment f length 10 m. where t is time, x is psitin, and Δt is called the prper time. But psitin and time must be measured in the same units, such as minutes, where a minute f distance is hw far light travels in a time f ne minute. Fr example, light takes 8 minutes t travel frm the sun t Earth, s the distance is 8 minutes. Nte that the speed f light c equals ne minute per minute. This equatin is valid nly in systems f units like this, where c = 1. Using these units, let s lk at an example. Suppse we have a rcket that mves alng the x axis. We are interested in tw events, the first event is when the rcket has a psitin f 3, and then later when it has a psitin f 6. We als nte that the first event ccurs at a time f 2 and the secnd ccurs at a time f 7. x We calculate the speed f the rcket: 0.6 t 72 5 And the prper time is t t x minutes. This means that nly 4 minutes f time passes fr peple abard the rcket! The result f a calculatin like this is s unusual that it causes us t scratch ur heads and wnder hw we can understand it. We have t use care in explaining what s happening. The axis we used t measure the rcket s psitin is at rest relative t us. Likewise the clck we used t measure time is als at rest relative t us. They frm what we call ur frame f reference. Nte that in ur frame f reference we are at rest. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 1

2 Figure 2 shws such a frame f reference. We imagine an bject such as a rcket mving alng the x axis. When the bject is lcated at sme particular psitin we call it an event. We use the clck t measure the time t f the event and the x axis t measure the psitin x f the event. We say that x and t are the crdinates f the event. Fig 2: Viewing a frame f reference used t measure psitin x and time t. The viewer is at rest relative t the frame f reference. The example we just lked at invlved tw events. In ur frame f reference the first event ccurred at psitin x = 3 and time t = 2, where we are measuring psitin in minutes and time in minutes. The secnd event ccurred at x = 6 and t = 7. Shwn in Figure 3 is a spacetime diagram f this situatin. T make clear that spacetime gemetry is nt like Euclidean gemetry, let s take anther lk at what s being shwn in Figure 3. Figure 4 shws the triangle we re using t d ur Fig 3: A prper time f 4 minutes. spacetime calculatin. Nte that it s a strange lking triangle because the hyptenuse is nt the lngest side! The type f thinking that we re used t ding just wn t help us ut here. The gemetry is nt Euclidean. We d nt use the Pythagrean therem and Euclidean gemetry. We use the definitin f prper time and spacetime gemetry. t t x Nte that peple abard the rcket have their frame f reference. It s a frame f reference in which they are at rest. It s the clck in their frame f reference that reads an elapsed time f 4 minutes between the events. Fig 4: A spacetime triangle. One f the things that makes this seem strange is that we are used t lking at things frm ur wn frame f reference, and thinking that what we measure wuld be the same when measured in every frame f reference. We culd instead think f ur planet Earth as ur rcket, traveling thrugh space at a speed f 0.6c relative t sme bserver. Suppse tw events ccur at the same psitin, and are separated by 4 minutes f time, in ur frame f reference. Thse same tw events are separated by 5 minutes f time in that bserver s frame f reference. Is there anther way f relating these clck readings that desn t invlve gemetry? Yes, it s the familiar frmula fr time dilatin. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 2

3 In ur example we have 0.6 and therefre t t S the time in ur frame f reference is t t Mun Decay: A Wrked Example Muns are particles that were first discvered in The mun is a much heavier cusin f the electrn, and it s unstable with a mean lifetime f 2.2 μs. Sme will live fr a lnger time, sme shrter, but n average, they live fr 2.2 μs befre decaying. Muns are created in Earth s upper atmsphere, and sme f them subsequently travel tward Earth s surface. Fr this example we ll suppse we have a single mun mving tward Earth s surface at a typical speed f c, and that it lives fr 2.2 μs befre decaying. T calculate the lifetime f this mun in ur frame f reference we first calculate the value f Giving us a lifetime f t t μs μs. Therefre the distance it travels in ur frame f reference is 8 m 6 s xvt s m. If we had instead used the prper time f 2.2 μs t d this calculatin we d have a distance traveled f nly 660 meters. Such a difference is s huge that it s easy t detect experimentally. Earth s atmsphere reaches t a height f mre than meters, s if the average mun decayed after traveling nly 660 meters we d expect very few t make it t Earth s surface. Abut 90 years has passed since the discvery f muns in ur atmsphere, but even then physicists knew that a large fractin f the muns created near the tp f the atmsphere wuld make it t Earth s surface. This is just ne f many experimental results and measurements that cnfirm the validity f Einstein s relativity. Let s lk at the spacetime gemetry. Remember that we need a system f units where c = 1, s we shall chse micrsecnds fr units f time and micrsecnds fr units f distance. A micrsecnd f 8 m 6 distance, the distance light travels in ne micrsecnd, is (310 s )(110 s) 300 m. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 3

4 First, we calculate that x t Finally, we verify that the definitin f prper time is satisfied: t t x Fig 5: Spacetime diagram fr flight f mun. Figure 5 shws the spacetime diagram fr the flight f the mun. Nte that the tw legs f the right triangle are almst identical in length (because the mun s speed is very near the speed f light) and that again the hyptenuse is nt the lngest side. The Twin Paradx: A Wrked Example Tw twins, Tess and Sam, test ut the validity f relativity by ding an experiment. Sam stays at hme while Tess travels. Tess spends ne hur traveling at a speed f 0.6c, then turns arund and heads back hme at the same speed. When she gets back 2 hurs have elapsed n her clck, but 2.5 hurs have elapsed n Sam s clck. Scratching their heads, they try t use the definitin f prper time t figure ut the discrepancy in their clck readings. T d their spacetime calculatins they decide t measure time in minutes and distance in minutes. A 8 m 10 minute f distance is hw far light travels in a minute f time: (310 s )(60s) m. First, we lk at things frm Sam s frame f reference. He imagines an x axis with the rigin at his huse n planet Earth. It stretches ut int space fr several minutes f distance. Each minute representing the distance a light beam wuld travel in ne minute f time. Sam realizes that half f 2.5 hurs is 75 minutes f time. He uses this t calculate the distance t the turn arund pint: x t (0.6)(75) 45. S there are tw events, Event 1 is when Tess leaves, and Event 2 is when Tess reaches the turn arund pint. In Sam s frame f reference they ccur at x = 1 and x = 45, respectively. Fig 6: Event 1 is the departure, ccurs at x = 1. Event 2 is the turn arund, ccurs at x = 45. Nw let s lk at things frm Tess s frame f reference. These tw events ccur at the same place in her frame f reference, s the amunt f time that elapses between them is a prper time. The relatinship between her crdinates and his is given by 2 t t x Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 4

5 Figure 7 is the spacetime diagram Sam drew f the utbund half f Tess s trip. We can verify the calculatin f the elapsed time using the time dilatin frmula. Recalling that = 1.25 when = 0.6. t t S, frm Sam s perspective Tess takes 60 minutes t g ut. And then she wuld take 60 minutes t cme back, fr a ttal time f 2 hurs. But fr Sam it s 75 minutes f waiting fr Tess t g ut, 75 minutes fr her return, fr a ttal f 2.5 hurs! Fig 7: Sam s spacetime diagram f Tess s utbund trip. The questin arises, and this is the apparent paradx, why can t we lk at things frm Tess s frame f reference, and cnclude the reverse? In ther wrds, Tess imagines an x axis stretching ut frm her space ship. In her frame f reference she stays at the rigin, but she sees Sam as mving away frm her alng that x axis until he gets t the turnarund pint. Then it wuld be her that experiences 75 minutes f time while he experiences 60. In fact, that is a perfectly valid way t lk at it! Figure 6 culd just as well be Tess s spacetime diagram f Sam s utbund jurney. It seems strange t think f Sam as ging anywhere when he stays at hme, but frm Tess s pint f view that s exactly what she sees Sam ding. Yu culd imagine a trip like this ccurring at a much faster speed and fr a much lnger time, s the difference is mre prnunced. The traveling twin culd be gne fr 50 years, and be nly 5 years lder upn return. The stay at hme twin is 50 years lder and is nw a grandfather. It seems perfectly reasnable t ask why the reverse desn t happen s that the traveling twin is 50 years lder and the stay at hme twin is nly 5 years lder. The reslutin f the paradx lies in lking at the cmplete spacetime diagram f bth the utbund and the return legs f the jurney, as shwn in Figure 9. This has t be a diagram f Tess s jurney frm Sam s perspective, it cannt be a diagram f Sam s jurney. The reasn is the kink at the half way pint where Tess turns arund and heads back hme. Such a change in directin is smething that nly Tess experiences. Even if she had her eyes clsed she wuld ntice the acceleratin frm turning arund. She can feel it. Sam experiences n such thing at the half way pint. He culd be relaxing in a chair, eyes clsed r pen, and he d feel nthing. N acceleratin. It s nt he wh turns arund, it s she. That s what breaks the symmetry and lets us realize that the scenari where the stay at hme twin ages mre is the crrect ne. Fig 9: Sam s spacetime diagram f Tess s cmplete trip. A mre in depth analysis is required t wrk things ut frm Tess s tw different frames f reference, the ne she s in during the utbund half, and the ne she switches t fr the return half. Ding ne f these mre cmplete analyses gives f curse the same result. Tess experiences a prper time f 2 hurs, Sam experiences a prper time f 2.5 hurs. The difference in their ages is a difference in prper times. Nte that this is nt the same thing as the difference between a prper time and a dilated time. A difference in prper times is smething all bservers will agree n, whereas the difference between an bserved dilated time and a prper time depends n the bserver s frame f reference. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 5

6 Exercises 1. A subatmic particle lives fr 7 micrsecnds befre decaying, as measured in its wn frame f reference. (a) Calculate its lifetime as measured by an bserver with a relative speed f 0.96c. (b) Sketch a spacetime diagram f the particle s mtin frm the frame f reference f the bserver. Draw the diagram t scale. Label the axes with numbers. Label the length f the lines drawn n the diagram. 2. One twin leaves hme and travels in a straight line at a speed f 0.96c fr 7 years, as measured in his wn frame f reference. He then turns arund and cmes back hme at the same speed, having aged 14 years during the trip. (a) His twin sister, wh stayed at hme, ages hw much during her brther s absence? (b) Sketch a spacetime diagram f the traveling twin s mtin frm the frame f reference f the stay at hme twin. Draw the diagram t scale. Label the axes with numbers. Label the lengths f the lines drawn n the diagram. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 6

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

Einstein's special relativity the essentials

Einstein's special relativity the essentials VCE Physics Unit 3: Detailed study Einstein's special relativity the essentials Key knwledge and skills (frm Study Design) describe the predictin frm Maxwell equatins that the speed f light depends nly

More information

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y= Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents

More information

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b . REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but

More information

Physics 2010 Motion with Constant Acceleration Experiment 1

Physics 2010 Motion with Constant Acceleration Experiment 1 . Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin

More information

Lab #3: Pendulum Period and Proportionalities

Lab #3: Pendulum Period and Proportionalities Physics 144 Chwdary Hw Things Wrk Spring 2006 Name: Partners Name(s): Intrductin Lab #3: Pendulum Perid and Prprtinalities Smetimes, it is useful t knw the dependence f ne quantity n anther, like hw the

More information

SPH3U1 Lesson 06 Kinematics

SPH3U1 Lesson 06 Kinematics PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.

More information

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.

More information

Fall 2013 Physics 172 Recitation 3 Momentum and Springs

Fall 2013 Physics 172 Recitation 3 Momentum and Springs Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.

More information

INSTRUCTIONAL PLAN Day 2

INSTRUCTIONAL PLAN Day 2 INSTRUCTIONAL PLAN Day 2 Subject: Trignmetry Tpic: Other Trignmetric Ratis, Relatinships between Trignmetric Ratis, and Inverses Target Learners: Cllege Students Objectives: At the end f the lessn, students

More information

Trigonometric Ratios Unit 5 Tentative TEST date

Trigonometric Ratios Unit 5 Tentative TEST date 1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin

More information

READING STATECHART DIAGRAMS

READING STATECHART DIAGRAMS READING STATECHART DIAGRAMS Figure 4.48 A Statechart diagram with events The diagram in Figure 4.48 shws all states that the bject plane can be in during the curse f its life. Furthermre, it shws the pssible

More information

Interference is when two (or more) sets of waves meet and combine to produce a new pattern.

Interference is when two (or more) sets of waves meet and combine to produce a new pattern. Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme

More information

Differentiation Applications 1: Related Rates

Differentiation Applications 1: Related Rates Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm

More information

37 Maxwell s Equations

37 Maxwell s Equations 37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut

More information

20 Faraday s Law and Maxwell s Extension to Ampere s Law

20 Faraday s Law and Maxwell s Extension to Ampere s Law Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet

More information

Getting Involved O. Responsibilities of a Member. People Are Depending On You. Participation Is Important. Think It Through

Getting Involved O. Responsibilities of a Member. People Are Depending On You. Participation Is Important. Think It Through f Getting Invlved O Literature Circles can be fun. It is exciting t be part f a grup that shares smething. S get invlved, read, think, and talk abut bks! Respnsibilities f a Member Remember a Literature

More information

1 Course Notes in Introductory Physics Jeffrey Seguritan

1 Course Notes in Introductory Physics Jeffrey Seguritan Intrductin & Kinematics I Intrductin Quickie Cncepts Units SI is standard system f units used t measure physical quantities. Base units that we use: meter (m) is standard unit f length kilgram (kg) is

More information

AP Physics Kinematic Wrap Up

AP Physics Kinematic Wrap Up AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x

More information

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) = P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when

More information

Physics 101 Math Review. Solutions

Physics 101 Math Review. Solutions Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t

More information

Lifting a Lion: Using Proportions

Lifting a Lion: Using Proportions Overview Students will wrk in cperative grups t slve a real-wrd prblem by using the bk Hw D yu Lift a Lin? Using a ty lin and a lever, students will discver hw much wrk is needed t raise the ty lin. They

More information

Being able to connect displacement, speed, and acceleration is fundamental to working

Being able to connect displacement, speed, and acceleration is fundamental to working Chapter The Big Three: Acceleratin, Distance, and Time In This Chapter Thinking abut displacement Checking ut speed Remembering acceleratin Being able t cnnect displacement, speed, and acceleratin is undamental

More information

Putting Scientific Notation to Work

Putting Scientific Notation to Work 10 Putting Scientific Ntatin t Wrk Physics deals with sme very large and very small numbers. T wrk with such numbers, yu use scientific ntatin. Scientific ntatin is expressed as a number multiplied by

More information

The Law of Total Probability, Bayes Rule, and Random Variables (Oh My!)

The Law of Total Probability, Bayes Rule, and Random Variables (Oh My!) The Law f Ttal Prbability, Bayes Rule, and Randm Variables (Oh My!) Administrivia Hmewrk 2 is psted and is due tw Friday s frm nw If yu didn t start early last time, please d s this time. Gd Milestnes:

More information

Study Guide Physics Pre-Comp 2013

Study Guide Physics Pre-Comp 2013 I. Scientific Measurement Metric Units S.I. English Length Meter (m) Feet (ft.) Mass Kilgram (kg) Pund (lb.) Weight Newtn (N) Ounce (z.) r pund (lb.) Time Secnds (s) Secnds (s) Vlume Liter (L) Galln (gal)

More information

Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN

Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 321-3424 Rm: N-110 Fax : (901) 321-3402

More information

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving. Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus

More information

TP1 - Introduction to ArcGIS

TP1 - Introduction to ArcGIS TP1 - Intrductin t ArcGIS During this practical, we will use ArcGIS (ArcMap and ArcCatalg) t create maps f predictrs that culd explain the bserved bird richness in Switzerland. ArcMap is principally used

More information

Thermodynamics Partial Outline of Topics

Thermodynamics Partial Outline of Topics Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)

More information

WYSE Academic Challenge Regional Mathematics 2007 Solution Set

WYSE Academic Challenge Regional Mathematics 2007 Solution Set WYSE Academic Challenge Reginal Mathematics 007 Slutin Set 1. Crrect answer: C. ( ) ( ) 1 + y y = ( + ) + ( y y + 1 ) = + 1 1 ( ) ( 1 + y ) = s *1/ = 1. Crrect answer: A. The determinant is ( 1 ( 1) )

More information

1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south?

1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south? 1.2.1 Vectrs Definitins Vectrs are represented n paper by arrws directin = magnitude = Examples f vectrs: Examples What is the reference vectr angle fr a vectr that pints 50 degrees east f suth? What is

More information

Activity Guide Loops and Random Numbers

Activity Guide Loops and Random Numbers Unit 3 Lessn 7 Name(s) Perid Date Activity Guide Lps and Randm Numbers CS Cntent Lps are a relatively straightfrward idea in prgramming - yu want a certain chunk f cde t run repeatedly - but it takes a

More information

We can see from the graph above that the intersection is, i.e., [ ).

We can see from the graph above that the intersection is, i.e., [ ). MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with

More information

2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS

2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 6. An electrchemical cell is cnstructed with an pen switch, as shwn in the diagram abve. A strip f Sn and a strip f an unknwn metal, X, are used as electrdes.

More information

Physics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1

Physics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1 Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs

More information

Hubble s Law PHYS 1301

Hubble s Law PHYS 1301 1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and

More information

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems. Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define

More information

Example 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W

Example 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W Eample 1 rbt has a mass f 60 kg. Hw much des that rbt weigh sitting n the earth at sea level? Given: m Rbt = 60 kg ind: Rbt Relatinships: Slutin: Rbt =589 N = mg, g = 9.81 m/s Rbt = mrbt g = 60 9. 81 =

More information

Trigonometric Functions. Concept Category 3

Trigonometric Functions. Concept Category 3 Trignmetric Functins Cncept Categry 3 Gals 6 basic trig functins (gemetry) Special triangles Inverse trig functins (t find the angles) Unit Circle: Trig identities a b c 2 2 2 The Six Basic Trig functins

More information

, which yields. where z1. and z2

, which yields. where z1. and z2 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin

More information

Five Whys How To Do It Better

Five Whys How To Do It Better Five Whys Definitin. As explained in the previus article, we define rt cause as simply the uncvering f hw the current prblem came int being. Fr a simple causal chain, it is the entire chain. Fr a cmplex

More information

CHAPTER 6 -- ENERGY. Approach #2: Using the component of mg along the line of d:

CHAPTER 6 -- ENERGY. Approach #2: Using the component of mg along the line of d: Slutins--Ch. 6 (Energy) CHAPTER 6 -- ENERGY 6.) The f.b.d. shwn t the right has been prvided t identify all the frces acting n the bdy as it mves up the incline. a.) T determine the wrk dne by gravity

More information

Section 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law

Section 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law Sectin 5.8 Ntes Page 1 5.8 Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential

More information

Math 9 Year End Review Package. (b) = (a) Side length = 15.5 cm ( area ) (b) Perimeter = 4xside = 62 m

Math 9 Year End Review Package. (b) = (a) Side length = 15.5 cm ( area ) (b) Perimeter = 4xside = 62 m Math Year End Review Package Chapter Square Rts and Surface Area KEY. Methd #: cunt the number f squares alng the side ( units) Methd #: take the square rt f the area. (a) 4 = 0.7. = 0.. _Perfect square

More information

Pipetting 101 Developed by BSU CityLab

Pipetting 101 Developed by BSU CityLab Discver the Micrbes Within: The Wlbachia Prject Pipetting 101 Develped by BSU CityLab Clr Cmparisns Pipetting Exercise #1 STUDENT OBJECTIVES Students will be able t: Chse the crrect size micrpipette fr

More information

Our Lady Star of the Sea Religious Education CIRCLE OF GRACE LESSON PLAN - Grade 1

Our Lady Star of the Sea Religious Education CIRCLE OF GRACE LESSON PLAN - Grade 1 Our Lady Star f the Sea Religius Educatin CIRCLE OF GRACE LESSON PLAN - Grade 1 Opening Prayer: (ech prayer) Hly Spirit (ech) Shw us the way (ech) Be with us in all we think.. d and say (ech) Amen GETTING

More information

Preparation work for A2 Mathematics [2017]

Preparation work for A2 Mathematics [2017] Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will

More information

Lab 1 The Scientific Method

Lab 1 The Scientific Method INTRODUCTION The fllwing labratry exercise is designed t give yu, the student, an pprtunity t explre unknwn systems, r universes, and hypthesize pssible rules which may gvern the behavir within them. Scientific

More information

CONSTRUCTING STATECHART DIAGRAMS

CONSTRUCTING STATECHART DIAGRAMS CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6

More information

AP Statistics Notes Unit Two: The Normal Distributions

AP Statistics Notes Unit Two: The Normal Distributions AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres).

More information

ASTRODYNAMICS. o o o. Early Space Exploration. Kepler's Laws. Nicolaus Copernicus ( ) Placed Sun at center of solar system

ASTRODYNAMICS. o o o. Early Space Exploration. Kepler's Laws. Nicolaus Copernicus ( ) Placed Sun at center of solar system ASTRODYNAMICS Early Space Explratin Niclaus Cpernicus (1473-1543) Placed Sun at center f slar system Shwed Earth rtates n its axis nce a day Thught planets rbit in unifrm circles (wrng!) Jhannes Kepler

More information

EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017

EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017 OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a

More information

Information for Physics 1201 Midterm I Wednesday, February 20

Information for Physics 1201 Midterm I Wednesday, February 20 My lecture slides are psted at http://www.physics.hi-state.edu/~humanic/ Infrmatin fr Physics 1201 Midterm I Wednesday, February 20 1) Frmat: 10 multiple chice questins (each wrth 5 pints) and tw shw-wrk

More information

Lecture 5: Equilibrium and Oscillations

Lecture 5: Equilibrium and Oscillations Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if

More information

Department of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets

Department of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0

More information

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the

More information

Project CONVERGE 1/13/15. How to Read CONVERGE CODAR Imagery Data Primer

Project CONVERGE 1/13/15. How to Read CONVERGE CODAR Imagery Data Primer Hw t Read CONVERGE CODAR Imagery Data Primer Overall Ntes abut the data: Everything is in Greenwich Mean Time (GMT), which is 5 hurs ahead f the east cst. All time is presented in military time, which

More information

Study Guide: PS. 10 Motion, Forces, Work & Simple Machines DESCRIBING MOTION SPEED

Study Guide: PS. 10 Motion, Forces, Work & Simple Machines DESCRIBING MOTION SPEED DESCRIBING MOTION Distance: hw far smething has mved; SI unit meters (m) Reference pint: nn-mving bject used as a cmparisn pint t detect an bject s mtin. Displacement: the distance between the starting

More information

Chapter 23 Electromagnetic Waves Lecture 14

Chapter 23 Electromagnetic Waves Lecture 14 Chapter 23 Electrmagnetic Waves Lecture 14 23.1 The Discvery f Electrmagnetic Waves 23.2 Prperties f Electrmagnetic Waves 23.3 Electrmagnetic Waves Carry Energy and Mmentum 23.4 Types f Electrmagnetic

More information

Lesson Plan. Recode: They will do a graphic organizer to sequence the steps of scientific method.

Lesson Plan. Recode: They will do a graphic organizer to sequence the steps of scientific method. Lessn Plan Reach: Ask the students if they ever ppped a bag f micrwave ppcrn and nticed hw many kernels were unppped at the bttm f the bag which made yu wnder if ther brands pp better than the ne yu are

More information

Module M3: Relative Motion

Module M3: Relative Motion Mdule M3: Relative Mtin Prerequisite: Mdule C1 Tpics: Nninertial, classical frames f reference Inertial, relativistic frames f reference References: Fwles and Cassiday Analytical Mechanics 7th ed (Thmsn/Brks/Cle

More information

Thermodynamics and Equilibrium

Thermodynamics and Equilibrium Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,

More information

Group Color: Subgroup Number: How Science Works. Grade 5. Module 2. Class Question: Scientist (Your Name): Teacher s Name: SciTrek Volunteer s Name:

Group Color: Subgroup Number: How Science Works. Grade 5. Module 2. Class Question: Scientist (Your Name): Teacher s Name: SciTrek Volunteer s Name: Grup Clr: Subgrup Number: Hw Science Wrks Grade 5 Mdule 2 Class Questin: Scientist (Yur Name): Teacher s Name: SciTrek Vlunteer s Name: VOCABULARY Science: The study f the material wrld using human reasn.

More information

Equilibrium of Stress

Equilibrium of Stress Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small

More information

Chemistry 20 Lesson 11 Electronegativity, Polarity and Shapes

Chemistry 20 Lesson 11 Electronegativity, Polarity and Shapes Chemistry 20 Lessn 11 Electrnegativity, Plarity and Shapes In ur previus wrk we learned why atms frm cvalent bnds and hw t draw the resulting rganizatin f atms. In this lessn we will learn (a) hw the cmbinatin

More information

Solution to HW14 Fall-2002

Solution to HW14 Fall-2002 Slutin t HW14 Fall-2002 CJ5 10.CQ.003. REASONING AND SOLUTION Figures 10.11 and 10.14 shw the velcity and the acceleratin, respectively, the shadw a ball that underges unirm circular mtin. The shadw underges

More information

Work, Energy, and Power

Work, Energy, and Power rk, Energy, and Pwer Physics 1 There are many different TYPES f Energy. Energy is expressed in JOULES (J 419J 4.19 1 calrie Energy can be expressed mre specifically by using the term ORK( rk The Scalar

More information

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y ) (Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well

More information

B. Definition of an exponential

B. Definition of an exponential Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.

More information

Computational modeling techniques

Computational modeling techniques Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical

More information

NUMBERS, MATHEMATICS AND EQUATIONS

NUMBERS, MATHEMATICS AND EQUATIONS AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t

More information

Chapter 5: Force and Motion I-a

Chapter 5: Force and Motion I-a Chapter 5: rce and Mtin I-a rce is the interactin between bjects is a vectr causes acceleratin Net frce: vectr sum f all the frces n an bject. v v N v v v v v ttal net = i = + + 3 + 4 i= Envirnment respnse

More information

Pages with the symbol indicate that a student should be prepared to complete items like these with or without a calculator. tan 2.

Pages with the symbol indicate that a student should be prepared to complete items like these with or without a calculator. tan 2. Semester Eam Review The semester A eaminatin fr Hnrs Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr will be allwed.

More information

Curriculum Development Overview Unit Planning for 8 th Grade Mathematics MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2

Curriculum Development Overview Unit Planning for 8 th Grade Mathematics MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2 Unit Title It s All Greek t Me Length f Unit 5 weeks Fcusing Lens(es) Cnnectins Standards and Grade Level Expectatins Addressed in this Unit MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2 Inquiry Questins (Engaging-

More information

Manifesta Mediation Cards

Manifesta Mediation Cards Manifesta Mediatin Cards Manifesta 11 Mediatin cards If yu dn t have time t jin the guided tur, these art mediatin cards can help yu get int a dialgue with cntemprary art, t reflect and t frmulate yur

More information

CHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India

CHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce

More information

Chapter 8 Predicting Molecular Geometries

Chapter 8 Predicting Molecular Geometries Chapter 8 Predicting Mlecular Gemetries 8-1 Mlecular shape The Lewis diagram we learned t make in the last chapter are a way t find bnds between atms and lne pais f electrns n atms, but are nt intended

More information

Student Exploration: Cell Energy Cycle

Student Exploration: Cell Energy Cycle Name: Date: Student Explratin: Cell Energy Cycle Vcabulary: aerbic respiratin, anaerbic respiratin, ATP, cellular respiratin, chemical energy, chlrphyll, chlrplast, cytplasm, glucse, glyclysis, mitchndria,

More information

If (IV) is (increased, decreased, changed), then (DV) will (increase, decrease, change) because (reason based on prior research).

If (IV) is (increased, decreased, changed), then (DV) will (increase, decrease, change) because (reason based on prior research). Science Fair Prject Set Up Instructins 1) Hypthesis Statement 2) Materials List 3) Prcedures 4) Safety Instructins 5) Data Table 1) Hw t write a HYPOTHESIS STATEMENT Use the fllwing frmat: If (IV) is (increased,

More information

CHAPTER 8b Static Equilibrium Units

CHAPTER 8b Static Equilibrium Units CHAPTER 8b Static Equilibrium Units The Cnditins fr Equilibrium Slving Statics Prblems Stability and Balance Elasticity; Stress and Strain The Cnditins fr Equilibrium An bject with frces acting n it, but

More information

Name: Period: Date: ATOMIC STRUCTURE NOTES ADVANCED CHEMISTRY

Name: Period: Date: ATOMIC STRUCTURE NOTES ADVANCED CHEMISTRY Name: Perid: Date: ATOMIC STRUCTURE NOTES ADVANCED CHEMISTRY Directins: This packet will serve as yur ntes fr this chapter. Fllw alng with the PwerPint presentatin and fill in the missing infrmatin. Imprtant

More information

PHYS 314 HOMEWORK #3

PHYS 314 HOMEWORK #3 PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des

More information

Use a lens holder fabricated from SiC. SiC has a larger CTE than C-C, i.e. it is better matched to the SFL6.

Use a lens holder fabricated from SiC. SiC has a larger CTE than C-C, i.e. it is better matched to the SFL6. Frm: Steve Sctt, Jinsek K, Syun ichi Shiraiwa T: MSE enthusiasts Re: MSE mem 101b: allwable thickness f Vitn sheet Nvember 25, 2008 Update frm MSE Mem 101b Let s assume: Vitn thickness = 1 mm Vitn mdulus

More information

5 th grade Common Core Standards

5 th grade Common Core Standards 5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin

More information

CHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.

CHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came. MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the

More information

Physics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018

Physics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018 Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and

More information

Exam Review Trigonometry

Exam Review Trigonometry Exam Review Trignmetry (Tyler, Chris, Hafsa, Nasim, Paniz,Tng) Similar Triangles Prving Similarity (AA, SSS, SAS) ~ Tyler Garfinkle 3 Types f Similarities: 1. Side Side Side Similarity (SSS) If three pairs

More information

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,

More information

A solution of certain Diophantine problems

A solution of certain Diophantine problems A slutin f certain Diphantine prblems Authr L. Euler* E7 Nvi Cmmentarii academiae scientiarum Petrplitanae 0, 1776, pp. 8-58 Opera Omnia: Series 1, Vlume 3, pp. 05-17 Reprinted in Cmmentat. arithm. 1,

More information

Edexcel GCSE Physics

Edexcel GCSE Physics Edexcel GCSE Physics Tpic 10: Electricity and circuits Ntes (Cntent in bld is fr Higher Tier nly) www.pmt.educatin The Structure f the Atm Psitively charged nucleus surrunded by negatively charged electrns

More information

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) > Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);

More information

Phys. 344 Ch 7 Lecture 8 Fri., April. 10 th,

Phys. 344 Ch 7 Lecture 8 Fri., April. 10 th, Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, 009 Fri. 4/0 8. Ising Mdel f Ferrmagnets HW30 66, 74 Mn. 4/3 Review Sat. 4/8 3pm Exam 3 HW Mnday: Review fr est 3. See n-line practice test lecture-prep is t

More information

Chapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion.

Chapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion. Chapter Kinematics in One Dimensin Kinematics deals with the cncepts that are needed t describe mtin. Dynamics deals with the effect that frces have n mtin. Tgether, kinematics and dynamics frm the branch

More information

4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression

4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression 4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw

More information

Figure 1a. A planar mechanism.

Figure 1a. A planar mechanism. ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,

More information

CESAR Science Case The differential rotation of the Sun and its Chromosphere. Introduction. Material that is necessary during the laboratory

CESAR Science Case The differential rotation of the Sun and its Chromosphere. Introduction. Material that is necessary during the laboratory Teacher s guide CESAR Science Case The differential rtatin f the Sun and its Chrmsphere Material that is necessary during the labratry CESAR Astrnmical wrd list CESAR Bklet CESAR Frmula sheet CESAR Student

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time

More information

Homology groups of disks with holes

Homology groups of disks with holes Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.

More information