General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation"

Transcription

1 General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton = d d to solve or d o you should rst rewrte t as = d d (1) and then begn to put n numbers such as: d = 5m 6m = 30m Then you'll know that what you have just got s not yet 1/ d. o you must stll do one more step to nd that o d (2) but s only d = 30m (3) as the nal answer. Now you don't have a good habt you would wrte only 1 1 = wthout sayng what t s really equal to. You would then orget that you have one more step to do and smply gve your answer as d = 1 m. 30 Comparng the correct answer (3) wth the wrong answer (5) I hope you can now see where you are lkely to make mstakes. It all bols down to the bad habt o omttng the step gven as Eq. (1) and also beng lazy n not payng attenton to unts n Eq. (4) and not explctly notng that t s actually 1/ d you have calculated there not yet d. That s you should have wrtten Eqs. (1) and (2) nstead o Eq. (4). Wrtng Eq. (4) nstead o Eqs. (1) and (2) means that you have a bad habt and you are lazy. You can not blame others you get wrong answers ths way. (4) (5)

2 (2) I you are to draw a gure to help you gure out the answer avod puttng nto the gure normaton that s not gven by the problem. For example an angle s nvolved avod makng t a rght angle or 45 or 30 or 60 unless the problem says so. These are specal angles wth specal propertes whch are not shared by a general angle. I the problem dd not say that you have such a specal angle then you should not draw such a specal angle. That s n your drawng you should avod drawng any such specal angles. Another example s: I a rectangle s nvolved n the problem you should avod drawng t as a square (unless t s a square). It should clearly be a rectangle. In act you should avod drawng t n such a way that one sde appears to be twce or three tmes etc. as long as the other sde. Ther rato should not be a smple nteger. It can mslead you nto thnkng that you have ths extra normaton n the problem that you really don't have. Now these are just examples. I can not pont out all thngs you are lkely to do. The general gudelne here s that you should avod dong thngs to ool yoursel nto thnkng that you have thngs you really don't have. (3) Do not mprovse. Physcs laws do not allow you to mody them. For example n optcs there s the equaton or double-slt ntererence: = (*) d sn θ mλ. I you are gven one m and one d but two derent λ s then usng the rst λ you can nd the rst θ and usng the second λ you can nd the second θ but you can not put n the derence n λ and hope to drectly compute the derence n θ. That would be a msuse o the ormula. That s what I mean by mprovse! A ormula does not allow you to use t n a way that t really doesn t allow. On the other hand n the example ormula gven when d s much larger than λ so you know that both θ s are very small then you can approxmate sn θ by θ n radans. We then have dθ1= mλ1 and dθ 2 = mλ 2 so takng the derence o the two equatons does gve you d( θ θ ) = m( λ λ ) or d θ = m λ o n ths case you can ndeed put n the derence o λ and obtan drectly the derence n θ. But ths s an excepton. It can be done

3 n ths case only because both λ and θ enter the equaton lnearly. Ths s not true wth the orgnal equaton (*)! I the change n λ and θ are very small then you can derentate both sdes o the orgnal e- quaton and obtan d cosθ dθ = m d λ. From ths equaton you can ndeed put n the very small quantty dλ and obtan drectly the very small quantty dθ and vce versa. But notce that t has cos θ n t whereas the orgnal equaton has sn θ. Another example s the equaton or Doppler sht o a sound wave requency when the source s movng wth the speed v and the lstener s movng wth the speed v :. - = v v v -v where v s the speed o sound s the actual requency o the sound emtted by the source and s the apparent requency o the sound heard by the lstener. When solvng such problems you can not work wth a relatve speed as you are movng wth the source so only the lstener s movng wth the speed v - v or movng wth the lstener (who s not you) so only the source s movng wth the speed v -v. I you do that you must also change v because v s dened wth respect to statonary carrer o the wave and you are movng wth respect to the statonary carrer o the wave then the sound speed s no longer v. I you don t take care o ths pont and stll work wth a relatve speed then you are mprovsng snce you have changed the content o the equaton. I the problem s about the Doppler sht o a lght wave then the ormula changes to: R = c -v c + v where c s the speed o lght and v v -v s the relatve speed. That s only the relatve speed matters n ths equaton. Ths pont

4 actually has very sophstcated reasons behnd t and s the great dscovery o Ensten. Namely lght s movng at the same speed o lght no matter how ast you are movng. It also has to do wth the act that lght waves have no carrer so you can t move wth respect to that non-exstent carrer to change ts speed. (4) Use dmensonal analyss to help you nd some o your mstakes. For example n wave theory we have λ = c. That s the wavelength o the wave tmes the requency o the wave s equal to the speed o the wave. The dmenson or unt o λ s meter or m. The dmenson or unt o s Hz or 1/s. Hence the product on the let hand sde o the equaton has the dmenson or unt o m/s whch s precsely the unt o the wave speed c. Now suppose you care- λ = c lessly put t as. You should be able to nd ths mstake easly by lookng at the dmenson or unt on the two sdes o the equaton. The dmenson or unt o λ s m. But the dmenson or unt o c s (m/s) (1/s) = m/s 2 whch s not what you have on the let hand sde o the equaton Thereore t must be wrong. As a matter o act ths dmensonal analyss mmedately suggests to you that the correct ormula should be λ = c and not λ = c. Every ormula n physcs allows you to analyzng t ths way and the dmenson or unt must be the same on ts two sdes. (5) Many ormulas allow you to have some eelng on why certan quanttes are n the numerator and why certan other quanttes are n the denomnator. Take the ormula or the speed o transverse wave on a strng. F v= µ where F s the tenson n the strng and µ s the lnear mass densty o the strng. In ths ormula t s not dcult to see why F s n the numerator and µ s n the denomnator and not the other way a- round. It s because ncreasng F can clearly make v larger because neghborng molecules n the strng can nluence each other more easly when F s larger and ncreasng µ wll clearly make v smaller because now neghborng molecules n the strng wll have a harder tme to nluence each other when µ s larger because larger µ

5 means larger nerta. o you can now see that you should never wrte t as v = µ / F. (As to why there s a square root sgn dmensonal analyss can help you see t.) (6) I possble you should use a gure to help you get thngs straght. For example take the ormula I = I 2 0 cos ( φ / 2). You should remember the phasor dagram assocated wth t: φ / 2 φ Ths phasor dagram not only let you see why t s φ / 2 and not just φ as the argument o the cosne uncton t also let you see that ths ormula s assocated wth the ntererence o two narrow lght beams wth a phase derence o φ between them so t can t be used or the dracton o a sngle wde slt nor the dracton o N narrow slts wth N > 2. You wll then not use t n the wrong place. On the other hand you wll also be able to see that you can also use ths ormula or the ntererence o two sound or rado waves generated at two pont sources A and B when the recever s at dstance d 1 orm source A and d 2 rom source B. Then all you need s to also see that n ths case φ = 2π(d 1 - d 1 ) / λ. On the other hand two lght beams are ntererng n phase to begn wth and a thn glass plate wth ndex o reracton n and thckness t s nserted nto one beam perpendcular to the beam leavng the other beam undsturbed then the above ormula can also be used except that now t t φ = 2π = 2 π( n 1) t λ / n λ λ because t s the change o phase o one beam (that s a new phase 2 π[ t/( λ / n)] mnus an old phase 2 π[ t / λ ]) wth the phase o the other beam unchanged. o t s also the derence between the phases o the two beams ater the glass plate has been nserted they are n phase beore the glass plate s nserted. Note that n all these cases when the ormula can be used there are always two narrow beams o some wave ntererng whle there s a phase derence between them.

6 Many ormulas allow you to understand them usng some sort o gures. o you should learn such ormulas ths way and not try to just memorze them. (7) I the problem gves you a quantty and you dd not use t to get your answer then you should ask yoursel the ollowng queston: hould the answer be ndependent o ths quantty? Assumng that t s true then you should be able to change the value o ths quantty to say zero or nnty ( allowed). But the correct answer or these cases mght be very easy to see. I t dsagrees wth your answer then you know that your answer must be wrong and the correct answer should depend on ths quantty. For example take the problem o nsertng a glass plate nto one beam and you are asked the ntensty change when t s ntererng wth another beam. I your answer does not depend on the thckness t o the glass plate then you should see that t has to be wrong snce or t equal to zero we have no glass plate and the phase sht must be zero! o how can you nd a nonzero answer that s ndependent o t? (8) Even when a quantty does enter your soluton you can stll ask whether t has entered correctly. et us agan take the glass-platenserton problem as an example. I you gve the phase sht as t t φ = 2π = 2πn λ / n λ then you can ask whether n has entered correctly. Well you can look at the case when n =1. The glass plate has been replaced by a layer o ar. That s nothng s now nserted nto the beam path. The phase sht should clearly be zero n that case. But s your answer zero n that case? It s not φ s gven by the above equaton. o you know t has to be wrong! It mmedately remnds you that you orgot to subtract the phase due to a layer o ar o the same thckness. That s you dd not compute the change. O course you know nothng about what s a phase all these remarks are useless to you. You must learn the general concepts rst.

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

PHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014

PHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014 PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 Work-Knetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Work is the change in energy of a system (neglecting heat transfer). To examine what could

Work is the change in energy of a system (neglecting heat transfer). To examine what could Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes

More information

Physics 2A Chapters 6 - Work & Energy Fall 2018

Physics 2A Chapters 6 - Work & Energy Fall 2018 Physcs A Chapters 6 - Work & Energy Fall 018 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on

More information

Physics 2A Chapters 6 - Work & Energy Fall 2017

Physics 2A Chapters 6 - Work & Energy Fall 2017 Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on

More information

Modeling motion with VPython Every program that models the motion of physical objects has two main parts:

Modeling motion with VPython Every program that models the motion of physical objects has two main parts: 1 Modelng moton wth VPython Eery program that models the moton o physcal objects has two man parts: 1. Beore the loop: The rst part o the program tells the computer to: a. Create numercal alues or constants

More information

Computational Biology Lecture 8: Substitution matrices Saad Mneimneh

Computational Biology Lecture 8: Substitution matrices Saad Mneimneh Computatonal Bology Lecture 8: Substtuton matrces Saad Mnemneh As we have ntroduced last tme, smple scorng schemes lke + or a match, - or a msmatch and -2 or a gap are not justable bologcally, especally

More information

FE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS)

FE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS) FE EIEW OPEATIONAL AMPLIFIES (OPAMPS) 1 The Opamp An opamp has two nputs and one output. Note the opamp below. The termnal labeled wth the () sgn s the nvertng nput and the nput labeled wth the () sgn

More information

I have not received unauthorized aid in the completion of this exam.

I have not received unauthorized aid in the completion of this exam. ME 270 Sprng 2013 Fnal Examnaton Please read and respond to the followng statement, I have not receved unauthorzed ad n the completon of ths exam. Agree Dsagree Sgnature INSTRUCTIONS Begn each problem

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

Solutions to Problem Set 6

Solutions to Problem Set 6 Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:

More information

Period & Frequency. Work and Energy. Methods of Energy Transfer: Energy. Work-KE Theorem 3/4/16. Ranking: Which has the greatest kinetic energy?

Period & Frequency. Work and Energy. Methods of Energy Transfer: Energy. Work-KE Theorem 3/4/16. Ranking: Which has the greatest kinetic energy? Perod & Frequency Perod (T): Tme to complete one ull rotaton Frequency (): Number o rotatons completed per second. = 1/T, T = 1/ v = πr/t Work and Energy Work: W = F!d (pcks out parallel components) F

More information

Four Bar Linkages in Two Dimensions. A link has fixed length and is joined to other links and also possibly to a fixed point.

Four Bar Linkages in Two Dimensions. A link has fixed length and is joined to other links and also possibly to a fixed point. Four bar lnkages 1 Four Bar Lnkages n Two Dmensons lnk has fed length and s oned to other lnks and also possbly to a fed pont. The relatve velocty of end B wth regard to s gven by V B = ω r y v B B = +y

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Spring Force and Power

Spring Force and Power Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems

More information

PHYS 1441 Section 002 Lecture #16

PHYS 1441 Section 002 Lecture #16 PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Non-conservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!

More information

Chapter 3 Differentiation and Integration

Chapter 3 Differentiation and Integration MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton

More information

MAE140 - Linear Circuits - Fall 13 Midterm, October 31

MAE140 - Linear Circuits - Fall 13 Midterm, October 31 Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

Use these variables to select a formula. x = t Average speed = 100 m/s = distance / time t = x/v = ~2 m / 100 m/s = 0.02 s or 20 milliseconds

Use these variables to select a formula. x = t Average speed = 100 m/s = distance / time t = x/v = ~2 m / 100 m/s = 0.02 s or 20 milliseconds The speed o a nere mpulse n the human body s about 100 m/s. I you accdentally stub your toe n the dark, estmatethe tme t takes the nere mpulse to trael to your bran. Tps: pcture, poste drecton, and lst

More information

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16 0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03 -- Lecture 7 0/4/03

More information

36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to

36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

ECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals

ECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy

More information

Lecture 16. Chapter 11. Energy Dissipation Linear Momentum. Physics I. Department of Physics and Applied Physics

Lecture 16. Chapter 11. Energy Dissipation Linear Momentum. Physics I. Department of Physics and Applied Physics Lecture 16 Chapter 11 Physcs I Energy Dsspaton Lnear Momentum Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Department o Physcs and Appled Physcs IN IN THIS CHAPTER, you wll learn

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

Spring 2002 Lecture #13

Spring 2002 Lecture #13 44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

More information

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous

More information

12. The Hamilton-Jacobi Equation Michael Fowler

12. The Hamilton-Jacobi Equation Michael Fowler 1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

More information

PHYS 1441 Section 002 Lecture #15

PHYS 1441 Section 002 Lecture #15 PHYS 1441 Secton 00 Lecture #15 Monday, March 18, 013 Work wth rcton Potental Energy Gravtatonal Potental Energy Elastc Potental Energy Mechancal Energy Conservaton Announcements Mdterm comprehensve exam

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-rotation coupling of the angularly accelerated rigid body Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Physics 11 Chapter 9 HW Solutions

Physics 11 Chapter 9 HW Solutions Physcs Chapter 9 HW Solutons Chapter 9 Conceptual Queston:, 4, 8, 3 Problems: 3, 8,, 5, 3, 40, 5, 6 Q9.. Reason: We can nd the change n momentum o the objects by computng the mpulse on them and usng the

More information

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1) Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For

More information

EMU Physics Department

EMU Physics Department Physcs 0 Lecture 8 Potental Energy and Conservaton Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aovgun.com Denton o Work W q The work, W, done by a constant orce on an object s dened as the product

More information

Force = F Piston area = A

Force = F Piston area = A CHAPTER III Ths chapter s an mportant transton between the propertes o pure substances and the most mportant chapter whch s: the rst law o thermodynamcs In ths chapter, we wll ntroduce the notons o heat,

More information

Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Section 1-5 Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

More information

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular

More information

ONE-DIMENSIONAL COLLISIONS

ONE-DIMENSIONAL COLLISIONS Purpose Theory ONE-DIMENSIONAL COLLISIONS a. To very the law o conservaton o lnear momentum n one-dmensonal collsons. b. To study conservaton o energy and lnear momentum n both elastc and nelastc onedmensonal

More information

Physics 30 Lesson 31 The Bohr Model of the Atom

Physics 30 Lesson 31 The Bohr Model of the Atom Physcs 30 Lesson 31 The Bohr Model o the Atom I. Planetary models o the atom Ater Rutherord s gold ol scatterng experment, all models o the atom eatured a nuclear model wth electrons movng around a tny,

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming

OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or

More information

Physics for Scientists and Engineers. Chapter 9 Impulse and Momentum

Physics for Scientists and Engineers. Chapter 9 Impulse and Momentum Physcs or Scentsts and Engneers Chapter 9 Impulse and Momentum Sprng, 008 Ho Jung Pak Lnear Momentum Lnear momentum o an object o mass m movng wth a velocty v s dened to be p mv Momentum and lnear momentum

More information

Note on EM-training of IBM-model 1

Note on EM-training of IBM-model 1 Note on EM-tranng of IBM-model INF58 Language Technologcal Applcatons, Fall The sldes on ths subject (nf58 6.pdf) ncludng the example seem nsuffcent to gve a good grasp of what s gong on. Hence here are

More information

Linear Momentum. Equation 1

Linear Momentum. Equation 1 Lnear Momentum OBJECTIVE Obsere collsons between two carts, testng or the conseraton o momentum. Measure energy changes durng derent types o collsons. Classy collsons as elastc, nelastc, or completely

More information

PHYSICS 203-NYA-05 MECHANICS

PHYSICS 203-NYA-05 MECHANICS PHYSICS 03-NYA-05 MECHANICS PROF. S.D. MANOLI PHYSICS & CHEMISTRY CHAMPLAIN - ST. LAWRENCE 790 NÉRÉE-TREMBLAY QUÉBEC, QC GV 4K TELEPHONE: 48.656.69 EXT. 449 EMAIL: smanol@slc.qc.ca WEBPAGE: http:/web.slc.qc.ca/smanol/

More information

Math1110 (Spring 2009) Prelim 3 - Solutions

Math1110 (Spring 2009) Prelim 3 - Solutions Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

More information

Integrals and Invariants of Euler-Lagrange Equations

Integrals and Invariants of Euler-Lagrange Equations Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,

More information

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.

More information

Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)

Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017) Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed

More information

PHYS 1443 Section 002

PHYS 1443 Section 002 PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS

More information

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law: CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam. Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: Instructor s Name and Secton: (Crcle Your Secton) Sectons:

More information

G = G 1 + G 2 + G 3 G 2 +G 3 G1 G2 G3. Network (a) Network (b) Network (c) Network (d)

G = G 1 + G 2 + G 3 G 2 +G 3 G1 G2 G3. Network (a) Network (b) Network (c) Network (d) Massachusetts Insttute of Technology Department of Electrcal Engneerng and Computer Scence 6.002 í Electronc Crcuts Homework 2 Soluton Handout F98023 Exercse 21: Determne the conductance of each network

More information

Experiment 5 Elastic and Inelastic Collisions

Experiment 5 Elastic and Inelastic Collisions PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 1 Experment 5 Elastc and Inelastc Collsons Readng: Bauer&Westall: Chapter 7 (and 8, or center o mass deas) as needed Homework 5: turn n the

More information

Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen

Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem

More information

Physics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V.

Physics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V. Physcs 4 Solutons to Chapter 3 HW Chapter 3: Questons:, 4, 1 Problems:, 15, 19, 7, 33, 41, 45, 54, 65 Queston 3-1 and 3 te (clockwse), then and 5 te (zero), then 4 and 6 te (counterclockwse) Queston 3-4

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam. NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Physics 4C. Chapter 19: Conceptual Questions: 6, 8, 10 Problems: 3, 13, 24, 31, 35, 48, 53, 63, 65, 78, 87

Physics 4C. Chapter 19: Conceptual Questions: 6, 8, 10 Problems: 3, 13, 24, 31, 35, 48, 53, 63, 65, 78, 87 Physcs 4C Solutons to Chater 9 HW Chater 9: Concetual Questons: 6, 8, 0 Problems:,, 4,,, 48,, 6, 6, 78, 87 Queston 9-6 (a) 0 (b) 0 (c) negate (d) oste Queston 9-8 (a) 0 (b) 0 (c) negate (d) oste Queston

More information

Prof. Dr. I. Nasser T /16/2017

Prof. Dr. I. Nasser T /16/2017 Pro. Dr. I. Nasser T-171 10/16/017 Chapter Part 1 Moton n one dmenson Sectons -,, 3, 4, 5 - Moton n 1 dmenson We le n a 3-dmensonal world, so why bother analyzng 1-dmensonal stuatons? Bascally, because

More information

PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76

PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76 PHYS 1101 Practce problem set 1, Chapter 3: 1,, 4, 57, 61, 83 Chapter 33: 7, 1, 3, 38, 44, 49, 76 3.1. Vsualze: Please reer to Fgure Ex3.1. Solve: Because B s n the same drecton as the ntegraton path s

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

MAE140 - Linear Circuits - Winter 16 Final, March 16, 2016

MAE140 - Linear Circuits - Winter 16 Final, March 16, 2016 ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3. r r. Position, Velocity, and Acceleration Revisited Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

More information

P A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that

P A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that Backward Error Analyss for House holder Reectors We want to show that multplcaton by householder reectors s backward stable. In partcular we wsh to show fl(p A) = P (A) = P (A + E where P = I 2vv T s the

More information

MAE140 - Linear Circuits - Winter 16 Midterm, February 5

MAE140 - Linear Circuits - Winter 16 Midterm, February 5 Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

Physics 2A Chapter 9 HW Solutions

Physics 2A Chapter 9 HW Solutions Phscs A Chapter 9 HW Solutons Chapter 9 Conceptual Queston:, 4, 8, 13 Problems: 3, 8, 1, 15, 3, 40, 51, 6 Q9.. Reason: We can nd the change n momentum o the objects b computng the mpulse on them and usng

More information

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding. Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #16 Scribe: Yannan Wang April 3, 2014

COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #16 Scribe: Yannan Wang April 3, 2014 COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #16 Scrbe: Yannan Wang Aprl 3, 014 1 Introducton The goal of our onlne learnng scenaro from last class s C comparng wth best expert and

More information

Answers Problem Set 2 Chem 314A Williamsen Spring 2000

Answers Problem Set 2 Chem 314A Williamsen Spring 2000 Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

Lecture 2 Solution of Nonlinear Equations ( Root Finding Problems )

Lecture 2 Solution of Nonlinear Equations ( Root Finding Problems ) Lecture Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Root Fndng

More information

MEASUREMENT OF MOMENT OF INERTIA

MEASUREMENT OF MOMENT OF INERTIA 1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us

More information

You will analyze the motion of the block at different moments using the law of conservation of energy.

You will analyze the motion of the block at different moments using the law of conservation of energy. Physcs 00A Homework 7 Chapter 8 Where s the Energy? In ths problem, we wll consder the ollowng stuaton as depcted n the dagram: A block o mass m sldes at a speed v along a horzontal smooth table. It next

More information

Conservation of Energy

Conservation of Energy Lecture 3 Chapter 8 Physcs I 0.3.03 Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcsall.html 95.4, Fall 03,

More information

and problem sheet 2

and problem sheet 2 -8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,

More information

Physics 240: Worksheet 30 Name:

Physics 240: Worksheet 30 Name: (1) One mole of an deal monatomc gas doubles ts temperature and doubles ts volume. What s the change n entropy of the gas? () 1 kg of ce at 0 0 C melts to become water at 0 0 C. What s the change n entropy

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order: 68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse

More information

find (x): given element x, return the canonical element of the set containing x;

find (x): given element x, return the canonical element of the set containing x; COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:

More information

I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam. ME 270 Fall 2012 Fnal Exam Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem

More information

Important Instructions to the Examiners:

Important Instructions to the Examiners: Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model

More information

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52 ontents ommonly Used Formulas. Estmaton, pproxmaton and Errors. Percentages. Polynomals and Formulas 8. Identtes and Factorzaton. Equatons and Inequaltes 66 6. Rate and Rato 8 7. Laws of Integral Indces

More information

Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georgia Tech PHYS 6124 Mathematical Methods of Physics I Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends

More information

Name: PHYS 110 Dr. McGovern Spring 2018 Exam 1. Multiple Choice: Circle the answer that best evaluates the statement or completes the statement.

Name: PHYS 110 Dr. McGovern Spring 2018 Exam 1. Multiple Choice: Circle the answer that best evaluates the statement or completes the statement. Name: PHYS 110 Dr. McGoern Sprng 018 Exam 1 Multple Choce: Crcle the answer that best ealuates the statement or completes the statement. #1 - I the acceleraton o an object s negate, the object must be

More information