Module M3: Relative Motion

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Module M3: Relative Motion"

Transcription

1 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 2005) Chapter 5 Thrntn and Marin Classical Dynamics f Particles and Systems 5th ed (Thmsn/Brks/Cle 2004) Chapters 10 and 14 Taylr Classical Mechanics (University Science Bks 2005) Chapters 9 and 15 Read thrugh A Graphing Checklist fr Physics Graduate Students befre submitting any plts in this mdule. See the curse website fr the link.

2 MODULE M3 Name: Term: Prblem Max 1 st 2 nd Final 0 textbk TOTAL 80

3 I. Nninertial, classical frames f reference A. The general case What fllws is a cndensed versin f the theretical backgrund fr prblems in this sectin. Yu may find it useful t supplement this material by reading frm ne f the abve references. Cnsider tw crdinate systems, ne the unprimed system and ne the primed system. Assume initially that their crdinate axes are parallel t each ther but that their rigins, O and O, d nt necessarily cincide. Let r be the vectr that pints frm O t a given pint P; let r " be the vectr that pints frm O t the same pint P, and let R be the vectr that pints frm O t O. That is, a persn lcating the pint P in the unprimed system wuld use r while a persn lcating P with respect t the primed system wuld use r ". R describes the relative psitin f the tw crdinate systems. Assuming that measuring devices behave the same way in bth frames f reference (an assumptin that we will questin in the next sectin), vectr additin implies: r = r " + R (1) Differentiating twice with respect t time, again assuming that differentiatin is viewed the same in each frame f reference, v = v " + V ( 2) a = a " + A 3 where v = d r a = d v ( ) v " = d r " a " = d v " V = d R A = d V Let us assume that the unprimed frame f reference is inertial (it des nt accelerate). Newtn s 2 nd Law applies t inertial frames f reference, and hence F = ma = m a " + ma (4) Equatin (4) applies in a straightfrward manner as lng as the tw sets f crdinate axes remain parallel. If the primed system rtates with respect t the unprimed system, things get a bit mre cmplicated. T see this, suppse that the rigins f the unprimed and primed crdinate systems cincide, and they remain the same, but that the primed crdinate system rtates with respect t the unprimed system. Let ω be the angular

4 velcity f the primed system. Fr instance, imagine painting a crdinate system n a disk and then setting the disk spinning with angular velcity ω. The disk wuld represent the primed frame f reference while the grund wuld be the unprimed frame. Since the rigins f the crdinate systems cincide, R = 0, s that by (1) r = r ". That is, xˆ i + yˆ j + zk ˆ = x " i ˆ " + y " ˆ j " + z " k ˆ " Once again, assuming the unprimed crdinate system is inertial (and hence nt rtating), then differentiating yields dx i ˆ + dy ˆ j + dz k ˆ = d x " i ˆ " + d y " ˆ j " + d z " k ˆ " + x dˆ i " " + y dˆ j " dˆ " + z k " " v = v " + x " dˆ i " + y " dˆ j " + z " d k ˆ " Here v is the velcity f pint P as measured in the unprimed frame f reference; v " is the velcity f that same pint but measured in the primed (rtating) frame f reference; and the final three terms in the equatin crrect fr the fact that the measurement is made in a rtating frame f reference. In a prblem belw, yu will shw that dˆ i " = ω i ˆ " and similarly fr the ther unit vectrs, s that v = v " + ω r " (5) Repeating this prcess yields the acceleratin transfrmatin: a = a " + d ω r " + 2 ω v " + ω ω r " ( ) (6) The left hand side represents the acceleratin as measured in the nn-rtating (inertial) frame. The first term n the right hand side represents the acceleratin an bserver in the rtating frame wuld measure, the secnd crrects fr the pssibility that the rtatin rate f the frame is nt cnstant, the third term is the Crilis term, and the furth term is the centripetal acceleratin term. These last three terms can be thught f as the price yu pay fr measuring the acceleratin with respect t a rtating set f crdinate axes. We can cmbine equatins (4) and (6) t arrive at a general transfrmatin between crdinate systems, where the primed system is bth accelerating and rtating with respect t the unprimed system: a = a " + d ω r " + 2 ω v " + ω ω r " ( ) + A (7).

5 Prblems: 1. A pendulum is held at rest with respect t a subway car. When the car accelerates, it is bserved that the pendulum is deflected backward by an angle, θ. a. Shw that Newtn s Secnd Law des nt apply when measurements are made in the frame f the subway car. b. Determine the acceleratin f the subway car as a functin f g and θ. 2. Fill in the gaps in the derivatin f the equatins fr measuring velcity and acceleratin in rtating crdinate systems. In particular, a. shw that dˆ i " = ω i ˆ ". b. derive equatin (5). c. derive equatin (6). Hint: the derivatin in Fwles and Cassiday is, in my pinin, uncnvincing since it relies n an undefined term and an unprven intermediate step. It is straight frward t prve equatin (6) by ging thrugh the same type f direct differentiatin that is necessary t derive equatin (5). 3. A rtating restaurant is typically set t cmplete a revlutin in ne hur. Cnsider a patrn walking 1.2 m/s radially ut frm the center at a distance f 10m frm the center. Estimate the size f the centripetal acceleratin and the Crilis acceleratin fr the patrn. Will either acceleratin likely be nticed by the patrn? On what basis can yu decide that? B. Prjectile Mtin Fllwing the apprach in Fwles and Cassiday s Analytical Mechanics, nte that fr an bject suspended by a string and at rest in a frame f reference attached t the surface f the earth, Newtn s 2 nd Law cmbined with equatin (7) yields F = ma T + mg = ma T + m( g A ) = 0 where g is the acceleratin due t gravity at the earth s surface as cmputed by the Universal Law f Gravitatin and T is the tensin in the string. Had we ignred the fact that that the earth s surface is nn-inertial, we wuld have written (as is cmmn in first year physics bks) T + mg = 0 Thus what we typically measure as the acceleratin due t gravity is g = g A Frm nw n, we will use g t represent the measured acceleratin due t gravity. It has incrprated in it a crrectin fr the fact that any pint n the earth s surface is accelerating. Treating the center f the earth as inertial, then A just represents the centripetal acceleratin f a pint n the earth s surface, whse directin pints in

6 twards the axis f rtatin (nt generally the center f the earth) and whse magnitude is ω 2 ρ where ω is the rtatinal velcity f the earth and ρ is the distance frm the pint n the surface t the axis f rtatin. Nw cnsider a prjectile near the earth s surface. Neglecting air resistance, F = ma mg & = m a # + d ω r # + 2ω v # + ω ( ω r #) + A ) ( ' + * g A ( ) = a # + d ω r # + 2 ω v # + ω ( ω r #) g = a # + d ω r # + 2 ω v # + ω ( ω r #) The secnd term n the right hand side is zer, assuming the earth rtates at a cnstant rate. The last term n the right hand side is generally smaller than the thers and hence can be drpped. Thus we have a " = g 2ω v " (8) The angular velcity vectr fr the earth pints frm the suth ple t the nrth ple. Hwever, when making measurements n the surface f the earth, we generally use a lcal crdinate system. Let the x axis pint east, the y axis pint nrth (bth tangent t the earth s surface) and the z axis pint vertically upward (equivalently, utward alng a radius f the earth). Let λ be the latitude f the rigin f ur primed crdinate system (that is, ur bservatin pint n the surface f the earth). In terms f this angle, ω = ω csλˆ j $ +ω sinλˆ k $ s that (8) can be rewritten ( ) x " = 2ω z " csλ y " sinλ y " = 2ω x " sinλ z " = g + 2ω x " csλ (9) Fr an bject launched frm the rigin with initial velcity v " = x " ˆ i " + y " ˆ j " + z " ˆ k ", these equatins may be integrated nce with respect t time t yield ( ) + " x " = 2ω z " csλ y " sinλ y " = 2ω x " sinλ + " y z " = gt + 2ω x " csλ + " x z (10) Using equatins (10) as substitutins int equatins (9) and drpping terms f rder ω 2, equatins (9) can be integrated twice t yield

7 ( ) = 1 3 ωgt 3 csλ ωt 2 " x " t ( ) = " y " t y t ω x " t 2 sinλ ( z csλ y " sinλ) + x " t (11) ( ) = 1 2 gt 2 + " z " t z t +ω x " t 2 csλ Prblems: 4. Derive equatins (11) frm equatins (9) and (10) as utlined abve. 5. Suppse we were t cnstruct a 100m tall twer and drp a rck frm the tp f the twer. The Crilis effect will cause the rck t land ff-center. Hw far away frm the center f the twer base will the rck land, and in which gegraphic directin? In additin, prvide a qualitative explanatin fr why the directin is what it is. Nte that the latitude f Ypsilanti is abut Inspectin f equatins (11) shws that the maximum height achieved by a prjectile is influenced by the Crilis effect. a. In which directin (nrth, suth, east r west) shuld an bject be launched in rder t maximize the peak f the trajectry, and hw high will it rise? Assume the angle f launch (with respect t the hrizntal) and the launch speed are nt changing. b. In which directin shuld an bject be launched in rder t minimize the peak f the trajectry, and hw high will it rise? c. Fr an bject launched with a speed f 50 m/s and at an angle f 45, what is the difference in heights between (a) and (b)? Take the latitude at the launch pint t be 42, and neglect air resistance. II. Inertial, Relativistic Transfrmatins I will nt re-derive sme f the basic equatins here, nr g in any detail int hw the pstulates f relativity came t be. This material is easily fund in almst every Mdern Physics textbk as well as in many intrductry textbks. The purpse f this sectin is t intrduce the transfrmatins in the cntext f 4-vectrs and t lk at energy and mmentum relatins in mre detail. Cnsider tw frames f reference, S and S, such that their rigins cincide at t=0 and S mves in the +x directin at a velcity V with respect t x. An event bserved t take place at time t and psitin (x,y,z) in frame S will be bserved in frame S t take place at time t and psitin (x, y, z ), where

8 ( ) x " = γ x Vt y " = y z " = z % t " = γ t Vx ( ' * & ) γ = 1 c 2 1 ( V c) 2 (12) If these equatins are nt familiar t yu, nw wuld be a gd time t get a Mdern Physics text bk and read its intrductin t Special Relativity. The rest f this may nt make much sense therwise. I will nw recast these equatins in terms f the cmpnents f the spacetime 4-vectr: x 1 = x x 2 = y x 3 = z x 4 = ct (13) It is als useful t intrduce β=v/c s that γ = ( 1 β 2 ) 1 2. The transfrmatin equatins are nw x 1 " = γx 1 γβx 4 x " 2 = x 2 x " 3 = x 3 x " 4 = γβx 1 +γx 4 (14) I will intrduce the fllwing ntatin: x refers t the cnventinal 3-cmpnent psitin vectr. x refers t the 4-vectr defined by " x 1 % $ ' x 2 x = $ ' $ x 3 ' $ ' # & x 4 Equatins (14) can then be written cmpactly as x " = Λx (15) prvided we define

9 & γ 0 0 γβ) ( Λ = ( + ( ( + ' γβ 0 0 γ * (16) We will adpt as ur mre general definitin f a 4-vectr any quantity which, when measured by bservers in frames S and S as defined abve, transfrms accrding t q " = Λq (17) where Λ is as given by equatin (16). Fr a cnventinal vectr, we calculate its length by a mdified Pythagrean therem: x = x x x 3 2 A prperty f this definitin f length is that it is invariant under rtatins. We can view equatin (17) as the 4-vectr generalizatin f a rtatin. In this case, hwever, the spatial axes are nt being rtated. Rather, the #1 and #4 axes are being rtated. It can be shwn that if we define the 4-vectr length as q = q q q 3 2 q 4 2 (18) then this length is invariant under the transfrmatin described by equatin (17). The final negative sign makes this equatin lk a little dd. As a result, sme authrs will incrprate a factr f i int the 4 th cmpnent, which then prduces a negative sign when squared. This gets rid f the negative sign in (18) but requires yur 4 th axis t be imaginary. Prblems: 7. Verify that q " = q fr any tw 4-vectrs related t each ther by equatin (17). 8. Suppse yu set up an bservatry in space with tw synchrnized clcks m apart. A miniature prbe has been cnstructed s that it will emit a signal nce every µs. This prbe is launched such that it passes parallel t the line determined by the clcks, and it des s at a cnstant speed f 0.200c. Hw many times will the prbe emit a signal as it passes between yur tw clcks? Answer this questin using the transfrmatin equatins (14) (nt using simplistic time dilatin arguments).

10 Other 4-vectrs Starting frm the transfrmatin, % t " = γ t Vx ( ' * & ) c 2 we cnsider tw nearby pints in spacetime: % d t " = γ Vdx ( ' * & c 2 ) If in the unprimed frame f reference, dx=0, then this frame f reference wuld crrespnd t ne in which the tw events take place at the same psitin. Such a situatin wuld arise if yu are lking at tw events which take place n the same bject, and the bject is at rest. In this case, d t " = γ prper = d t " γ = 1 β2 d t " Time measured in the rest frame f an bject is knwn as prper time. This time is relevant t a clck munted n a mving bject (the clck reads the bject s prper time) as well as t the time that gverns the decay prbability fr an unstable particle. It can be shwn that if yu differentiate the spacetime 4-vectr f a particle with respect t its prper time, the result is a new 4-vectr: # γv x & % ( % γv y ( % γv z ( % ( $ γc ' If we multiply this by the mass f the particle, we btain the relativistic 4-mmentum: # γmv x & % ( γmv y p = % ( % γmv z ( % ( $ γmc ' The first three cmpnents are γ times the nn-relativistic mmentum. In the limit that v<<c, these reduce t the n-relativistic mmentum. The furth cmpnent can be interpreted as the particle energy, E, divided by c. In particular, if yu d an expansin fr v<<c, yu find that E=γmc 2 has as its lead term mc 2 and its secnd term, the classical kinetic energy, 1/2 mv 2. The lead term mc 2 is interpreted as the rest energy f the particle, r its mass-energy equivalent. Anything in excess f that is the particle s kinetic energy. Thus we have: p = γmv (19) E = γmc 2 (20)

11 KE = ( γ 1)mc 2 (21) With these definitins, then the principles f mmentum cnservatin (in the absence f unbalanced external frces) and energy cnservatin apply. These will be explred in prblems belw. Prblems: 9. Shw that fr v<<c, the energy E can be expanded in the frm E = A 1 + A 2 v 2 + A 3 v 4. Determine the cnstants A 1, A 2, A 3, and cmment n the significance/interpretatin f each term. 10. Shw that a. p p = - mc ( ) 2 ( ) 2 + p 2 c 2. This equatin implies that fr a massless particle, E=pc, a result b. E 2 = mc 2 that will be helpful in prblem # A prtn is accelerated in a lab t the pint that its kinetic energy is 537MeV. a. Calculate γ fr this prtn. b. The prtn cllides with a secnd prtn that is at rest in the lab frame. Shw that the center f mass frame f reference, defined as the ne in which ttal mmentum f the system is zer, has a velcity with respect t the lab frame given by v T = γ 1+γ v x. Here v x and γ refer t values assciated with the accelerated prtn in the lab frame (that is, values yu dealt with in part a). Nte that yu cannt slve this prblem by calculating the center f mass velcity in the nnrelativistic apprximatin. Yu shuld begin by seeing what it will take fr the ttal mmentum t be zer. Als nte, that by the time yu cmplete the next part f this prblem, there will be several γ s and β s invlved, depending n the particle and frame f reference. An additinal set is assciated with transfrming between the tw frames f reference. Identify thse with the T subscript. c. Hw much kinetic energy des each prtn have in the center f mass frame? d. Discuss why the ttal kinetic energy in the center f mass frame puts an upper limit n the energy available t create new particles, and why this limit is better than using the kinetic energy in the lab frame. 12. A particle f mass m mves in the +x directin with a speed f 0.75c. Determine its speed in a frame f reference mving in the x directin at 0.50c. D this by calculating the 4-mmentum in the riginal frame f reference, transfrming the 4-mmentum int the new frame, and determining the speed frm the transfrmed 4-mmentum.

12 13. A pin (π - ) can decay int a mun (µ - ) and a mun antineutrin (ν µ ). The rest energy f the pin (mc 2 ) is 140 MeV, while that f the mun is 105.7MeV. The mun antineutrin can be treated as massless (see prblem 10). a. Shw that principles f energy cnservatin and mmentum cnservatin require that in the center f mass frame, the energy f the neutrin must be 30.1 MeV. Make sure yu use the relativistic energy expressin when applying energy cnservatin. b. Suppse a pin is mving at 0.68c in the +x directin in the labratry frame f reference when it decays. The mun is bserved t leave at an angle f 12.2 belw the x axis. In what directin (in the lab frame) will the neutrin be traveling? Hint: there are tw pssible slutins. As a check, verify the cnservatin f the 4-mmentum vectrs in the lab frame. 14. In this mdified versin f the previus prblem, cntinue t assume the pin is mving in the +x directin at 0.68c, but d nt assume anything is knwn abut the directin f the utging mun. a. Determine the range f pssible utging angles fr the mun. Yu may either d this analytically r numerically, but whichever way yu apprach it, make sure yu explain yur reasning. b. Plt the energy f the utging mun as a functin f its utging angle, as measured in the lab frame.

Chapter 3 Kinematics in Two Dimensions; Vectors

Chapter 3 Kinematics in Two Dimensions; Vectors Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs

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

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

Introduction to Spacetime Geometry

Introduction to Spacetime Geometry 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

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

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

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

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

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

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

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

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

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

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle? Physics 0 Public Exam Questins Unit 1: Circular Mtin NAME: August 009---------------------------------------------------------------------------------------------------------------------- 1. Which describes

More information

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle? Physics 3204 Public Exam Questins Unit 1: Circular Mtin NAME: August 2009---------------------------------------------------------------------------------------------------------------------- 12. Which

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

ENGI 4430 Parametric Vector Functions Page 2-01

ENGI 4430 Parametric Vector Functions Page 2-01 ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr

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

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

Phys101 Final Code: 1 Term: 132 Wednesday, May 21, 2014 Page: 1

Phys101 Final Code: 1 Term: 132 Wednesday, May 21, 2014 Page: 1 Phys101 Final Cde: 1 Term: 1 Wednesday, May 1, 014 Page: 1 Q1. A car accelerates at.0 m/s alng a straight rad. It passes tw marks that are 0 m apart at times t = 4.0 s and t = 5.0 s. Find the car s velcity

More information

Phys101 First Major-131 Zero Version Coordinator: Dr. A. A. Naqvi Wednesday, September 25, 2013 Page: 1

Phys101 First Major-131 Zero Version Coordinator: Dr. A. A. Naqvi Wednesday, September 25, 2013 Page: 1 Phys11 First Majr-11 Zer Versin Crdinatr: Dr. A. A. Naqvi Wednesday, September 5, 1 Page: 1 Q1. Cnsider tw unifrm slid spheres A and B made f the same material and having radii r A and r B, respectively.

More information

CHAPTER 6 WORK AND ENERGY

CHAPTER 6 WORK AND ENERGY CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS 16. REASONING AND SOLUTION A trapeze artist, starting rm rest, swings dwnward n the bar, lets g at the bttm the swing, and alls reely t the net. An assistant,

More information

= m. Suppose the speed of a wave on a string is given by v = Κ τμ

= m. Suppose the speed of a wave on a string is given by v = Κ τμ Phys101 First Majr-11 Zer Versin Sunday, Octber 07, 01 Page: 1 Q1. Find the mass f a slid cylinder f cpper with a radius f 5.00 cm and a height f 10.0 inches if the density f cpper is 8.90 g/cm 3 (1 inch

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

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

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

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

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

PHYSICS LAB Experiment 10 Fall 2004 ROTATIONAL DYNAMICS VARIABLE I, FIXED

PHYSICS LAB Experiment 10 Fall 2004 ROTATIONAL DYNAMICS VARIABLE I, FIXED ROTATIONAL DYNAMICS VARIABLE I, FIXED In this experiment we will test Newtn s Secnd Law r rtatinal mtin and examine hw the mment inertia depends n the prperties a rtating bject. THE THEORY There is a crrespndence

More information

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.

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

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

Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm

Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm Faculty f Engineering and Department f Physics Engineering Physics 131 Midterm Examinatin February 27, 2006; 7:00 pm 8:30 pm N ntes r textbks allwed. Frmula sheet is n the last page (may be remved). Calculatrs

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

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

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

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

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

AP Physics Laboratory #4.1: Projectile Launcher

AP Physics Laboratory #4.1: Projectile Launcher AP Physics Labratry #4.1: Prjectile Launcher Name: Date: Lab Partners: EQUIPMENT NEEDED PASCO Prjectile Launcher, Timer, Phtgates, Time f Flight Accessry PURPOSE The purpse f this Labratry is t use the

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

Kinetics of Particles. Chapter 3

Kinetics of Particles. Chapter 3 Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between

More information

Rigid Body Dynamics (continued)

Rigid Body Dynamics (continued) Last time: Rigid dy Dynamics (cntinued) Discussin f pint mass, rigid bdy as useful abstractins f reality Many-particle apprach t rigid bdy mdeling: Newtn s Secnd Law, Euler s Law Cntinuus bdy apprach t

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

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

( ) + θ θ. ω rotation rate. θ g geographic latitude - - θ geocentric latitude - - Reference Earth Model - WGS84 (Copyright 2002, David T.

( ) + θ θ. ω rotation rate. θ g geographic latitude - - θ geocentric latitude - - Reference Earth Model - WGS84 (Copyright 2002, David T. 1 Reference Earth Mdel - WGS84 (Cpyright, David T. Sandwell) ω spherid c θ θ g a parameter descriptin frmula value/unit GM e (WGS84) 3.9864418 x 1 14 m 3 s M e mass f earth - 5.98 x 1 4 kg G gravitatinal

More information

and the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are:

and the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are: Algrithm fr Estimating R and R - (David Sandwell, SIO, August 4, 2006) Azimith cmpressin invlves the alignment f successive eches t be fcused n a pint target Let s be the slw time alng the satellite track

More information

f = µ mg = kg 9.8m/s = 15.7N. Since this is more than the applied

f = µ mg = kg 9.8m/s = 15.7N. Since this is more than the applied Phsics 141H lutins r Hmewrk et #5 Chapter 5: Multiple chice: 8) (a) he maimum rce eerted b static rictin is µ N. ince the blck is resting n a level surace, N = mg. the maimum rictinal rce is ( ) ( ) (

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

Finding the Earth s magnetic field

Finding the Earth s magnetic field Labratry #6 Name: Phys 1402 - Dr. Cristian Bahrim Finding the Earth s magnetic field The thery accepted tday fr the rigin f the Earth s magnetic field is based n the mtin f the plasma (a miture f electrns

More information

Q x = cos 1 30 = 53.1 South

Q x = cos 1 30 = 53.1 South Crdinatr: Dr. G. Khattak Thursday, August 0, 01 Page 1 Q1. A particle mves in ne dimensin such that its psitin x(t) as a functin f time t is given by x(t) =.0 + 7 t t, where t is in secnds and x(t) is

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

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

Kinematic transformation of mechanical behavior Neville Hogan

Kinematic transformation of mechanical behavior Neville Hogan inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized

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

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

Kinematics. Describing Motion. Reference Frames. Measurements of position, distance or speed must be with respect to a frame of reference.

Kinematics. Describing Motion. Reference Frames. Measurements of position, distance or speed must be with respect to a frame of reference. Kinematics Describing Mtin Reference Frames Measurements f psitin, distance r speed must be with respect t a frame f reference. What is the speed f a persn with respect t the grund if she walks tward the

More information

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic. Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage

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

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

EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016

EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016 OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a

More information

Kinetic Model Completeness

Kinetic Model Completeness 5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins

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

Conceptual Dynamics SDC. An Interactive Text and Workbook. Kirstie Plantenberg Richard Hill. Better Textbooks. Lower Prices.

Conceptual Dynamics SDC. An Interactive Text and Workbook. Kirstie Plantenberg Richard Hill. Better Textbooks. Lower Prices. Cnceptual Dynamics An Interactive Text and Wrkbk Kirstie Plantenberg Richard Hill SDC P U B L I C AT I O N S Better Textbks. Lwer Prices. www.sdcpublicatins.cm Pwered by TCPDF (www.tcpdf.rg) Visit the

More information

I understand the new topics for this unit if I can do the practice questions in the textbook/handouts

I understand the new topics for this unit if I can do the practice questions in the textbook/handouts 1 U n i t 6 11U Date: Name: Sinusidals Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will learn hw trignmetry can be used t mdel wavelike relatinships. These wavelike functins are called

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

February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA

February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal

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

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

Introduction to Smith Charts

Introduction to Smith Charts Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary

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

, 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

Phys101 Second Major-061 Zero Version Coordinator: AbdelMonem Saturday, December 09, 2006 Page: 1

Phys101 Second Major-061 Zero Version Coordinator: AbdelMonem Saturday, December 09, 2006 Page: 1 Crdinatr: AbdelMnem Saturday, December 09, 006 Page: Q. A 6 kg crate falls frm rest frm a height f.0 m nt a spring scale with a spring cnstant f.74 0 3 N/m. Find the maximum distance the spring is cmpressed.

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

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

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

. (7.1.1) This centripetal acceleration is provided by centripetal force. It is directed towards the center of the circle and has a magnitude

. (7.1.1) This centripetal acceleration is provided by centripetal force. It is directed towards the center of the circle and has a magnitude Lecture #7-1 Dynamics f Rtatin, Trque, Static Equilirium We have already studied kinematics f rtatinal mtin We discussed unifrm as well as nnunifrm rtatin Hwever, when we mved n dynamics f rtatin, the

More information

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007 CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is

More information

Physics 321 Solutions for Final Exam

Physics 321 Solutions for Final Exam Page f 8 Physics 3 Slutins fr inal Exa ) A sall blb f clay with ass is drpped fr a height h abve a thin rd f length L and ass M which can pivt frictinlessly abut its center. The initial situatin is shwn

More information

Professional Development. Implementing the NGSS: High School Physics

Professional Development. Implementing the NGSS: High School Physics Prfessinal Develpment Implementing the NGSS: High Schl Physics This is a dem. The 30-min vide webinar is available in the full PD. Get it here. Tday s Learning Objectives NGSS key cncepts why this is different

More information

1 PreCalculus AP Unit G Rotational Trig (MCR) Name:

1 PreCalculus AP Unit G Rotational Trig (MCR) Name: 1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will

More information

Momentum 1. MOMENTUM. An object of mass m traveling at velocity v has a linear momentum (or just B. momentum) p, given by. p m B v. (6.

Momentum 1. MOMENTUM. An object of mass m traveling at velocity v has a linear momentum (or just B. momentum) p, given by. p m B v. (6. Mmentum 6 In this chapter we begin ur study f mre realistic systems in which the bjects are n lnger pint particles but have extensin in space. Up until nw we ve generally limited urselves t the dynamics

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

CHAPTER 1 -- MATH REVIEW

CHAPTER 1 -- MATH REVIEW Slutins--Ch. 1 (Math Review) CHAPTER 1 -- MATH REVIEW 1.1) Bth Parts a and b are straight vectr additin, graphical stle. Yu need a centimeter stick and prtractr t d them. a.) In this case, the velcit cmpnent

More information

Electric Current and Resistance

Electric Current and Resistance Electric Current and Resistance Electric Current Electric current is the rate f flw f charge thrugh sme regin f space The SI unit f current is the ampere (A) 1 A = 1 C / s The symbl fr electric current

More information

NGSS High School Physics Domain Model

NGSS High School Physics Domain Model NGSS High Schl Physics Dmain Mdel Mtin and Stability: Frces and Interactins HS-PS2-1: Students will be able t analyze data t supprt the claim that Newtn s secnd law f mtin describes the mathematical relatinship

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

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax .7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical

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

Phy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1

Phy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1 Ph 13: General Phsics III 6/14/007 Chapter 8 Wrksheet 1 Magnetic Fields & Frce 1. A pint charge, q= 510 C and m=110-3 m kg, travels with a velcit f: v = 30 ˆ s i then enters a magnetic field: = 110 T ˆj.

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

Lyapunov Stability Stability of Equilibrium Points

Lyapunov Stability Stability of Equilibrium Points Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),

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

A Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture

A Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu

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

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

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

Lim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?

Lim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)? THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques,

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

Review for the final exam (Math 127)

Review for the final exam (Math 127) . Evaluate 3 tan tan 4 3 (b) (c) cs cs 4 7 3 sec cs 4 4 (d) cs tan 3 Review fr the final eam (Math 7). If sec, and 7 36, find cs, sin, tan, ct, csc tan (b) If, evaluate cs, sin 7 36 (c) Write the csc in

More information