Physics 235 Chapter 8. Chapter 8 Central-Force Motion

Size: px
Start display at page:

Download "Physics 235 Chapter 8. Chapter 8 Central-Force Motion"

Transcription

1 Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion of two-body systems on which centra force are acting. We wi encounter important exampes from astronomy and from nucear physics. Two-Body Systems with a Centra Force Consider the motion of two objects that are effected by a force acting aong the ine connecting the centers of the objects. To specify the state of the system, we must specify six coordinates (for exampe, the (x, y, z) coordinates of their centers). The Lagrangian for this system is given by L = 1 m 1 1 r 1 + m r U ( r1 r ) Note: here we have assumed that the potentia depends on the position vector between the two objects. This is not the ony way to describe the system; we can for exampe aso specify the position of the center-of-mass, R, and the three components of the reative position vector r. In this case, we choose a coordinate system such that the center-of-mass is at rest, and ocated at the origin. This requires that R = The reative position vector is defined as m 1 m r 1 + r = 0 m 1 + m m 1 + m r = r 1 r The position vectors of the two masses can be expressed in terms of the reative position vector: r 1 = m m 1 + m r The Lagrangian can now be rewritten as r = m 1 m 1 + m r - 1 -

2 Physics 35 Chapter 8 L = 1 m m 1 m 1 + m r + 1 m m 1 m 1 + m r U ( r ) = 1 µ r U ( r ) where µ is the reduced mass of the system: µ = m 1m m 1 + m Two-Body Systems with a Centra Force: Conserved Quantities Since we have assumed that the potentia U depends ony on the reative position between the two objects, the system poses spherica symmetry. As we have seen in Chapter 7, this type of symmetry impies that the anguar momentum of the system is conserved. As a resut, the momentum and position vector wi ay in a pane, perpendicuar to the anguar momentum vector, which is fixed in space. The three-dimensiona probem is thus reduced to a twodimensiona probem. We can express the Lagrangian in terms of the radia distance r and the poar ange θ: L = 1 µ ( r + r θ ) U ( r) The generaized momenta for this Lagrangian are p r = L r = µ r p θ = L θ = µr θ The Lagrange equations can be used to determine the derivative of these momenta with respect to time: p r = d dt L r = L r = µr θ U r p θ = d dt L θ = L θ = 0 The ast equation tes us that the generaized momentum p θ is constant: - -

3 Physics 35 Chapter 8 = µr θ = constant The constant is reated to the area veocity. Consider the situation in Figure 1. During the time interva dt, the radius vector sweeps an area da where da = 1 r dθ The area veocity, da/dt, is thus equa to Figure 1. Cacuation of the area veocity. da dt = 1 dθ r dt = 1 r µr = µ = constant This resut is aso known as Keper's Second Law. The Lagrangian for the two-body system does not depend expicity on time. In Chapter 7 we showed that in that case, the energy of the system is conserved. The tota energy E of the system is equa to E = T +U = 1 µ ( r + r θ ) +U r = 1 µ r + 1 µr +U r ( ) µr ( ) = 1 µ r + r +U r ( ) = Two-Body Systems with a Centra Force: Equations of Motion If the potentia energy is specified, we can use the expression for the tota energy E to determine dr/dt: dr dt = ± µ E U ( ) µ r - 3 -

4 Physics 35 Chapter 8 This equation can be used to find the time t as function of r: t = dt = ± 1 µ E U r ( ( )) µ r dr However, in many cases, the shape of the trajectory, θ(r), is more important than the time dependence. We can express the change in the poar ange in terms of the change in the radia distance: dθ = dθ dt dt dr dr = θ r dr Integrating both sides we obtain the foowing orbita equation θ ( r) = θ r dr = ± µr ( µ E U ) dr = ± µ r r µ E U µr dr The extremes of the orbit can be found in genera by requiring that dr/dt = 0, or ( µ E U ) µ r = µ E U r ( ) µr = 0 In genera, this equation has two soutions, and the orbit is confined between a minimum and maximum vaue of r. Under certain conditions, there is ony a singe soution, and in that case the orbit is circuar. Using the orbita equation we can determine the change in the poar ange when the radius changes from r min to r max. During one period, the poar ange wi change by Δθ = r max r min r µ E U µr dr If the change in the poar ange is a rationa fraction of π then after a number of compete orbits, the system wi have returned to its origina position. In this case, the orbit is cosed. In a other cases, the orbit is open

5 Physics 35 Chapter 8 The orbita motion is specified above in terms of the potentia U. Another approach to study the equations of motion is to start from the Lagrange equations. In this case we obtain an equation of motion that incudes the force F instead of the potentia U: d 1 dθ r + 1 r = µr F( r) This version of the equations of motion is usefu when we can measure the orbit and want to find the force that produces this orbit. Exampe: Probem 8.8 Investigate the motion of a partice repeed by a force center according to the aw F(r) = kr. Show that the orbit can ony be hyperboic. The genera expression for θ(r) is [see Eq. (8.17) in the text book] θ ( r) = ( r )dr (8.8.1) µ E U µr where U = kr dr = kr in the present case. Substituting x = r and dx = rdr into (8.8.1), we have θ ( r) = 1 dx x µk x + µe (8.8.) x 1 Using Eq. (E.10b), Appendix E, dx = 1 bx + c x ax + bx + c c sin 1 (8.8.3) x b 4ac - 5 -

6 Physics 35 Chapter 8 and expressing again in terms of r, we find θ ( r) = 1 sin 1 µe r 1 µ E r + µk 4 + θ 0 (8.8.4) or, sin ( θ θ 0 ) = k µe 1 µe r 1 + k µe (8.8.5) In order to interpret this resut, we set 1 + k µe ε µe α (8.8.6) and specifying θ 0 = π/4, (8.8.5) becomes α r = 1 + ε cos θ (8.8.7) or, α = r + ε r ( cos θ sin θ) (8.8.8) Rewriting (8.8.8) in x-y coordinates, we find α = x + y + ε x y ( ) (8.8.9) or, - 6 -

7 Physics 35 Chapter 8 1 = x + y α α 1 + ε 1 ε (8.8.10) Since α ' > 0, ε ' > 1 from the definition, (8.8.10) is equivaent to 1 = x + α 1 + ε y α 1 ε (8.8.11) which is the equation of a hyperboa. Soving the Orbita Equation The orbita equation can ony be soved anayticay for certain force aws. Consider for exampe the gravitationa force. The corresponding potentia is -k/r and the poar ange θ is thus equa to ( ) = ± θ r / r µ E + k r µr dr Consider the change of variabes from r to u = /r: ( ) = ± θ r ( / r) / d u / µ E + k r 1 µ u = ± r µ E + k u 1 u du = µ u = ± 1 µ E + k u 1 µ u du The integra can be soved using one of the integras found in Appendix E (see E8.c): - 7 -

8 Physics 35 Chapter 8 ( ) = ± θ r k u + µ 1 u + µ k du = ±sin 1 u + µe µ k µ k = ±sin 1 u µ k + µe µk = ±sin 1 r µk ( ) + µ E + 8µE + C = µ k + C = ±sin 1 r + C = µ k + µe + C This equation can be rewritten as sin( θ + constant) = µk r ( µk) + µ E We can aways choose our reference position such that the constant is equa to π/ and we thus find the foowing soution: cos( θ) = µk r ( µk) + µ E We can rewrite this expression such that we can determine the distance r as function of the poar ange: r = µk ( µk) + µ E cos( θ) = µk 1 1+ µ k E cos( θ) Since cosθ varies between -1 and +1, we see that the minimum (the pericenter) and the maximum (the apocenter) positions are - 8 -

9 Physics 35 Chapter 8 r min = r max = µk µ k E µk 1 1+ µ k E The equation for the orbit is in genera expressed in terms of the eccentricity ε and the atus rectum α: ε = 1+ µ k E α = The possibe orbits are usuay parameterized in terms of the eccentricity, and exampes are shown Figure. µk Figure. Possibe orbits in the gravitationa fied. The period of the orbita motion can be found by integrating the expression for dt over one compete period: - 9 -

10 Physics 35 Chapter 8 τ = dt = µ da = µ πab ( ) = µ π k E µ E = π k µ E 3/ When we take the square of this equation we get Keper's third aw: µ τ = π k E 3 µ = π k a k 3 = 4π µ a 3 k The Centripeta Force and Potentia In the previous discussion it appears as if the potentia U is modified by the term /(µr ). This term depends ony on the position r since is constant, and it is interpreted as a potentia energy. The force associated with this potentia energy is ( ) F c = U c r = µr = µr θ = µ r θ 3 r This force is often caed the centripeta force (athough it is not a rea force), and the potentia is caed the centripeta potentia. This potentia is a fictitious potentia and it represents the effect of the anguar momentum about the origin. Figure 3 shows an exampe of the rea potentia, due to the gravitationa force in this case, and the centripeta potentia. The effective potentia is the sum of these two potentias and has a characteristic dip where the potentia energy has a minimum. The resut of this dip is that there are certain energies for which the orbit is bound (has a minimum and maximum distance). These turning points are caed the apsida distances of the orbit. Figure 3. The effective potentia for the gravitationa force when the system has an anguar momentum

11 Physics 35 Chapter 8 We aso note that at sma distances the force becomes repusive. Exampe: Probem 8. Discuss the motion of a partice moving in an attractive centra-force fied described by F(r) = k/r 3. Sketch some of the orbits for different vaues of the tota energy. For the given force the potentia is F ( r) = k r 3 U ( r) = k (8..1) r and the effective potentia is V ( r) = 1 µ k 1 (8..) r The equation of the orbit is [cf. Eq. (8.0) in the text book] d u dθ + u = µ ( ku 3 ) (8..3) u or, d u dθ + 1 µk u = 0 (8..4) Let us consider the motion for various vaues of. i) = µk : In this case the effective potentia V(r) vanishes and the orbit equation is

12 Physics 35 Chapter 8 d u dθ = 0 (8..5) with the soution u = 1 r = Aθ + B (8.6) and the partice spiras towards the force center. ii) > µk : In this case the effective potentia is positive and decreases monotonicay with increasing r. For any vaue of the tota energy E, the partice wi approach the force center and wi undergo a reversa of its motion at r = r 0 ; the partice wi then proceed again to an infinite distance. Setting 1 µk β > 0 equation (8..4) becomes d u dθ + β u = 0 (8..7) with the soution u = 1 r = A cos ( βθ δ ) (8..8) Since the minimum vaue of u is zero, this soution corresponds to unbounded motion, as expected from the form of the effective potentia V(r). iii) < µk : For this case we set µk 1 G > 0-1 -

13 Physics 35 Chapter 8 and the orbit equation becomes d u dθ G u = 0 (8..9) with the soution u = 1 r = A cosh ( βθ δ ) (8..10) so that the partice spiras in towards the force center. Orbita Motion The understanding of orbita dynamics is very important for space trave. The orbit in which a spaceship traves is determined by the energy of the spaceship. When we change the energy of the ship, we wi change the orbit from for exampe a spherica orbit to an eiptica orbit. By changing the veocity at the appropriate point, we can contro the orientation of the new orbit. The Hofman transfer represents the path of minimum energy expenditure to move from one soar-based orbit to another. Consider trave from earth to mars (see Figure 4). The goa is to get our spaceship in an orbit that has apsida distances that correspond to the distance between the earth and the sun and between mars and the sun. This requires that and The eccentricity of such an orbit is thus equa to r 1 = a( 1 ε) r = a( 1+ ε) ε = r r 1 a The tota energy of an orbit with a major axis of a = (r 1 + r )/ is equa to E = k a = k ( ) r 1 + r

14 Physics 35 Chapter 8 Since the space ship starts from a circuar orbit with a major axis a = r 1, its initia energy is equa to E = k r 1 Figure 4. The Hofman transfer to trave from earth to mars. The increase in the tota energy is thus equa to k ΔE = ( r 1 + r ) k r 1 = k 1 r 1 ( ) r 1 + r = k r r 1 r 1 r 1 + r ( ) This energy must be provided by the thrust of the engines that increase the veocity of the space ship (note: the potentia energy does not change at the moment of burn, assuming the thrusters are ony fired for a short period of time). The probem with the Hofman transfer mechanism is that the conditions have to be just right, and ony of the panets are in the proper position wi the transfer work. There are many other ways to trave between earth and mars. Many of these require ess time than the time required for the Hofman transfer, but they require more fue (see Figure 5)

15 Physics 35 Chapter 8 Figure 5. Different ways to get from earth to mars. SECTIONS 8.9 AND 8.10 WILL BE SKIPPED!

Cluster modelling. Collisions. Stellar Dynamics & Structure of Galaxies handout #2. Just as a self-gravitating collection of objects.

Cluster modelling. Collisions. Stellar Dynamics & Structure of Galaxies handout #2. Just as a self-gravitating collection of objects. Stear Dynamics & Structure of Gaaxies handout # Custer modeing Just as a sef-gravitating coection of objects. Coisions Do we have to worry about coisions? Gobuar custers ook densest, so obtain a rough

More information

Separation of Variables and a Spherical Shell with Surface Charge

Separation of Variables and a Spherical Shell with Surface Charge Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation

More information

Parallel-Axis Theorem

Parallel-Axis Theorem Parae-Axis Theorem In the previous exampes, the axis of rotation coincided with the axis of symmetry of the object For an arbitrary axis, the paraeaxis theorem often simpifies cacuations The theorem states

More information

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s

More information

Forces of Friction. through a viscous medium, there will be a resistance to the motion. and its environment

Forces of Friction. through a viscous medium, there will be a resistance to the motion. and its environment Forces of Friction When an object is in motion on a surface or through a viscous medium, there wi be a resistance to the motion This is due to the interactions between the object and its environment This

More information

Term Test AER301F. Dynamics. 5 November The value of each question is indicated in the table opposite.

Term Test AER301F. Dynamics. 5 November The value of each question is indicated in the table opposite. U N I V E R S I T Y O F T O R O N T O Facuty of Appied Science and Engineering Term Test AER31F Dynamics 5 November 212 Student Name: Last Name First Names Student Number: Instructions: 1. Attempt a questions.

More information

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed

More information

Legendre Polynomials - Lecture 8

Legendre Polynomials - Lecture 8 Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha

More information

Bohr s atomic model. 1 Ze 2 = mv2. n 2 Z

Bohr s atomic model. 1 Ze 2 = mv2. n 2 Z Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated

More information

12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes

12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.

More information

Quantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18

Quantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18 Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms

More information

Solution Set Seven. 1 Goldstein Components of Torque Along Principal Axes Components of Torque Along Cartesian Axes...

Solution Set Seven. 1 Goldstein Components of Torque Along Principal Axes Components of Torque Along Cartesian Axes... : Soution Set Seven Northwestern University, Cassica Mechanics Cassica Mechanics, Third Ed.- Godstein November 8, 25 Contents Godstein 5.8. 2. Components of Torque Aong Principa Axes.......................

More information

$, (2.1) n="# #. (2.2)

$, (2.1) n=# #. (2.2) Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier

More information

Previous Years Problems on System of Particles and Rotional Motion for NEET

Previous Years Problems on System of Particles and Rotional Motion for NEET P-8 JPME Topicwise Soved Paper- PHYSCS Previous Years Probems on Sstem of Partices and otiona Motion for NEET This Chapter Previous Years Probems on Sstem of Partices and otiona Motion for NEET is taken

More information

Applied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation

Applied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation 22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements

More information

Candidate Number. General Certificate of Education Advanced Level Examination January 2012

Candidate Number. General Certificate of Education Advanced Level Examination January 2012 entre Number andidate Number Surname Other Names andidate Signature Genera ertificate of Education dvanced Leve Examination January 212 Physics PHY4/1 Unit 4 Fieds and Further Mechanics Section Tuesday

More information

Module 22: Simple Harmonic Oscillation and Torque

Module 22: Simple Harmonic Oscillation and Torque Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque

More information

Nuclear Size and Density

Nuclear Size and Density Nucear Size and Density How does the imited range of the nucear force affect the size and density of the nucei? Assume a Vecro ba mode, each having radius r, voume V = 4/3π r 3. Then the voume of the entire

More information

221B Lecture Notes Notes on Spherical Bessel Functions

221B Lecture Notes Notes on Spherical Bessel Functions Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,

More information

Physics 505 Fall Homework Assignment #4 Solutions

Physics 505 Fall Homework Assignment #4 Solutions Physics 505 Fa 2005 Homework Assignment #4 Soutions Textbook probems: Ch. 3: 3.4, 3.6, 3.9, 3.0 3.4 The surface of a hoow conducting sphere of inner radius a is divided into an even number of equa segments

More information

Jackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.

More information

Physics 116C Helmholtz s and Laplace s Equations in Spherical Polar Coordinates: Spherical Harmonics and Spherical Bessel Functions

Physics 116C Helmholtz s and Laplace s Equations in Spherical Polar Coordinates: Spherical Harmonics and Spherical Bessel Functions Physics 116C Hemhotz s an Lapace s Equations in Spherica Poar Coorinates: Spherica Harmonics an Spherica Besse Functions Peter Young Date: October 28, 2013) I. HELMHOLTZ S EQUATION As iscusse in cass,

More information

Section 6: Magnetostatics

Section 6: Magnetostatics agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The

More information

More Scattering: the Partial Wave Expansion

More Scattering: the Partial Wave Expansion More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction

More information

PHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization

More information

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant

More information

Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.

Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n. Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier

More information

MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES

MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 151 Lectue 6 Kepe Pobem (Chapte 3) What We Did Last Time Discussed enegy consevation Defined enegy function h Conseved if Conditions fo h = E Stated discussing Centa Foce Pobems Reduced

More information

Physics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27

Physics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27 Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that

More information

V.B The Cluster Expansion

V.B The Cluster Expansion V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over

More information

Lecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential

Lecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia 16.1 4.4 3.6 9.9 Meson Theory of Nucear potentia 4.5 3.11 9.10 I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider

More information

Agenda Administrative Matters Atomic Physics Molecules

Agenda Administrative Matters Atomic Physics Molecules Fromm Institute for Lifeong Learning University of San Francisco Modern Physics for Frommies IV The Universe - Sma to Large Lecture 3 Agenda Administrative Matters Atomic Physics Moecues Administrative

More information

LECTURE 10. The world of pendula

LECTURE 10. The world of pendula LECTURE 0 The word of pendua For the next few ectures we are going to ook at the word of the pane penduum (Figure 0.). In a previous probem set we showed that we coud use the Euer- Lagrange method to derive

More information

THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE

THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on

More information

In Coulomb gauge, the vector potential is then given by

In Coulomb gauge, the vector potential is then given by Physics 505 Fa 007 Homework Assignment #8 Soutions Textbook probems: Ch. 5: 5.13, 5.14, 5.15, 5.16 5.13 A sphere of raius a carries a uniform surface-charge istribution σ. The sphere is rotate about a

More information

Lecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String

Lecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary

More information

14 Separation of Variables Method

14 Separation of Variables Method 14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt

More information

Strauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.

Strauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t. Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes

More information

Lecture 8 February 18, 2010

Lecture 8 February 18, 2010 Sources of Eectromagnetic Fieds Lecture 8 February 18, 2010 We now start to discuss radiation in free space. We wi reorder the materia of Chapter 9, bringing sections 6 7 up front. We wi aso cover some

More information

Physics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions

Physics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p

More information

Math 124B January 31, 2012

Math 124B January 31, 2012 Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat

More information

V.B The Cluster Expansion

V.B The Cluster Expansion V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f(q ) = exp ( βv( q )) 1, which is obtained by summing over

More information

Elements of Kinetic Theory

Elements of Kinetic Theory Eements of Kinetic Theory Diffusion Mean free path rownian motion Diffusion against a density gradient Drift in a fied Einstein equation aance between diffusion and drift Einstein reation Constancy of

More information

4 Separation of Variables

4 Separation of Variables 4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE

More information

Strauss PDEs 2e: Section Exercise 1 Page 1 of 7

Strauss PDEs 2e: Section Exercise 1 Page 1 of 7 Strauss PDEs 2e: Section 4.3 - Exercise 1 Page 1 of 7 Exercise 1 Find the eigenvaues graphicay for the boundary conditions X(0) = 0, X () + ax() = 0. Assume that a 0. Soution The aim here is to determine

More information

Copyright information to be inserted by the Publishers. Unsplitting BGK-type Schemes for the Shallow. Water Equations KUN XU

Copyright information to be inserted by the Publishers. Unsplitting BGK-type Schemes for the Shallow. Water Equations KUN XU Copyright information to be inserted by the Pubishers Unspitting BGK-type Schemes for the Shaow Water Equations KUN XU Mathematics Department, Hong Kong University of Science and Technoogy, Cear Water

More information

1. Measurements and error calculus

1. Measurements and error calculus EV 1 Measurements and error cacuus 11 Introduction The goa of this aboratory course is to introduce the notions of carrying out an experiment, acquiring and writing up the data, and finay anayzing the

More information

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I 6 n terms of moment of inertia, equation (7.8) can be written as The vector form of the above equation is...(7.9 a)...(7.9 b) The anguar acceeration produced is aong the direction of appied externa torque.

More information

Central Force Problem

Central Force Problem Central Force Problem Consider two bodies of masses, say earth and moon, m E and m M moving under the influence of mutual gravitational force of potential V(r). Now Langangian of the system is where, µ

More information

Lecture 17 - The Secrets we have Swept Under the Rug

Lecture 17 - The Secrets we have Swept Under the Rug Lecture 17 - The Secrets we have Swept Under the Rug Today s ectures examines some of the uirky features of eectrostatics that we have negected up unti this point A Puzze... Let s go back to the basics

More information

CS229 Lecture notes. Andrew Ng

CS229 Lecture notes. Andrew Ng CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view

More information

Dynamical Model of Binary Asteroid Systems Using Binary Octahedrons

Dynamical Model of Binary Asteroid Systems Using Binary Octahedrons Dynamica Mode of Binary Asteroid Systems Using Binary Octahedrons Yu Jiang 1,, Hexi Baoyin 1, Mo Yang 1 1. Schoo of Aerospace Engineering, singhua University, Beijing 100084, China. State Key Laboratory

More information

Midterm 2 Review. Drew Rollins

Midterm 2 Review. Drew Rollins Midterm 2 Review Drew Roins 1 Centra Potentias and Spherica Coordinates 1.1 separation of variabes Soving centra force probems in physics (physica systems described by two objects with a force between

More information

c 2007 Society for Industrial and Applied Mathematics

c 2007 Society for Industrial and Applied Mathematics SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),

More information

Elements of Kinetic Theory

Elements of Kinetic Theory Eements of Kinetic Theory Statistica mechanics Genera description computation of macroscopic quantities Equiibrium: Detaied Baance/ Equipartition Fuctuations Diffusion Mean free path Brownian motion Diffusion

More information

14-6 The Equation of Continuity

14-6 The Equation of Continuity 14-6 The Equation of Continuity 14-6 The Equation of Continuity Motion of rea fuids is compicated and poory understood (e.g., turbuence) We discuss motion of an idea fuid 1. Steady fow: Laminar fow, the

More information

Fourier series. Part - A

Fourier series. Part - A Fourier series Part - A 1.Define Dirichet s conditions Ans: A function defined in c x c + can be expanded as an infinite trigonometric series of the form a + a n cos nx n 1 + b n sin nx, provided i) f

More information

Chemical Kinetics Part 2. Chapter 16

Chemical Kinetics Part 2. Chapter 16 Chemica Kinetics Part 2 Chapter 16 Integrated Rate Laws The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates

More information

Strain Energy in Linear Elastic Solids

Strain Energy in Linear Elastic Solids Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua

More information

l Two observers moving relative to each other generally do not agree on the outcome of an experiment

l Two observers moving relative to each other generally do not agree on the outcome of an experiment Reative Veocity Two observers moving reative to each other generay do not agree on the outcome of an experiment However, the observations seen by each are reated to one another A frame of reference can

More information

Chemical Kinetics Part 2

Chemical Kinetics Part 2 Integrated Rate Laws Chemica Kinetics Part 2 The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates the rate

More information

HYDROGEN ATOM SELECTION RULES TRANSITION RATES

HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS

More information

Keywords: Rayleigh scattering, Mie scattering, Aerosols, Lidar, Lidar equation

Keywords: Rayleigh scattering, Mie scattering, Aerosols, Lidar, Lidar equation CEReS Atmospheric Report, Vo., pp.9- (007 Moecuar and aeroso scattering process in reation to idar observations Hiroaki Kue Center for Environmenta Remote Sensing Chiba University -33 Yayoi-cho, Inage-ku,

More information

Assignment 7 Due Tuessday, March 29, 2016

Assignment 7 Due Tuessday, March 29, 2016 Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges

More information

Kepler s Laws. force = mass acceleration = m q. m q = GmM q

Kepler s Laws. force = mass acceleration = m q. m q = GmM q Keper s Laws I. The equation of motion We consider the motion of a point mass under the infuence of a gravitationa fied created by point mass that is fixed at the origin. Newton s aws give the basic equation

More information

Supporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers

Supporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com . Two points A and B ie on a smooth horizonta tabe with AB = a. One end of a ight eastic spring, of natura ength a and moduus of easticity mg, is attached to A. The other end of the spring is attached

More information

Physics 566: Quantum Optics Quantization of the Electromagnetic Field

Physics 566: Quantum Optics Quantization of the Electromagnetic Field Physics 566: Quantum Optics Quantization of the Eectromagnetic Fied Maxwe's Equations and Gauge invariance In ecture we earned how to quantize a one dimensiona scaar fied corresponding to vibrations on

More information

FOURIER SERIES ON ANY INTERVAL

FOURIER SERIES ON ANY INTERVAL FOURIER SERIES ON ANY INTERVAL Overview We have spent considerabe time earning how to compute Fourier series for functions that have a period of 2p on the interva (-p,p). We have aso seen how Fourier series

More information

4 1-D Boundary Value Problems Heat Equation

4 1-D Boundary Value Problems Heat Equation 4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2012 DO NOT DISTRIBUTE THIS PAGE

AAPT UNITED STATES PHYSICS TEAM AIP 2012 DO NOT DISTRIBUTE THIS PAGE 2012 Semifina Exam 1 AAPT UNITED STATES PHYSICS TEAM AIP 2012 Semifina Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor This examination consists of two parts. Part A has

More information

Elements of Kinetic Theory

Elements of Kinetic Theory Eements of Kinetic Theory Statistica mechanics Genera description computation of macroscopic quantities Equiibrium: Detaied Baance/ Equipartition Fuctuations Diffusion Mean free path Brownian motion Diffusion

More information

Famous Mathematical Problems and Their Stories Simple Pendul

Famous Mathematical Problems and Their Stories Simple Pendul Famous Mathematica Probems and Their Stories Simpe Penduum (Lecture 3) Department of Appied Mathematics Nationa Chiao Tung University Hsin-Chu 30010, TAIWAN 23rd September 2009 History penduus: (hanging,

More information

Session : Electrodynamic Tethers

Session : Electrodynamic Tethers Session : Eectrodynaic Tethers Eectrodynaic tethers are ong, thin conductive wires depoyed in space that can be used to generate power by reoving kinetic energy fro their orbita otion, or to produce thrust

More information

Math 124B January 17, 2012

Math 124B January 17, 2012 Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia

More information

11.1 One-dimensional Helmholtz Equation

11.1 One-dimensional Helmholtz Equation Chapter Green s Functions. One-dimensiona Hemhotz Equation Suppose we have a string driven by an externa force, periodic with frequency ω. The differentia equation here f is some prescribed function) 2

More information

Gaussian Curvature in a p-orbital, Hydrogen-like Atoms

Gaussian Curvature in a p-orbital, Hydrogen-like Atoms Advanced Studies in Theoretica Physics Vo. 9, 015, no. 6, 81-85 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.5115 Gaussian Curvature in a p-orbita, Hydrogen-ike Atoms Sandro-Jose Berrio-Guzman

More information

TAM 212 Worksheet 9: Cornering and banked turns

TAM 212 Worksheet 9: Cornering and banked turns Name: Group members: TAM 212 Worksheet 9: Cornering and banked turns The aim of this worksheet is to understand how vehices drive around curves, how sipping and roing imit the maximum speed, and how banking

More information

Physics 506 Winter 2006 Homework Assignment #6 Solutions

Physics 506 Winter 2006 Homework Assignment #6 Solutions Physics 506 Winter 006 Homework Assignment #6 Soutions Textbook probems: Ch. 10: 10., 10.3, 10.7, 10.10 10. Eectromagnetic radiation with eiptic poarization, described (in the notation of Section 7. by

More information

Fourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form

Fourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier

More information

Lecture 22: Gravitational Orbits

Lecture 22: Gravitational Orbits Lecture : Gravitational Orbits Astronomers were observing the motion of planets long before Newton s time Some even developed heliocentric models, in which the planets moved around the sun Analysis of

More information

Electron-impact ionization of diatomic molecules using a configuration-average distorted-wave method

Electron-impact ionization of diatomic molecules using a configuration-average distorted-wave method PHYSICAL REVIEW A 76, 12714 27 Eectron-impact ionization of diatomic moecues using a configuration-average distorted-wave method M. S. Pindzoa and F. Robicheaux Department of Physics, Auburn University,

More information

CONIC SECTIONS DAVID PIERCE

CONIC SECTIONS DAVID PIERCE CONIC SECTIONS DAVID PIERCE Contents List of Figures 1 1. Introduction 2 2. Background 2 2.1. Definitions 2 2.2. Motivation 3 3. Equations 5 3.1. Focus and directrix 5 3.2. The poar equation 6 3.3. Lines

More information

LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS

LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL

More information

An approximate method for solving the inverse scattering problem with fixed-energy data

An approximate method for solving the inverse scattering problem with fixed-energy data J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999

More information

VI.G Exact free energy of the Square Lattice Ising model

VI.G Exact free energy of the Square Lattice Ising model VI.G Exact free energy of the Square Lattice Ising mode As indicated in eq.(vi.35), the Ising partition function is reated to a sum S, over coections of paths on the attice. The aowed graphs for a square

More information

Theory and implementation behind: Universal surface creation - smallest unitcell

Theory and implementation behind: Universal surface creation - smallest unitcell Teory and impementation beind: Universa surface creation - smaest unitce Bjare Brin Buus, Jaob Howat & Tomas Bigaard September 15, 218 1 Construction of surface sabs Te aim for tis part of te project is

More information

Nonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles

Nonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles ISSN 002-3640, JETP Letters, 20, Vo. 94, No., pp. 5. Peiades Pubishing, Inc., 20. Origina Russian Text V.I. Matveev, D.N. Makarov, 20, pubished in Pis ma v Zhurna Eksperimenta noi i Teoreticheskoi Fiziki,

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations .615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f =

More information

THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS

THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Sevie, Spain, -6 June 04 THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS M. Wysocki a,b*, M. Szpieg a, P. Heström a and F. Ohsson c a Swerea SICOMP

More information

OSCILLATIONS. dt x = (1) Where = k m

OSCILLATIONS. dt x = (1) Where = k m OSCILLATIONS Periodic Motion. Any otion, which repeats itsef at reguar interva of tie, is caed a periodic otion. Eg: 1) Rotation of earth around sun. 2) Vibrations of a sipe penduu. 3) Rotation of eectron

More information

First-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries

First-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische

More information

Demonstration of Ohm s Law Electromotive force (EMF), internal resistance and potential difference Power and Energy Applications of Ohm s Law

Demonstration of Ohm s Law Electromotive force (EMF), internal resistance and potential difference Power and Energy Applications of Ohm s Law Lesson 4 Demonstration of Ohm s Law Eectromotive force (EMF), interna resistance and potentia difference Power and Energy Appications of Ohm s Law esistors in Series and Parae Ces in series and Parae Kirchhoff

More information

Traffic data collection

Traffic data collection Chapter 32 Traffic data coection 32.1 Overview Unike many other discipines of the engineering, the situations that are interesting to a traffic engineer cannot be reproduced in a aboratory. Even if road

More information

Self Inductance of a Solenoid with a Permanent-Magnet Core

Self Inductance of a Solenoid with a Permanent-Magnet Core 1 Probem Sef Inductance of a Soenoid with a Permanent-Magnet Core Kirk T. McDonad Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (March 3, 2013; updated October 19, 2018) Deduce the

More information

APPENDIX C FLEXING OF LENGTH BARS

APPENDIX C FLEXING OF LENGTH BARS Fexing of ength bars 83 APPENDIX C FLEXING OF LENGTH BARS C.1 FLEXING OF A LENGTH BAR DUE TO ITS OWN WEIGHT Any object ying in a horizonta pane wi sag under its own weight uness it is infinitey stiff or

More information

1.2 Partial Wave Analysis

1.2 Partial Wave Analysis February, 205 Lecture X.2 Partia Wave Anaysis We have described scattering in terms of an incoming pane wave, a momentum eigenet, and and outgoing spherica wave, aso with definite momentum. We now consider

More information

DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE

DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE 3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses

More information

The Hydrogen Atomic Model Based on the Electromagnetic Standing Waves and the Periodic Classification of the Elements

The Hydrogen Atomic Model Based on the Electromagnetic Standing Waves and the Periodic Classification of the Elements Appied Physics Research; Vo. 4, No. 3; 0 ISSN 96-9639 -ISSN 96-9647 Pubished by Canadian Center of Science and ducation The Hydrogen Atomic Mode Based on the ectromagnetic Standing Waves and the Periodic

More information