Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom


 Morgan Webb
 2 years ago
 Views:
Transcription
1 Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in equal time. The peiod of evolution T of a planet about the sun is elated to the majo axis A of the ellipse by 3 T = k A whee k is the same fo all planets. 7. Planetay Obits: The Keple Poblem Intoduction Since Johannes Keple fist fomulated the laws that descibe planetay motion, scientists endeavoed to solve fo the equation of motion of the planets. In his hono, this poblem has been named The Keple Poblem. When thee ae moe than two bodies, the poblem becomes impossible to solve exactly. The most impotant theebody poblem at the time involved finding the motion of the moon, since the moon inteacts gavitationally with both the sun and the eath. Newton ealized that if the exact position of the moon wee known, the longitude of any obseve on the eath could be detemined by measuing the moon s position with espect to the stas. In the eighteenth centuy, Leonhad Eule and othe mathematicians spent many yeas tying to solve the theebody poblem, and they aised a deepe question. Do the small contibutions fom the gavitational inteactions of all the planets make the planetay system unstable ove long peiods of time? At the end of 8th centuy, Piee Simon Laplace and othes found a seies solution to this stability question, but it was unknown whethe o not the seies solution conveged afte a long peiod of time. Heni Poincaé poved that the seies actually diveged. Poincaé went on to invent new mathematical methods that poduced the moden fields of diffeential geomety and topology in ode to answe the stability question using geometic aguments, athe than analytic methods. Poincaé and othes did manage As stated in An Intoduction to Mechanics, Daniel Kleppne and Robet Kolenkow, McGawHill, 973, p 4. //9 7 
2 to show that the theebody poblem was indeed stable, due to the existence of peiodic solutions. Just as in the time of Newton and Leibniz and the invention of calculus, unsolved poblems in celestial mechanics became the expeimental laboatoy fo the discovey of new mathematics. 7. Reducing the TwoBody Poblem into a OneBody Poblem We shall begin ou solution of the twobody poblem by showing how the motion of two bodies inteacting via a gavitational foce (twobody poblem) is mathematically equivalent to the motion of a single body with a educed mass given by mm μ = m + m (7..) that is acted on by an extenal cental gavitational foce. Once we solve fo the motion of the educed body in this equivalent onebody poblem, we can then etun to the eal twobody poblem and solve fo the actual motion of the two oiginal bodies. The educed mass was intoduced in Section.7 of these notes. That section used simila but diffeent notation fom that used in this chapte. Conside the gavitational foce between two bodies with masses m and m as shown in Figue 7.. Figue 7. Gavitational foce between two bodies. Choose a coodinate system with a choice of oigin such that body has position and body has position (Figue 7.). The elative position vecto pointing fom body to body is =. We denote the magnitude of by =, whee is the distance between the bodies, and ˆ is the unit vecto pointing fom body to body, so that = ˆ (7..) //9 7 
3 Figue 7. Coodinate system fo the twobody poblem. The foce on body (due to the inteaction of the two bodies) can be descibed as mm F ˆ, = F, = G ˆ. (7..3) Recall that Newton s Thid Law equies that the foce on body is equal in magnitude and opposite in diection to the foce on body, F = F,, Newton s Second Law can be applied individually to the two bodies: F,. (7..4) d = m, (7..5) dt F d = m dt,. (7..6) Dividing though by the mass in each of Equations (7..5) and (7..6) yields F d, = dt m F d, = dt m, (7..7). (7..8) Subtacting the expession in Equation (7..8) fom that in Equation (7..7) gives //9 73
4 F F d d d = =. (7..9) m m dt dt dt,, Using Newton s Thid Law as given in Equation (7..4), Equation (7..9) becomes F d + =. (7..), m m dt Using the educed mass μ, as defined in Equation (7..),, μ = m + m (7..) Equation (7..) becomes F, d = μ dt d F, = μ dt (7..) whee F is given by Equation (7..3)., Ou esult has a special intepetation using Newton s Second Law. Let μ be the educed mass of a educed body with position vecto = ˆ with espect to an oigin O, whee ˆ is the unit vecto pointing fom the oigin O to the educed body. Then the equation of motion, Equation (7..), implies that the body of educed mass μ is unde the influence of an attactive gavitational foce pointing towad the oigin. So, the oiginal twobody gavitational poblem has now been educed to an equivalent onebody poblem, involving a educed body with educed mass μ unde the influence of a cental foce F ˆ,. Note that in this efomulation, thee is no body located at the cental point (the oigin O ). The paamete in the twobody poblem is the elative distance between the oiginal two bodies, while the same paamete in the onebody poblem is the distance between the educed body and the cental point. 7.3 Enegy and Angula Momentum, Constants of the Motion The equivalent onebody poblem has two constants of the motion, enegy E and the angula momentum L about the oigin O. Enegy is a constant because thee ae no //9 74
5 extenal foces acting on the educed body, and angula momentum is constant about the oigin because the only foce is diected towads the oigin, and hence the toque about the oigin due to that foce is zeo (the vecto fom the oigin to the educed body is antipaallel to the foce vecto and sinπ = ). Since angula momentum is constant, the obit of the educed body lies in a plane with the angula momentum vecto pointing pependicula to this plane. In the plane of the obit, choose pola coodinates (, θ ) fo the educed body (see Figue 7.3), whee is the distance of the educed body fom the cental point that is now taken as the oigin, and θ is the angle that the educed body makes with espect to a chosen diection, and which inceases positively in the counteclockwise diection. Figue 7.3 Coodinate system fo the obit of the educed body. Thee ae two appoaches to descibing the motion of the educed body. We can ty to find both the distance fom the oigin, t ( ) and the angle, θ ( t), as functions of the paamete time, but in most cases explicit functions can t be found analytically. We can also find the distance fom the oigin, ( θ ), as a function of the angle θ. This second appoach offes a spatial desciption of the motion of the educed body (see Appendix 7.A). The Obit Equation fo the Reduced Body Conside the educed body with educed mass given by Equation (7..), obiting about a cental point unde the influence of a adially attactive foce given by Equation (7..3). Since the foce is consevative, the potential enegy with choice of zeo efeence point U ( ) = is given by U () Gm m =. (7.3.) The total enegy enegy is E is constant, and the sum of the kinetic enegy and the potential //9 75
6 E v Gm m = μ. (7.3.) The kinetic enegy tem, μv /, has the educed mass and the elative speed v of the two bodies. As in Chaptes 5 and 7, we will use the notation v = v ˆ ˆ ad + vtanθ, d (7.3.3) v = v =, dt whee v = d dt and vtan = ( dθ / dt). Equation (7.3.) then becomes ad / d dθ G m m E μ = + dt dt. (7.3.4) The magnitude of the angula momentum with espect to the cente of mass is dθ dt = μ tan = μ. (7.3.5) L v We shall explicitly eliminate the θ dependence fom Equation (7.3.4) by using ou expession in Equation (7.3.5), dθ L =. (7.3.6) dt μ The mechanical enegy as expessed in Equation (7.3.4) then becomes d L G mm dt E = μ + μ. (7.3.7) Equation (7.3.7) is a sepaable diffeential equation involving the vaiable function of time t and can be solved fo the fist deivative d / dt, as a d L G m m = E + dt μ μ. (7.3.8) Equation (7.3.8) can in pinciple be integated diectly fo t (). In fact, in doing the integal no fewe than six cases need to be consideed, and even then the solution is of the fom t ( ) instead of t ( ). These integals ae pesented in Appendix 7.E. The function //9 76
7 t () can then, in pinciple, be substituted into Equation (7.3.6) and can then be integated to find θ ( t). Instead of solving fo the position of the educed body as a function of time, we shall find a geometic desciption of the obit by finding ( θ ). We fist divide Equation (7.3.6) by Equation (7.3.8) to obtain dθ L dθ dt μ = = d d L dt E μ + μ Gm m. (7.3.9) The vaiables and θ ae sepaable; L d μ dθ = L E μ + μ Gmm ( L/ ) d = μ L E + Gmm μ (7.3.). Equation (7.3.) can be integated to find the adius as a function of the angle θ ; see Appendix 7.A fo the exact integal solution. The esult is called the obit equation fo the educed body and is given by ε cosθ = (7.3.) whee L = (7.3.) μ Gm m is a constant (known as the semilatus ectum) and ε = + μ EL ( Gm m ) (7.3.3) //9 77
8 is the eccenticity of the obit. The two constants of the motion in tems of L= ( μ Gm m ) Gmm E = ( ε ). and ε ae (7.3.4) An altenate deivation of Equation (7.3.) is given in Appendix 7.F. The obit equation as given in Equation (7.3.) is a geneal conic section and is pehaps somewhat moe familia in Catesian coodinates. Let x= cosθ and y = sinθ, with = x + y. The obit equation can be ewitten as = + ε cosθ. (7.3.5) Using the Catesian substitutions fo x and y, ewite Equation (7.3.5) as Squaing both sides of Equation (7.3.6), ( ) / x + y = + ε x. (7.3.6) x + y = + ε x + ε x. (7.3.7) Afte eaanging tems, Equation (7.3.7) is the geneal expession of a conic section with axis on the x axis, ( ε ) x ε x y + = (7.3.8) (we now see that the dotted axis in Figue 7.3 can be taken to be the x axis). Fo a given >, coesponding to a given nonzeo angula momentum as in Equation (7.3.), thee ae fou cases detemined by the value of the eccenticity. Case : When ε =, E = E min < and =. Equation (7.3.8) is the equation fo a cicle, x y + = (7.3.9) Case : When < ε <, E < E < and Equation (7.3.8) descibes an ellipse, min y + Ax Bx= k (7.3.) //9 78
9 whee A > and k is a positive constant. ( Appendix 7.C shows how this expession may be expessed in the moe taditional fom involving the coodinates of the cente of the ellipse and the semimajo and semimino axes.) Case 3: When ε =, E = and Equation (7.3.8) descibes a paabola, x y =. (7.3.) Case 4: When ε >, E > and Equation (7.3.8) descibes a hypebola, whee A > and k is a positive constant. y Ax Bx= k (7.3.) 7.4 Enegy Diagam, Effective Potential Enegy, and Obits of Motion The enegy (Equation (7.3.7)) of the educed body moving in two dimensions can be eintepeted as the enegy of a educed body moving in one dimension, the adial diection, in an ective potential enegy given by two tems, U L Gm m μ =. (7.4.) The total enegy is still the same, but ou intepetation has changed; d L G m m E = K + U = μ + dt μ, (7.4.) whee the ective kinetic enegy K associated with the onedimensional motion is K d = μ dt. (7.4.3) The gaph of U as a function of = /, whee as given in Equation (7.3.), is shown in Figue 7.4. The uppe cuve (ed, if you can see this in colo) is popotional to L /( μ ) /. The lowe blue cuve is popotional to Gm m / /. The sum U is epesented by the geen cuve. The minimum value of U is at =, as will be shown analytically below. The vetical scale is in units of U. ( ) //9 79
10 Figue 7.4 Gaph of ective potential enegy. Wheneve the onedimensional kinetic enegy is zeo, K =, the enegy is equal to the ective potential enegy, E L Gm m μ = U =. (7.4.4) Recall that the potential enegy is defined to be the negative integal of the wok done by the foce. Fo ou eduction to a onebody poblem, using the ective potential, we will intoduce an ective foce such that U U F d F d B B, B, A = = A A (7.4.5) The fundamental theoem of calculus (fo one vaiable) then states that the integal of the deivative of the ective potential enegy function between two points is the ective potential enegy diffeence between those two points, B du U, B U, A d d A = (7.4.6) Compaing Equation (7.4.6) to Equation (7.4.5) shows that the adial component of the ective foce is the negative of the deivative of the ective potential enegy, //9 7 
11 F du d = (7.4.7) The ective potential enegy descibes the potential enegy fo a educed body moving in one dimension. (Note that the ective potential enegy is only a function of the vaiable and is independent of the vaiable θ ). Thee ae two contibutions to the ective potential enegy, and the total adial component of the foce is F d d L Gm m (7.4.8) d d μ = U = Thus thee ae two foces acting on the educed body, with an ective centifugal foce given by F = F + F, centifugal, gavity, (7.4.9) F d L L = = d μ μ 3,centifugal (7.4.) and the conventional gavitational foce F Gm m =. (7.4.), gavity With this nomenclatue, let s eview the fou cases pesented in Section 7.3. Case : Cicula Obit E = E min The lowest enegy state, enegy, E zeo since = ( U ) min min E = K + U E min, coesponds to the minimum of the ective potential. When this condition is satisfied the ective kinetic enegy is. The condition K d = μ = dt (7.4.) implies that the adial velocity is zeo, so the distance fom the cental point is a constant. This is the condition fo a cicula obit. The condition fo the minimum of the ective potential enegy is = du L G m m 3 d = μ +. (7.4.3) //9 7 
12 We can solve Equation (7.4.3) fo, L =, (7.4.4) Gmm epoducing Equation (7.3.). Case : Elliptic Obit Emin < E < When K =, the mechanical enegy is equal to the ective potential enegy, E = U, as in Equation (7.4.4). Having d / dt = coesponds to a point of closest o futhest appoach as seen in Figue 7.4. This condition coesponds to the minimum and maximum values of fo an elliptic obit, E L Gm m μ = (7.4.5) Equation (7.4.5) is a quadatic equation fo the distance, + Gm m L E μe = (7.4.6) with two oots E E μe Gmm Gmm L = ± + Equation (7.4.7) may be simplified somewhat as /. (7.4.7) Gm m LE E μ( Gmm) = ± + / (7.4.8) Fom Equation (7.3.3), the squae oot is the eccenticity ε, ε = + μ EL ( Gm m ), (7.4.9) //9 7 
13 and Equation (7.4.8) becomes Gm m E ( = ±. (7.4.) ε ) A little algeba shows that L / μ Gmm = ε LE + μ( Gmm) = LE/ μ( Gmm ) Gm m = E L / μgmm. (7.4.) Substituting the last expession in (7.4.) into Equation (7.4.) gives an expession fo the points of closest and futhest appoach, ( ε ). (7.4.) ε = ± The minus sign coesponds to the distance of closest appoach, ε and the plus sign coesponds to the distance of futhest appoach, Case 3: Paabolic Obit E = min = (7.4.3) + ε max =. (7.4.4) The ective potential enegy, as given in Equation (7.4.), appoaches zeo ( U ) when the distance appoaches infinity ( ). Since the total enegy is zeo, when the kinetic enegy also appoaches zeo, K. This coesponds to a paabolic obit (see Equation (7.3.)). Recall that in ode fo a body to escape fom a planet, the body must have a total enegy E = (we set U = at infinity). This escape velocity condition coesponds to a paabolic obit. Fo a paabolic obit, the body also has a distance of closest appoach. This distance pa can be found fom the condition //9 73
14 L Gmm E = U = =. (7.4.5) μ Solving Equation (7.4.5) fo yields L = = ; (7.4.6) pa μ Gmm the fact that the minimum distance to the oigin (the focus of a paabola) is half the semilatus ectum is a wellknown popety of a paabola. Case 4: Hypebolic Obit E > When E >, in the limit as the kinetic enegy is positive, K >. This coesponds to a hypebolic obit (see Equation (7.3.)). The condition fo closest appoach is simila to Equation (7.4.5) except that the enegy is now positive. This implies that thee is only one positive solution to the quadatic Equation (7.4.6), the distance of closest appoach fo the hypebolic obit The constant ε hyp =. (7.4.7) + is independent of the enegy and fom Equation (7.3.3) as the enegy of the educed body inceases, the eccenticity inceases, and hence fom Equation (7.4.7), the distance of closest appoach gets smalle. 7.5 Obits of the Two Bodies The obit of the educed body can be cicula, elliptical, paabolic o hypebolic, depending on the values of the two constants of the motion, the angula momentum and the enegy. Once we have the explicit solution (in this discussion, ( θ ) ) fo the educed body, we can find the actual obits of the two bodies. Choose a coodinate system as we did fo the eduction of the twobody poblem (Figue 7.5). //9 74
15 Figue 7.5 Cente of mass coodinate system. The cente of mass of the system is given by R cm m+ m =. (7.5.) m + m Let be the vecto fom the cente of mass to body and cente of mass to body. Then, by the geomety in Figue 7.5, = = be the vecto fom the (7.5.) and hence m + m m ( ) μ R. (7.5.3) = cm = = = m+ m m+ m m A simila calculation shows that μ =. (7.5.4) m Thus each body undegoes a motion about the cente of mass in the same manne that the educed body moves about the cental point given by Equation (7.3.). The only diffeence is that the distance fom eithe body to the cente of mass is shotened by a facto μ / mi. When the obit of the educed body is an ellipse, then the obits of the two bodies ae also ellipses, as shown in Figue 7.6. //9 75
16 Figue 7.6 The elliptical motion of bodies unde mutual gavitation. When one mass is much smalle than the othe, fo example mass is appoximately the smalle mass, m m, then the educed mm mm m + μ = = m m m (7.5.5) The cente of mass is located appoximately at the position of the lage mass, body of mass. Thus body moves accoding to m μ = m (7.5.6) and body is appoximately stationay, μ m = m m (7.5.7) 7.6 Keple s Laws Elliptic Obit Law Each planet moves in an ellipse with the sun at one focus. When the enegy is negative, E <, and accoding to Equation (7.3.3), //9 76
17 ε = + μ EL ( Gm m ) (7.6.) and the eccenticity must fall within the ange ε <. These obits ae eithe cicles o ellipses. Note the elliptic obit law is only valid if we assume that thee is only one cental foce acting. We ae ignoing the gavitational inteactions due to all the othe bodies in the univese, a necessay appoximation fo ou analytic solution. Equal Aea Law The adius vecto fom the sun to a planet sweeps out equal aeas in equal time. Using analytic geomety, the sum of the aeas of the tiangles in Figue 7.7 is given by ( Δθ) ( Δθ) Δ A= ( Δ θ) + Δ = ( Δ θ) + Δ (7.6.) in the limit of small Δ θ (the aea of the small piece on the ight, bounded on one side by the cicula segment, is appoximated by that of a tiangle). The aveage ate of the change of aea, Figue 7.7 Keple s equal aea law. Δ A, in time In the limit as Δt, Δθ, this becomes ( θ) ( θ) Δ t, is given by ΔA Δ Δ Δ = + Δt Δt Δ t. (7.6.3) da dt dθ dt =. (7.6.4) //9 77
18 Note that in this appoximation, we ae essentially neglecting the small piece on the ight in Figue 7.7 Recall that accoding to Equation (7.3.6) (epoduced below as Equation (7.6.5)), the angula momentum is elated to the angula velocity dθ / dt by dθ L = (7.6.5) dt μ and Equation (7.6.4) is then da L =. (7.6.6) dt μ Since L and μ ae constants, the ate of change of aea with espect to time is a constant. This is often familialy efeed to by the expession: equal aeas ae swept out in equal times (see Keple s Laws at the beginning of this chapte). Peiod Law The peiod of evolution T of a planet about the sun is elated to the majo axis the ellipse by 3 T = k A A of whee k is the same fo all planets. When Keple stated his peiod law fo planetay obits based on obsevation, he only noted the dependence on the lage mass of the sun. Since the mass of the sun is much geate than the mass of the planets, his obsevation is an excellent appoximation. Equation (7.6.6) can be ewitten in the fom Equation (7.6.7) can be integated as μ da = L. (7.6.7) dt μ da = L dt (7.6.8) obit T whee T is the peiod of the obit. Fo an ellipse, //9 78
19 Aea = da =π ab obit (7.6.9) whee a is the semimajo axis and b is the semimino axis. ( Appendix 7.D deives this esult fom Equation (7.3.).) Thus we have T μ π ab =. (7.6.) L Squaing Equation (7.6.) then yields T 4π μ ab =. (7.6.) L In Appendix 7.B, the angula momentum is given in tems of the semimajo axis and the eccenticity by Equation (B..). Substitution fo the angula momentum into Equation (7.6.) yields T 4π μ ab = μ Gm m a ( ε ). (7.6.) In Appendix 7.B, the semimino axis is given by Equation (B.3.7), which upon substitution into Equation (7.6.) yields T 3 4π μ a =. (7.6.3) μ Gm m Using Equation (7..) fo educed mass, the squae of the peiod of the obit is popotional to the semimajo axis cubed, T 4π a = G m 3 ( + m ). (7.6.4) 7.7 The Boh Atom Numeical values of physical constants ae fom the Paticle Data Goup tables, available fom //9 79
20 Conside the electic foce between two pointlike objects with chages q and q. The foce law is an invese squae law, like the gavitational foce. The diffeence is that the constant Gm m is eplaced by kq q whee 9 k = = N m C (7.7.) 4πε (the constant k is not a sping constant o the Boltzmann constant, meely a eflection of ou finite alphabet). The minus sign in the gavitational inteaction does not appea in the electic inteaction because thee ae two types of electic chage, positive and negative. The electic foce is attactive fo chages of opposite sign and epulsive fo chages of the same sign. Figue 7.8 Coulomb inteaction between two chages. q The foce on the chaged paticle of chage due to the electic inteaction between the two chaged paticles is given by Coulomb s law, qq F = ˆ ( ),, 4πε. (7.7.) Coulomb s Law povides an accuate desciption of the motion of chaged paticles when they ae not bound togethe. We cannot model the inteaction between the electon and the poton when the chaged paticles ae as close togethe as in the hydogen atom, since the Newtonian concept of foce is not welldefined at length scales associated with the size of atoms. Thus we need a new theoy, quantum mechanics, to explain the popeties of the atom. When the chaged paticles ae fa apat they ae essentially fee paticles and the quantum mechanical desciption of the bound system is not necessay. Theefoe the exact same method of solution that was used in the Keple Poblem fo obits of planets applies to the motion of chaged paticles. //9 7 
21 Fo a poton and an electon in a bound system, the hydogen atom, Niels Boh found a semiclassical agument that allows one to use the classical theoy of electic foces to pedict the obseved enegies of the hydogen atom. The following agument does not satisfy the pinciples of quantum mechanics, even though the esult is in easonable ageement with expeimentally detemined popeties of the hydogen atom. We begin ou discussion by ecalling ou esult, Equation (7.4.) fo the enegy of the gavitational system of two bodies when consideed as a single educed body of educed mass μ = mm /( m+ m) moving in one dimension, with the distance fom the cental point denoted by the vaiable, E = K + U (7.7.3) whee the ective potential enegy is L L U = U gavity G μ + = μ mm (7.7.4) and the ective kinetic enegy is K d = μ dt. (7.7.5) We can extend this desciption to the electic inteaction between the electon and poton of the hydogen atom by eplacing the constant Gmm with kqq, whee the chage of the poton is q = e, and the chage of the electon is q = e. The enegy is then given by d d L E = μ + U e = μ +. (7.7.6) dt dt μ 4πε Since the mass of the electon of the poton m p m e = kg is much smalle than the mass 7 =.6767 kg, the educed mass is appoximately the mass of the electon, μ m e. A schematic plot of the ective potential enegy as a function of the vaiable fo the hydogen atom is shown in Figue 7.9. //9 7 
22 Figue 7.9 Hydogen atom enegy diagam. Thee ae two impotant diffeences between the classical mechanical desciption of the possible obits unde a cental foce and a quantum mechanical desciption of the possible states of a hydogen atom. The fist diffeence is that the enegy E of the hydogen atom can only take on discete quantized values, unlike the classical case whee the enegy E can take on a continuous ange of values, negative fo cicula and elliptic bound obits, and positive fo hypebolic fee obits. The second diffeence is that thee ae additional states with the same value of enegy, but which have diffeent discete values of angula momentum, (and othe discete quantum popeties of the atom, fo example spin of the electon). Chemists identify these discete values of angula momentum by alphabetical labels ( s, p, d, f, g,...) while physicists label them by the obital angula momentum quantum numbes l =,,,.... We shall ty to estimate the enegy levels of the electon in the hydogen atom. We shall begin by assuming that the discete enegy states descibe cicula electon obits about the poton. (Quantum mechanics equies us to dop the notion that the electon can be thought of as a point paticle moving in an obit and eplace the paticle pictue with the idea that the electon s position can only be descibed by pobabilistic aguments.) Despite the unphysical natue of ou hypothesis, ou estimation of the enegy levels of the electon in the atom agee supisingly well with expeiment. The cicula obits coesponded to the situation descibed by This occus when E L e = ( U ) =. (7.7.7) μ 4πε min min //9 7 
23 du L e = 3 d = μ + 4πε. (7.7.8) Solving Equation (7.7.8) fo the adius of the obit, we find 4πε L =. (7.7.9) μ e We also note that the squae of the angula momentum is then μ e L =. (7.7.) 4πε We now make ou semiclassical assumption that the angula momentum assume discete values L can only h L= n. (7.7.) π 34 whee n is an intege, n =,,..., and h = kg m s is the Planck constant. With this assumption that the values L ae discete, Equation (7.7.) becomes nh μ e n = π 4πε (7.7.) whee n denotes the adii of the discete cicula obits, 4πε nh n =. (7.7.3) 4π μe The adii ae quantized, 4πε nh n = =n (7.7.4) 4π μe and equal to integal multiples of the gound state adius 4πεh = (7.7.5) 4π μe //9 73
24 whee, setting μ = me, the electon mass, is 34 ( 6.66 kg m s ) ( )( )( ) = 4π N m C 9.9 kg.6 C = 5.9 m to fou significant figues. The length as a o, pehaps counteintuitively, (7.7.6) is known as the Boh adius and is often given a, the latte notation indicating a nuclea mass of infinity, in which case, μ = me, as in the above calculation. We can substitute Equation (7.7.6) into Equation (7.7.7) and find that the enegy levels ae also quantized and given by E n L e μ e n e = = μn 4πε n 4πε μn n (7.7.7) e e. n πε n = = 4πε 4 Using Equation (7.7.5) fo E n, Equation (7.7.7) fo the enegy levels becomes e e = = = A (7.7.8) 4πε 4πε 4π μ 4πε 4 π μ nh e ( ) h n n whee, with μ = me, the constant A is given by A = π me 4 e h ( 4πε ) ( )( ) ( 34 ( 6.66 kg m s ) π 9.9 kg N m C.6 C = = 8.8 J ) (7.7.9) to fou significant figues. We can expess this enegy in tems of the enegy units of electonvolts ( ev ), whee an electonvolt is the enegy necessay to acceleate an electon with chage e acoss a potential enegy pe unit chage of volt ( volt = 9 9 ev =.6 C J C =.6 J and joule pe coulomb). Thus ( )( ) //9 74
25 ev A = = ( ) ( 9.6 J ) The enegy in Equation (7.7.8) can be witten as 8.8 J.36 ev. (7.7.) E n hc R = (7.7.) n whee R, the Rydbeg constant, is given by 8 (.8 J) 34 8 ( 6.66 kg m s )(.998 m s ) A R = = hc 7.97 m. = (7.7.) and c = m s is the speed of light. The fist few enegy levels ae shown in Figue 7., along with the enegy diffeence between the second and thid levels. The fist thee enegy levels ae E = 3.6 ev, E = 3.39 ev, E 3 =.5 ev. Figue 7. Enegy levels fo an electon in a hydogen atom Emission of light When an electon makes a tansition fom a highe enegy state E i to a lowe enegy state E f, light is emitted. The fequency of the emitted light is given by //9 75
26 f ΔE h E E f i = =. (7.7.3) h Using ou esult as given in Equation (7.7.8) fo the enegy levels, we have f = Rc n f n. (7.7.4) i The wavelength of the emitted light λ is elated to the fequency f of the light by f λ = c. (7.7.5) Thus the invese wavelength of the light is given by f = = R λ c. (7.7.6) nf n 7.7. Example: Calculate the wavelength of the light emitted when an electon in the enegy level n = 3 dops to the enegy level n =. Fom Equation (7.7.6) the wavelength is λ nn = = i f 3, R ni nf m. (7.7.7) The emitted light lies in the visible spectum and appeas ed to the human eye. //9 76
2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationmv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !
Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationA) 2 B) 2 C) 2 2 D) 4 E) 8
Page 1 of 8 CTGavity1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationSo we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)
Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body
More informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationGravitation and Kepler s Laws
3 Gavitation and Keple s Laws In this chapte we will ecall the law of univesal gavitation and will then deive the esult that a spheically symmetic object acts gavitationally like a point mass at its cente
More informationCopyright 2008 Pearson Education, Inc., publishing as Pearson AddisonWesley.
Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding
More informationGeneral Physics (PHY 2130)
Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationResources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics
3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gapsystem.og/~histoy/mathematicians/ Newton.html http://www.fga.com http://www.clke.com/clipat
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationCHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS
9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and
More information2008 QuarterFinal Exam Solutions
2008 Quatefinal Exam  Solutions 1 2008 QuateFinal Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationClassical Lifetime of a Bohr Atom
1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state,
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More informationPhysics 111 Fall 2007 Electrostatic Forces and the Electric Field  Solutions
Physics 111 Fall 007 Electostatic Foces an the Electic Fiel  Solutions 1. Two point chages, 5 µc an 8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the
More informationAnalytical Proof of Newton's Force Laws
Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue
More informationPY1052 Problem Set 3 Autumn 2004 Solutions
PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the
More informationLearning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.
Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of
More informationPhysics 202, Lecture 4. Gauss s Law: Review
Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential
More informationChapter 26  Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationChapter F. Magnetism. Blinn College  Physics Terry Honan
Chapte F Magnetism Blinn College  Physics 46  Tey Honan F.  Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.
More information2. An asteroid revolves around the Sun with a mean orbital radius twice that of Earth s. Predict the period of the asteroid in Earth years.
CHAPTR 7 Gavitation Pactice Poblems 7.1 Planetay Motion and Gavitation pages 171 178 page 174 1. If Ganymede, one of Jupite s moons, has a peiod of days, how many units ae thee in its obital adius? Use
More informationOrbital Motion & Gravity
Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationExam I. Spring 2004 Serway & Jewett, Chapters 15. Fill in the bubble for the correct answer on the answer sheet. next to the number.
Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 15 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE
More informationTALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS
4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length planoconvex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, planoconvex long focal length lens, monochomatic
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationSAMPLE CHAPTERS UNESCO EOLSS THE MOTION OF CELESTIAL BODIES. Kaare Aksnes Institute of Theoretical Astrophysics University of Oslo
THE MOTION OF CELESTIAL BODIES Kaae Aksnes Institute of Theoetical Astophysics Univesity of Oslo Keywods: celestial mechanics, twobody obits, theebody obits, petubations, tides, nongavitational foces,
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationGravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.
Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationSolutions to Homework Set #5 Phys2414 Fall 2005
Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More informationReview Module: Dot Product
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationPhysics 107 HOMEWORK ASSIGNMENT #14
Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The
More informationPhysics 505 Homework No. 5 Solutions S51. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.
Physics 55 Homewok No. 5 s S5. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationXIIth PHYSICS (C2, G2, C, G) Solution
XIIth PHYSICS (C, G, C, G) 6 Solution. A 5 W, 0 V bulb and a 00 W, 0 V bulb ae connected in paallel acoss a 0 V line nly 00 watt bulb will fuse nly 5 watt bulb will fuse Both bulbs will fuse None of
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More information6.2 Orbits and Kepler s Laws
Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that
More informationThe Gravity Field of the Earth  Part 1 (Copyright 2002, David T. Sandwell)
1 The Gavity Field of the Eath  Pat 1 (Copyight 00, David T. Sandwell) This chapte coves physical geodesy  the shape of the Eath and its gavity field. This is just electostatic theoy applied to the Eath.
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationSimple Harmonic Motion
Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium
More informationReview of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.
Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationL19 Geomagnetic Field Part I
Intoduction to Geophysics L191 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o
More informationSection 53 Angles and Their Measure
5 5 TRIGONOMETRIC FUNCTIONS Section 5 Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More information