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Chs 13 & 14 - I said “Why not let him see if...

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Unformatted text preview: I said, “Why not let him see if hrough a large angle?“ I may rot believe that they would be, vas a very fast massive particle, 1 you could show that if the tted effect of a number of small article’s being scattered back- :membered two or three days It excitement and saying, “We the a-particles coming back- incredible event that has ever : almost as incredible as if you tissue paper and it came back [ realized that this scattering at single collision, and when I as impossible to get anything 'ou took a system in which the atom was concentrated in a had the idea of an atom with charge. I worked out mathe- should obey, and I found that through a given angle should the scattering foil, the square 1y proportional to the fourth JCthflS were later verified by beautiful experimentsl luced some excerpts from rsden. It is interesting to :attering (> 90°) on the )mic model of 1910. This 1e negative electrons were a sphere of uniform posi- A passing alpha particle : repulsion of the positive mass of the atom.2 The :ounter was quite small. ;everal atomsrmi-ght occur :ctures by various scientists at nd W. Pagel, eds.), Cambridge to light compared to the alpha aside in a collision between PROBLEMS 617 in sufficiently thick foils, producing a net deflection which is large. For a gold foil 10"4 cm thick such as Geiger and Marsden used for some of their experiments, the Thomson theory pre- dicted that the fraction of alpha particles scattered at angles greater than 90° would be about one out of every 10””! That is tantamount to saying that it would never happen. (Recall, for the purposes of comparison, that the total number of all the electrons, protons, and neutrons in all the galaxies of the observ- able universe is only about 1080.) No wonder Rutherford was astonished when Geiger and Marsden observed for a foil of this thickness that approximately one out of every to4 alpha particles was deflected at angles greater than 90°. 13—] The circular orbits under the action of a certain central force F(r) are found all to have the same rate of sweeping out area by the radius vector, independent of the orbital radius. Determine how F varies with r. 13—2 In the Bohr model of the hydrogen atom an electron (mass m) moves in a circular orbit around an effectively stationary proton, under the central Coulomb force F(r) = —ke2/r2. (a) Obtain an expression for the speed c of the electron as a function of r. (b) Obtain an expression for the orbital angular momentum l as a function of r. (c) Introduce Bohr’s postulate (of the so-called “old quantum theory,” now superseded) that the angular momentum in a circular orbit is equal to nit/2w, where h is Planck’s constant. Obtain an expression for the permitted orbital radii. (d) Calculate the potential energy of the system from the equation I U(r) = —-/ F(r)dr no Hence find an expression for the total energy of the quantized system as a function of n. (e) For the lowest energy state of the atom (corresponding to n = 1) calculate the numerical values of the orbital radius and the energy, measured in electron volts, needed to ionize the atom. (k = 9 x109N—m2/C2;e = 1.6 x 10'19C;m = 9.1 x1o—3‘ kg; h = h/21r = 1.05 x 10'34 J-sec.) 13—3 A mass m is joined to a fixed point 0 by a string of length l. Initially the string is slack and the mass is moving with constant speed [)0 along a straight line. At its closest approach the distance of the mass from 0 is It. When the mass reaches a distance 1 from 0, the string becomes taut and the mass goes into a circular path around 0. Find the ratio of the final kinetic energy of the mass to its initial kinetic energy. (Neglect any effects of gravity.) Where did the energy go? 13—4 A particle A, of mass m, is acted on by the gravitational force from a second particle, B, which remains fixed at the origin. Initially, when A is very far from B (r = 00), A has a velocity v0 directed along the line shown in the figure. The perpendicular distance between B and this line is D. The particle A is deflected from its initial course by B and moves along the trajectory shown in the figure. The shortest distance betWeen this trajectory and B is found to be d. Deduce the mass of B in terms of the quantities given and the gravitational con- stant G. i Trajectory 13—5 A particle of mass m moves in the field of a repulsive central force Ant/r”, where A is a constant. At a very large distance from the force center the particle has speed 1:0 and its impact parameter is b. Show that the closest m comes to the center of force is given by ‘) ‘) rtnin = (b2 'l‘ A/UO")l/' 13—6 A nonrotating, spherical planet with no atmosphere has mass M and radius R. A particle is fired off from the surfaCe with a speed equal to three quarters of the escape speed. By considering conserva— tion of total energy and angular momentum, calculate the farthest distance that it reaches (measured from the center of the planet) il~ it is fired oil (a) radially and (b) tangentially. Sketch the ell‘eetive po— tential-energy curve, given by for case (b). Draw the line representing the total energy of the motion, and thus verify your result. 13—7 imagine a spherical, nonrotating planet of mass M, radius R, that has no atmosphere. A satellite is tired from the surface ol‘ the planet with speed (In at 30° to the local vertical. In its subsequent orbit the satellite reaches a maximum distance of 5Ry'2 from the center ig with constant speed :h the distance of the listance I from 0, the 'cular path around 0. he mass to its initial Where did the energy he gravitational force t the origin. Initially, tcity v0 directed along r distance between B 'rom its initial course e figure. The shortest to be d. Deduce the ,he gravitational con- 'ajec tory )f a repulsive central 'ge distance from the pact parameter is b. rce is given by tmosphere has mass surface with a speed onsidering conserva— Ilculate the farthest r of the planet) if it tch the effective po- 1ergy of the motion, mass M, radius R, l the surface of the In its subsequent R, '2 from the center 619 of the planet. Using the principles of conservation of energy and angular momentum, show that 00 = (SGM/4R)“2 13—8 A particle moves under the influence of a central attractive force, —k/r3. At a very large (effectively infinite) distance away, it has a nonzero velocity that does not point toward the center. Con- struct the effective potential-energy diagram for the radial component of the motion. What conclusions can you draw about the dependence on r of the radial component of velocity? 13—9 A satellite in a circular orbit around the earth fires a small rocket. Without going into detailedcalculations, consider how the orbit is changed according to whether the rocket is fired (a) forward; (b) backward; (c) toward the earth; and (d) perpendicular to the plane of the orbit. 13—10 Two spacecraft are coasting in exactly the same circular orbit around the earth, but one is a few hundred yards behind the other. An astronaut in the rear wants to throw a ham sandwich to his partner in the other craft. How can he do it? Qualitatively describe the various possible paths of transfer open to him. (This question was posed by Dr. Lee DuBridge in an after-dinner speech to the American Physical Society on April 27, 1960.) 13—] I The elliptical orbit of an earth satellite has major axis 2a and minor axis 2b. The distance between the earth’s center and the other focus is 2c. The period is T. (a) Verify that b = (a2 — 02)“2. (b) Consider the satellite at perigee (r; a — c) and apogee (r2 = a + c). At these two points its velocity vector and its radius vector are at right angles. Verify that conservation of energy implies that 2 GMm lmv] 2 GMm 2 ._ 1 =—mt:2 —————=E (1-0 2 a+c Verify also that conservation of angular momentum implies that b E = %(a - c)vi = %(a + c)02 T (c) From the above relationships, deduce the following results, corresponding to Eqs. (13—36) and (13—39) in the text: T2 = 41r2a3/GM and = -—GMm/2a 13—12 A satellite of mass m is in an elliptical orbit about the earth. When the satellite is at its perigee, a distance R0 from the center of Eriroitéiems the earth, it is traveling with a speed v0. The mass of the earth, M, is much greater than m. (a) If the length of the major axis of the elliptical orbit is 4R0, what is the speed of the satellite at its apogee (the maximum distance from the earth) in terms of G, M, and R0? (b) Show that the length of the minor axis of the elliptical orbit is 2V5 R0, and find the period of the satellite in terms of Do and R0. 13—13 A satellite of mass m is traveling at speed we in a circular orbit of radius ro under the gravitational force of a fixed mass at 0. ' (a) Taking the potential energy to be zero at r = co , show that the total mechanical energy of the satellite is -—%mvo2. (b) At a certain point B in the orbit (see the figure) the direction of motion of the satellite is suddenly changed without any change in the magnitude of the velocity. As a result the satellite goes into an elliptic orbit. Its closest distance of approach to 0 (at point P) is now ro/5. What is the speed of the satellite at P, expressed as a multiple of 00? (c) Through what angle a (see the figure) was the velocity of the satellite turned at the point B? 13—14 A small satellite is in a circular orbit of radius r1 around the earth. The direction of the satellite‘s velocity is now changed, causing it to move in an elliptical orbit around the earth. The change in velocity is made in such a manner that the satellite loses half its orbital angular momentum, but its total energy remains unchanged. Cal- culate, in terms of r1, the perigee and apogee distances of the new orbit (measured with respect to the earth's center). 13—15 An experimental rocket is fired from Cape Kennedy with an initial speed v0 and angle 0 to the horizontal (see the figure). Neglect- ing air friction and the earth’s rotational motion, calculate the maxi- mum distance from the center of the earth that the rocket achieves in terms of the earth’s mass and radius (M and R), the gravitational constant G, and 6 and v0. 13—16 A satellite of mass m is traveling in a perfectly circular orbit of radius r about the earth (mass M). An explosion breaks up the satellite into two equal fragments, each of mass m/2. Immediately mass of the earth, M, is ie elliptical orbit is 4R0, (the maximum distance xis of the elliptical orbit in terms of 00 and R0. ed 120 in a circular orbit fixed mass at 0. r0 at r = 00 , show that -%mv02. > V the figure) the direction without any change in satellite goes into an ) 0 (at point P) is now xpressed as a multiple 'e) was the velocity of ' radius r] around the now changed, causing arth. The change in te loses half its orbital ins unchanged. Cal- distances of the new r). tpe Kennedy with an : the figure). Neglect- n, calculate the maxi- the rocket achieves in R), the gravitational :rfectly circular orbit losion breaks up the s m/2. Immediately after the explosion the two fragments have radial components of velocity equal to 120/2, where no is the orbital speed of the satellite prior to the explosion; in the reference frame of the satellite at the instant of the explosion the fragments appear to separate along the line joining the satellite to the center of the earth. (a) In terms of G, M, m, and r, what are the energy and the angular momentum (with respect to the earth’s center) of each frag- ment? (b) Make a sketch showing the original circular orbit and the orbits of the two fragments. In making the sketch, use the fact that the major axis of the elliptic orbit of a satellite is inversely proportional to the total energy. 13—17 A spaceship is in an elliptical orbit around the earth. It has a certain amount of fuel for orbit alteration. Where in the orbit should this fuel be used to attain the greatest distance from earth? Do you notice any similarity between this problem and the one con— cerning a rocket ignited after falling down a chute (Problem 10~1 3)? 13—18 The commander of a spaceship that has shut down its engines and is coasting near a strange-appearing gas cloud notes that the ship is following a circular path that will lead directly into the cloud (see the figure). He also deduces from the ship’s motion that its angular momentum with respect to the cloud is not changing. What attractive (central) force could account for such an orbit? Spaceship \ 13—19 (a) Make an analysis of an earth-to-Mars orbit transfer similar to that carried out in the text for the transfer to Venus. Assume that earth and Mars are in circular orbits of radii l and 1.52 AU, re- spectively. (b) In part (a), and in the discussion in the text, the gravita-' tional fields of the planets are neglected. (The problem was taken to be simply that of shifting from one orbit to another, not from the surface of one planet to the surface of the other.) At what distance from the earth is the earth’s field equal in magnitude to that of the sun? Similarly, at what distance from Mars is the sun’s field equaled by that of the planet? Further, compare the work done against the sun’s gravity in the transfer with that done against the earth’s gravity, and with the energy gained from the gravitational field of Mars. 13—20 The problem of dropping a spacecraft into the sun from the earth’s orbit with the application of minimum possible impulse (given to the spacecraft by firing a rocket engine) is not solved by firing the rocket in a direction opposite to the earth's orbital motion so as to reduce the velocity of the spacecraft to zero. A two-step process can accomplish the goal with a smaller rocket. Assume the initial orbit to be a circle of radius r; with the sun at the center (see the figure). By means of a brief rocket burn the spacecraft is Speeded up tangentially in the direction of the orbit velocity, so that it assumes an elliptical orbit whose perihelion coincides with the firing point. At the aphelion of this orbit the spacecraft is given a backward impulse sufficient to reduce its space velocity to zero, so that it will subsequently fall into the sun. (As in the previous transfer problem, the effects of the earth‘s gravity are neglected.) (a) For a given value of the aphelion distance, r2 of the space- craft, calculate the required increment of speed given to it at first firing. (b) Find the speed of the spacecraft at its aphelion distance, and so find the sum of the speed increments that must be given to the spacecraft in the two steps to make it fall into the sun. This sum pro- vides a measure of the total impulse that the-rocket engine must be able to supply. Compare this sum with the speed of the spacecraft in its initial earth orbit for the case r2 = 10r1. [Notes This problem is discussed by E. Feenberg, “Orbit to the Sun," Am. J. Phys., 28, 497 (1960).] 13—2] The sun loses mass at the rate of about 4 X 106 tons/sec. What change in the length of the year should this have produced within the span of recorded history (~5000 yr)? Note that the equa- tion for circular motion can be employed (even though the earth spirals away from the sun) because the fractional yearly change in radius is so small. The other condition needed to describe the gradual shift is the over-all conservation of angular momentum about the CM of the system. (This problem was given in a simplified form as Prob- lem 8—19.) ,he sun’s field equaled done against the sun’s 1e earth’s gravity, and 1d of Mars. nto the sun from the ossible impulse (given at solved by firing the 'bital motion so as to . two—step process can me the initial orbit to er (see the figure). By peeded up tangentially t assumes an elliptical Joint. At the aphelion l impulse sufficient to subsequently fall into 1e effects of the earth’s tance, r2 of the space- iven to it at first firing. its aphelion distance, t must be given to the 1e sun. This sum pro- ‘ocket engine must be ed of the spacecraft in g, “Orbit to the Sun,” out 4 X 106tons/sec. d this have produced ? Note that the equa— ven though the earth anal yearly change in :0 describe the gradual nentum ab0ut the CM nplified form as Prob- 623 13—22 A particle of mass m moves about a massive center of force C, with the attraction given by —f(r)e,, where r is the position of the particle as measured from C. If the particle is also subjected to a retarding force —)\v, and initially has angular momentum Lo about C, find its angular momentum as a function of time. 13—23 Consider a central force in a horizontal plane given by F(r) = —kr, where k is a constant. (This provides a good description, for example, of the pendulum encountered in the laboratory. Rarely is a pendulum physically confined to swing in only one vertical plane.) (a) A particle of mass m is moving under the influence of such a force. Initially the particle has position vector re and velocity vo as measured from the stationary force center. Set up a Cartesian co- ordinate system with the xy plane containing re and v0, and find the time dependence of the position (x, y) of the particle. Does the orbit correspond to any particular geometric curve? (Keep in mind the differences between this interaction and the gravitational problem.) What physical quantities are conserved? (b) Suppose the particle is originally in a circular orbit of radius R. What is its orbital speed? If at some point its velocity is doubled, what will be the maximum value of r in its subsequent motion? 13—24 According to general relativity theory, the gravitational po- tential energy of a mass m orbiting about a mass M is modified by the addition of a term —GMmC2/c2r3, where C = r2 dB/dt and c is the speed of light. Thus the period of a circular orbit of radius r is slightly smaller than would be predicted by Newtonian theory. (a) Show that the fractional change in the period of a circular orbit of radius r due to this relativistic term is —(127r2r2/C2To2), where To is the period predicted by Newtonian theory. (b) Since, by Kepler's third law, we have T 02 ~ r3, the effect of this relativistic correction is greatest for the planet closest to the sun, i.e., Mercury. Consider the effect of the relativistic term on the radial and angular periods, and see if you can thereby arrive at the famous result that the perihelion of Mercury’s orbit precesses at the rate of about 43 seconds of are per century. You may find it useful to refer back to Problem 8—20, which also deals with this question. 13—25 A beam of atoms traveling in the positive x direction and passing through a medium containing n particles per unit volume suffers an attenuation given by dN(x) dx = -—'AnN(x) where A is the cross section for scattering of an atom in the beam by ii’méréienzs an atom of the medium. Therefore, if the beam contains No atoms at x = 0, the number still traveling in the beam at x is just N(x) = Noe—An: (a) Set up a simple model of beam attenuation that gives the results stated above. (b) The graph summarizes a set of measurements of the at- tenuation of a beam of potassium atoms by argon gas at various pressures (the pressures are given in millimeters of mercury; the temperature is 0°C throughout). (These data are from the film “The Size of Atoms from an Atomic Beam Experiment,” by J. G. King, Education Development Center, Newton, Mass.I 1961.) Deduce the cross section for the scattering of a potassium atom by an argon atom. (1 cm3 of a perfect gas at STP contains 2.7 X 1019 molecules.) Check Whether theresults for difl‘erent values of 'the'pressure'a’gre'e.’ (c) If the potassium and argon atoms are visualized simply as hard spheres of radii rK and rA, respectively, what is implied about rK and rA by the result of part (b)? 13—26 (a) In the Rutherford scattering problem one can calculate a distance of closest approach do for alpha particles of a given energy approaching a nucleus head on. Verify that do is given by do = quiqz/mvoz. (b) The force of repulsion between two protons, separated by 10"14 m, is 2.3 N. Use this to deduce the value of do for alpha...
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