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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 aparticles 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 atomsrmight 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 socalled “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 Jsec.) 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—
tentialenergy 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 potentialenergy 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 afterdinner 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
(10 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 strangeappearing 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 earthtoMars 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 twostep 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 therocket 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 overall 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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 Fall '07
 Harris
 Angular Momentum, Kinetic Energy, Mass, Moment Of Inertia, Velocity

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