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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 par
ticles (charge 2e) of kinetic energy 5 MeV approaching nuclei of gold
(charge 79e). (c) By introducing do, the expression for the fraction of in
cident alpha particles scattered into d9 at go becomes seam contains No atoms at
beam at x is just N(x) = attenuation that gives the f measurements of the at
; by argon gas at various
llimeters of mercury; the
ata are from the ﬁlm “The
periment,” by J. G. King,
Mass., 1961.) Deduce the
lm atom by an argon atom.
X 1019 molecules.) Check ie pressure agree. ns are visualized simply as
ely, what is implied about oblem one can calculate a
particles of a given energy
that do is given by do = two protons, separated by
value of do for alpha par
approaching nuclei of gold ion for the fraction of in
, becomes d9
Sin“ (so/2) where n is the number of nuclei per unit volume and As is the length
of the path through the foil. Putting d9 = 21r sin g0 dgo, show that the
fraction of alpha particles scattered through angles _>_ 900 is given by _ 1 2
df—16(nAs)d0 1r
— n As (1'02 cot2 $2 > =
f(_s0o) 4 2 (d) A foil of gold leaf 10—4 cm thick is bombarded with alpha
particles of energy 5 MeV. Out of 1 million alpha particles, incident
normally on the foil, how many would be deﬂected through 90° or
more? ODensity of gold = 1.9 X 10" kg/m3; atomic weight = 197.) PROBLEMS 6 x 10"""’m3
4 x 1033m3 M... m 7 x 1022 kg, r...3
~ 2 x 1030 kg, r? 22 S
2 22 Hence
9 z (2.5 X 10—9)(l.7 X 10—3) z 4 X lO—lzsec—l and so T = Z5 z 1.5 X 1012 sec z 50,000 years We could try to trim this result a little~for example, we have
someWhat underestimated the mean density of the earth’s crust
and (by treating the earth as a uniform sphere) we have some
what overestimated the moment of inertia. Both of these would
cause us to underestimate the precessional rate and obtain too
large a value for the precessional period. But in view of the
gross assumptions we have made elsewhere in the calculation,
we should not set any great store on making small reﬁnements
of this type. The important thing is that, by quite simple means,
we have veriﬁed that the precession of the equinoxes can indeed
be understood in terms of Newtonian dynamical principles. But
Newton got there ﬁrst! 14—] (a) Devise a criterion for whether there is external force acting on a system of two particles. Use this criterion on the following one
dimensional system. A particle of mass m is observed to follow the
path x(t) = A sin (wt) + L + 1;!
The other particle, of mass M, follows
X(t) = Bsin (wt) + Vt
The different constants are arbitrary except that mA = — MB.
(b) Try it on the system with
x(t) = A sin (wt) and X(t) = Bsin (wt + (p) where A and B are related as before, and go # 0. >< 1025m3
X 1033 m3 4 X 10~12 sec”1 )0 years .ittle—for example, we have
density of the earth’s crust
"orm sphere) we have some
nertia. Both of these would
:ssional rate and obtain too
)eriod. But in view of the
lsewhere in the calculation,
n making small reﬁnements
that, by quite simple means,
of the equinoxes can indeed
1 dynamical principles. But r there is external force acting
'riterion on the following one
s m is observed to follow the :pt that mA = —MB. = Bsin (0)! + (p) l<p5£0. 14—2 Consider a system of three particles, each of mass m, which
remain always in the same plane. The particles interact among them
selves, always in a manner consistent with Newton’s third law. If the
particles A, B, and C have positions at various times as given in the
table, determine whether any external forces are acting on the system. Time A B C
0 (1, 1) (2, 2) (3, 3)
1 (1, 0) (0, 1) (3, 3)
2 (O, 1) (1, 2) (2, 0) 14—3 Two skaters, each of mass 70 kg, skate at speeds of 4 m/sec in
opposite directions along parallel lines 1.5 m apart. As they are about
to pass one another they join hands and go into circular paths about
their common center of mass. (a) What is their total angular momentum? (b) A third skater is skating at 2 m/sec along a line parallel to
the initial directions of the other two and 6 m off to the side of the
track of the nearer one. From his standpoint, what is the total angular
momentum of the other two skaters as they rotate? 14—4 A molecule of carbon monoxide (CO) is moving along in a
straight line with a kinetic energy equal to the value of AT at room
temperature (A = Boltzmann’s constant = 1.38 X l0‘23 J/°K). The
molecule is also rotating about its center of mass with a total angular
momentum equal to h (= 1.05 X 10—34 Jsec). The internuclear
distance in the C0 molecule is 1.1 A. Compare the kinetic energy of
its rotational motion with its kinetic energy of translation. What does
this result suggest about the ease or difﬁculty of exciting such rotational
motion in a gas of C0 molecules at room temperature? 14—5 A uniform disk of mass Mand radius R is rotating freely about
a vertical axis with initial angular velocity w”. Then sand is poured
onto the disk in a thin stream so that it piles up on the disk at the
radius r (< R). The sand is added at the constant rate [.1 (mass per
unit time). (a) At what rate are the angular velocity and the rotational
kinetic energy varying with time at a given instant? (b) After what length of time is the rotational kinetic energy
reduced to half of its initial value? What has happened to this energy? 14—6 Two men, each of mass 100 kg, stand at opposite ends of the
diameter of a rotating turntable of mass 200 kg and radius 3 m. In
itially the turntable makes one revolution every 2 sec. The two men
make their way to the middle of the turntable at equal rates. (a) Calculate the ﬁnal rate of revolution and the factor by which the kinetic energy of rotation has been increased. (b) Analyze, at least qualitatively, the means by which the in
crease of rotational kinetic energy occurs. (c) At what radial distance from the axis of rotation do the
men experience the greatest centrifugal force as they make their way
to the center? 14—7 Estimate the kinetic energy in a hurricane. Take the density of
air as 1 kg/m3. 14—8 A useful way of calculating the approximate value of the moment
of inertia of a continuous object is to consider the object as if it were
built up of concentrated masses, and to calculate the value of Zmr2_
As an example, take the case of a long uniform bar of mass M and
lengthrL (with its transverse dimensions much less than L). We know
that its moment of inertia about one end is ML2/3. (a) The most primitive approximation is to consider the total
mass M to be concentrated at the midpoint, distant L/2 from the end.
You will not be surprised to ﬁnd that this is a poor approximation. (b) Next, treat the bar as being made up of two masses, each
equal to M/2, at distances L/4 and 3L/4 from one end. (c) Examine the improvements obtained from ﬁner subdivisions
—e.g., 3 parts, 5 parts, 10 parts. 14—9 (a) Calculate the moment of inertia of a thinwalled spherical
shell, of mass M and radius R, about an axis passing through its center.
Consider the shell as a set of rings deﬁned by the amounts of material
lying within angular ranges dB at the various angles 0 to the axis
(see the ﬁgure). (b) Verify the formula I = 2MR2/5 for the moment of inertia
of a solid sphere of uniform density about an axis through its center.
You can proceed just as in part (a), except that the system is to be
regarded as a stack of circular disks instead of rings. 14—10 (a) Calculate the moment of inertia of a thin square plate about
an axis through its center perpendicular to its plane. (Use the per
pendicularaxis theorem.)  (b) Making appropriate use of the theorems of parallel and
perpendicular axes, calculate the moment of inertia of a hollow cubical
box about an axis passing through the centers of two opposite faces. (c) Using the result of (a), deduce the moment of inertia of a
uniform, solid cube about an axis passing through the midpoints of
tw0 opposite faces. (d) For a cube of mass M and edge a, you should have obtained
the result Ma2/6. It is noteworthy that the moment of inertia has this
same value about any axis passing through the center of the cube.
See how far you can go toward verifying this result, perhaps by con
sidering other special axes—e.g., an axis through diagonally opposite
corners of the cube or an axis through the midpoints of opposite edges. increased. y, the means by which the in
rs. n the axis of rotation do the
force as they make their way iurricane. Take the density of proximate value of the moment
onsider the object as if it were
u calculate the value of Zmrr".
; uniform bar of mass M and
much less than L). We know
id is ML2/3. 1ation is to consider the total
)int, distant L/2 from the end.
this is a poor approximation.
made up of two masses, each
‘4 from one end. )tained from ﬁner subdivisions rtia of a thinwalled spherical
axis passing through its center.
ed by the amounts of material
various angles 0 to the axis 2/5 for the moment of inertia
am an axis through its center.
:cept that the system is to be
:ead of rings. ia of a thin square plate about
r to its plane. (Use the per the theorems of parallel and
t of inertia of a hollow cubical
centers of two opposite faces.
:e the moment of inertia of a
ing through the midpoints of ;e a, you should have obtained
the moment of inertia has this
ough the center of the cube.
g this result, perhaps by con
; through diagonally opposite
e midpoints of opposite edges. 14—11 Refer to Fig. 11—19(a), which shows the variation of density
with radial distance inside the earth. Using this graph, compare the
moment of inertia of the earth about its axis with the moment of inertia
of a sphere of the same mass and radius but of uniform density. You
can do quite well by considering the earth to be made up of a central
core and two thick concentric shells, each of approximately uniform
density. The boundaries between these three regions correspond to the
abrupt changes of density shown in the graph. [Alternatively, consider
the earth as built up of three superposed solid spheres—a basic one,
occupying the whole volume of the earth, with the density of the outer
most region (r > 0.54RE) and two other spheres with densities corre
sponding to the mean density differences between the successive regions] 14—12 (a) A hoop of mass WM and radius R rolls down a slope that
makes an angle 0 with the horizontal. This means that when the
linear velocity of its center is 1; its angular velocity is v/R. Show that
the kinetic energy of the rolling hoop is M122. (b) There is a traditional story about the camper—physicist who
has a can of bouillon and a can of beans, but the labels have come off,
so he lets them roll down a board to discover which is which. What
would you expect to happen? Does the method work? (Try it!) 14—13 A skier is enjoying the mountain air while standing on a 30°
snow slope when he suddenly notices a huge snowball rolling down at
him. By the time he notices the ball, it is only 100 m away and is
traveling at 25 m/sec. The skier gives himself a speed of 10 m/sec
almost instantaneously and proceeds to accelerate down the slope at
g sin 0 (= g/2). Does he get away? (Assume that the snowball has a
constant acceleration corresponding to that of a sphere of given radius
rolling, without slipping, down the slope. Assume that the moment of
inertia of the snowball about an axis through its center is 2MR2/5.) 14—14 The preceding problem suggests another one. If an object is
rolling down a slope, gathering material as it goes, how‘does its acceler
ation compare, in fact, with a similar object that is not adding material
in this way? To give yourself a relatively straightforward situation to
consider, take the case of a cylinder, rather than a sphere, that grows
in size as it rolls. Make whatever assumptions seem reasonable about
the way in which the rate of increase in radius depends on the existing
radius, R, and on the instantaneous speed, 12. 14—15 Two masses, of 9 kg and 1 kg, hang from the ends of a string
that passes around a pulley of mass 40 kg and radius 0.5 m (I = lMR2)
as shown in the diagram. The system is released from rest and the
9kg mass drops, starting the pulley rotating.
(a) What is the acceleration of the 9kg mass?
 (b) What is the angular velocity of 'the pulley after the 9kg mass has dropped 2 m?
(c) What is the tension in the part of the string which is between the pulley and the 9kg mass? Between the pulley and the lkg mass? (d) If the coeﬁﬁcient of friction between the string and the pulley
is 0.2, what is the least number of turns that the string must make
around the pulley to prevent slipping? (Cf. Problem 5—14.) 14—16 An amusement park has a downhill racetrack in which the
competitors ride down a 30° slope on small carts. Each cart has four
wheels, each of mass 20 kg and diameter 1 m. The frame of each cart
has a mass of 20 kg. (a) What is the acceleration of a cart if its rider has a mass of
50 kg? (Assume that the moment of inertia of a wheel is given by
0.8MR3, where R is its radius.) (b) If two riders, of masses 50 and 60 kg, respectively, start oﬂ’
simultaneously, what is the distance between them when the winner
passes the ﬁnishing line 60 m down the slope? 14—17 A uniform rod of length 31) swings as a pendulum about a
pivot a distance x from one end. For what value(s) of x does this
pendulum have the same period as a simple pendulum of length 21;? 14—18 (a) A piece of putty of mass m is stuck very near the rim of a
uniform disk of mass 2m and radius R. The disk is set on edge on a
table on which it can roll without slipping. The equilibrium position
is obviously that in which the piece of putty is closest to the table.
Find the period of smallamplitude oscillations about this position
and the length of the equivalent simple pendulum. (b) A circular hoop hangs over a nail on a wall. Find the period of its small—amplitude oscillations and the length of the equiva
lent simple pendulum.
(In these and similar problems, use the equation of conservation of
energy as a starting point. The more complicated the system, the
greater is the advantage that this method has over a direct application
of Newton’s law.) 14—19 A uniform cylinder of mass M and radius R can rotate about
a shaft but is restrained by a spiral spring (like the balance wheel of
your watch). When the cylinder is turned through an angle 0 from its
equilibrium position, the spring exerts a restoring torque M equal to
—c0. Set up an equation for the angular oscillations of this system
and ﬁnd the period, T. 14—20 Assuming that you let your legs swing more or less like rigid
pendulums, estimate the approximate time of one stride. Hence estimate your comfortable walking speed in miles per hour. How does
it compare with your actual pace? 14—2] A torsion balance to measure the momentum of electrons con
sists of a rectangular vane of thin aluminum foil, 10 by 2 by 0.005 cm,
attached to a very thin vertical ﬁber, as shown. The period of torsional
oscillation is 20 sec, and the density of aluminum is 2.7 times that of
water. :n the pulley and the lkg mass?
between the string and the pulley urns that the string must make
.’ (Cf. Problem 5—14.) 3wnhill racetrack in which the
small carts. Each cart has four
ter 1 m. The frame of each cart ‘a cart if its rider has a mass of
f inertia of a wheel is given by and 60 kg, respectively, start off
between them when the winner
e slope? :wings as a pendulum about a
)r what value(s) of x does this
simple pendulum of length 2b? is stuck very near the rim of a
E. The disk is set on edge on a
ping. The equilibrium position
3f putty is closest to the table.
oscillations about this position
e pendulum. :r a nail on a wall. Find the
ns and the length of the equiva he equation of conservation of
‘e complicated the system, the
0d has over a direct application and radius R can rotate about
ring (like the balance wheel of
1ed through an angle 0 from its
a restoring torque M equal to
.ular oscillations of this system 5 swing more or less like rigid
: time of one stride. Hence
:d in miles per hour. How does 1e momentum of electrons con
inum foil, 10 by 2 by 0.005 cm,
shown. The period of torsional
' aluminum is 2.7 times that of Electron—
beam spot (a) What is the torsion constant of the suspension, in mN/rad ? (b) What horizontal force, applied perpendicular to the surface
of the vane at a point 3 cm from the axis, will produce an angular
deﬂectipn of, 10°? , , ,, (c) A beam of 1 mA of electrons accelerated through 500 V
strikes the vane perpendicularly at a point 4 cm from the axis. What
steady angular deﬂection is produced, assuming that the electrons are
stopped in the vane? 14—22 The torsion constant of a wire or ﬁber of length I, and of circular
cross section of radius a, is given by c = Esra4/21, where E, is an
elastic constant of the material known as the shear modulus, measured
in N/m2. The maximum load that can be supported by such a ﬁber
is given by its crosssectional area, 7ra2, multiplied by the ultimate
tensile strength of the material, also measured in N/m2. For glass
ﬁbers the value of E, is about 2.5 X 10lo N/mz, and the ultimate
tensile strength is about 109 N/mz. (a) Calculate the diameter of the thinnest glass ﬁber that can
safely support two lead spheres, each of mass 20 g, in a gravity torsion
balance. Allow a safety factor of about 3. (b) If the spheres are at the ends of a light bar of length 20 cm,
and the length of the suspending ﬁber is also 20 cm, what is the period
of torsional oscillation of this system? (The measurement of this
period is the practical way of inferring the torsion constant of the
suspension.) (c) What angular deﬂection of this system is produced by
placing lead spheres of mass 2 kg with their centers 5 cm from the
centers of the small suspended spheres? What linear displacement
would this give in a spot of light reﬂected from a mirror on the torsion
arm to a scale 5 m away? Compare this result with the ﬁgures used
in Problem 5—3 on a Cavendish experiment. 14—23 A wheel of uniform thickness, of mass 10 kg and radius 10 cm,
is driven by a motor through a belt (see the ﬁgure). The drive wheel
on the motor is 2 cm in radius, and the motor is capable of delivering
a torque of 5 mN.
(a) Assuming that the belt does not slip on the wheel, how long
does it take to accelerate the large wheel from rest up to 100 rpm?
(b) If the coeﬂicient of friction between belt and wheel is Path of CM Drive wheel
T. T2
Driven wheel 0.3, what are the tensions in the belt on the two sides of the wheel?
(Assume that the belt touches the wheel over half its circumference.) . 14—24 A possible scheme for stopping the rotation of a spacecraft of
radius R is to let two small masses, m, swing out at the ends of strings
of length l, which are attached to the spacecraft at the points P and I"
(see the ﬁgure). Initially, the masses are held at the positions shown
and are rotating with the body of the spacecraft. When the masses
have swung out to their maximum distance, with the strings extending
radially straight out, the ends P and P’ of the strings are released from
the spacecraft. For given values of m, R, and I (the moment of inertia
of the spacecraft), what value of 1 will leave the spacecraft in a non
rotating state as a result of this operation? Apply the result to a
spacecraft that can be regarded as a uniform disk of mass M and radius R. (Put in some numbers, too, maybe.)
P P,
14—25 The technique of “pumping” a playground swing in order to
increase the amplitude of its motion can be learned by example or
(less easily) by trial and error. The mechanics of the procedure are
not trivial. According to one model of the process, the pumping is
taken to consist of a sudden elevation of the rider’s center of mass at
each passing of the vertical, or low point (the rider lifts and holds
himself above the seat), and a subsequent return to resting on the
swing seat at each turning point (see the ﬁgure). The support ropes
are assumed to be always straight, and the instantaneous changes of
effective length of the “pendulum” allow conservation of angular
momentum to be applied not only to the lowpoint pumping motions
but also to those at the turning points. (a) Carry out the analysis as indicated above and show that
increase of amplitude can be achieved. Note that the result agrees
with the qualitative experience that any given amount of increase is
more easily achieved as the amplitude increases. 3%, E: my; {been in the two sides of the wheel?
el over half its circumference.) . the rotation of a spacecraft of
:wing out at the ends of strings
racecraft at the points P and P’
re held at the positions shown
spacecraft. When the masses
.nce, with the strings extending
3f the strings are released from
t, and I (the moment of inertia
leave the spacecraft in a non
ttion? Apply the result to a
uniform disk of mass M and
maybe.) I7! P,
playground swing in order to
an be learned by example or
echanics of the procedure are
f the process, the pumping is
«f the rider’s center of mass at
lint (the rider lifts and holds
lent return to resting on the
Ie ﬁgure). The support ropes
the instantaneous changes of
low conservation of angular
e lowpoint pumping motions :licated above and show that Note that the result agrees
' given amount of increase is
creases. 707 (b) Consider in what ways the analysis indicated above may be
imperfect. Also, how well does this idealized technique match the
actual pumping method that children utilize every day? [The analysis
suggested above may be found in an article by P. L. Tea, Jr., and
H. Falk, “Pumping on a Swing,” Am. J. Phys., 36, 1165 (1968).] 14—26 Two gear wheels, A and B, of radii RA and R B, and of moments
of inertia IA and 13, respectively, are mounted on parallel shafts so
that they are not quite in contact (see the ﬁgure). Both wheels can rotate completely freely on their shafts. Initially, A is rotating with
angular velocity mo, and B is stationary. At a certain instant, one shaft
is moved slightly so that the gear wheels engage. Find the resulting
angular velocity of each in terms of the given quantities. (Warning: Do
not be tempted into a glib use of angular momentum conservation.
Consider the forces and torques resulting from the contact.) 14—27 A section of steel pipe of large diameter and relatively thin wall
is mounted as shown on a ﬂat—bed truck. The driver of the truck, not
realizing that the pipe has not been lashed in place, starts up the truck
with a constant acceleration of 0.5 g. As a result, the pipe rolls back
ward (relative to the truck bed) without slipping, and falls to the
ground. The length of the truck bed is 5 m. (a) With what horizontal velocity does the pipe strike the
ground? (b) What is its angular velocity at this instant? (c) How far does it skid before beginning to roll without slip
ping, if the coefﬁcient of friction between pipe and ground is 0.3? (d) What is its linear velocity when its motion changes to rolling
without slipping?
14—28 (a) How far above the center of a billiard ball or pool ball
should the ball be struck (horizontally) by the cue so that it will be
sure to begin rolling without slipping? (b) Analyze the consequences of striking the ball at the level
of its center if the coefﬁcient of friction between the ball and the table is p.
14—29 A man kicks sharply at the bottom end of a vertical uniform
post which is stuck in the ground so that 6 ft of it are above ground. 2? z" {:3 isle 2'; Unfortunately for him the post has rotted where it enters the ground
and breaks off at this point. To appreciate why “unfortunately” is the
appropriate word, consider the subsequent motion of the top end of the post. 14—30 Refer to page 673 for the discussion of a rectangular board
rotating about an axis in its plane. Using the notation and the method
of attack of that discussion, show that the angular momentum com
ponent L’ about an axis in the plane of the board and perpendicular to
w is given by combining the resolved parts of 1,0), and Iyw” in this
direction; i.e., L’ = Izwz sin 6 — Iva), cos 6 14—31 A ﬂywheel in the form of a uniform'disk of radius 5 cm is
mounted on an axle that just ﬁts along the diameter of a gimbal ring
of diameter 12 cm. The ﬂywheel is set rotating at 1000 rpm and the
gimbal ring is supported at the point where one end of the axle meets it.
Calculate the rate of precession in rpm. 14—32 In most cars the engine has its axis of rotation pointing fore
and aft along the car. The gyroscopic properties of the engine when
rotating at high speed are not negligible. Consider the tendency of
this gyroscopic property to make the front end of the car rise or fall as
the car follows a curve in the road. What about the corresponding
effects for a car with its engine mounted transversely? Try to make
some quantitative estimates of the importance of such effects. Con
sider whether a lefthand curve or a righthand curve might involve
the greater risk of losing control over the steering of the car. 14—33 See if you can pick up the challenge, given in the text, of making
a more respectable calculation of the precession of the equinoxes. ...
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This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.
 Fall '07
 Harris

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