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Unformatted text preview: PROBLEMS nitudes.) Taking the x components alone, we have 2
mu
.7 9 = __ __ 9
cos A cos ll F: 9 U ~ 0
ACOS ll (I: Thus the x component of the complete vector equation, F = ma
is F: = max, with the values of F1 and a, stated‘ above. In order to display the dynamical identity of this component
motion with SHM, we can take the expressions for F, and aI separately, introducing the angular velocity w and putting u =
«2/1. We then have 3 FI = —mw2A c050 = ——mw2X aI = ——ng c050 = ——w‘~’x The ﬁrst of these equations deﬁnes a restoring force proportional
to diSplacement, exactly in accord with our initial statement of
Hooke’s law [Eq. (7—38)]. The second corresponds exactly to
theequation [Eq. (7~40)] that was our starting point for the
kinematic analysis of the problem: ‘ 2
dx 2
———=—wx
(1/2 Thus we see that the dynamical correspondence is complete in
every respect. It tells us, moreover, that we could, if we wished,
go the other way and treat a uniform circular motion as a super
position of two simple harmonic motions at right angles. This
is, in fact, an extremely important and useful procedure in some contexts, although we shall not take time to follow it up here
and now. 7—) Two identical gliders, each of mass m, are being towed thr0ugh
the air in tandem, as shown. Initially they are traveling at a constant
speed and the tension in the tow rope A is T0. The tow plane then
begins to accelerate with an acceleration a. What are the tensions in
A and B immediately after this acceleration begins? alone, we have ete vector equation, F = ma
and a; stated' above. :al identity of this component 1e expressions for FI and a,
velocity to and putting v = a l restoring force proportional
with our initial statement of
cond corresponds exactly to
5 our starting point for the Irrespondence is complete in
that we could, if we wished,
n circular motion as a super
.otions at right angles. This
nd useful procedure in some
:e time to follow it up here iss m, are being towed through
.hey are traveling at a constant
A is To. The tow plane then
In a. What are the tensions in
tion begins? "I m, m._, 235 7—2 Two blocks, of masses M = 3 kg and m = 2 kg, are in contact
on a horizontal table. A constant horizontal force F = 5 N is applied
to block M as shown. There is a constant frictional force of 2 N be
tween the table and the block m but no frictional force between the
table and the ﬁrst block M. (a) Calculate the acceleration of the two blocks. (b) Calculate the force of contact between the blocks. 7—3 A sled of mass m is pulled by a force of magnitude P at angle 0
to the horizontal (see the ﬁgure). The sled slides over a horizontal
surface of snow. It experiences a tangential resistive force equal to
)1 times the perpendicular force N exerted on the sled by the snow. (a) Draw an isolation diagram showing all the forces exerted
on the sled.  (b) Write the equations corresponding to F = ma for the
horizontal and vertical components of the motion. (c) Obtain an expression for the horizontal acceleration in
terms of P, 0, m, u, and g. (d) For a given magnitude of P, ﬁnd what value of 0 gives the
biggest acceleration. 7—4 A block of mass In; rests on a frictionless horizontal surface;
it is connected by a massless string, passing through a frictionless
eyelet, to a second block o'fmass m2 that rests on a frictionless incline
(see the ﬁgure). (a) Draw isolation diagrams for the masses and write the equa
tion of motion for each one separately. (b) Find the tension in the string and the acceleration of mg. (c) Verify that, for 0 = 7r/2, your answers reduce to the ex
pected results. 7—5 In the ﬁgure, P is a pulley of negligible mass. An external force
F acts on it as indicated. (21) Find the relation between the tensions on the righthand
and lefthand sides of the pulley. Find also the relation between F
and the tensions. (b) What relation among the motions of m, M, and P is pro
vided by the presence of the string? (c) Use the above results and Newton's law as applied to each
block to ﬁnd the accelerations of m, M, and P in terms of m, M, g,
and F. Check 'that the results make sense for various specialized or
simpliﬁed situations. 7—6 A man is raising himself and the platform on which he stands
with a uniform acceleration of 5 m/sec2 by means of the r0peand
pulley arrangement shown. The man has mass 100 kg and the plat
form is 50 kg. Assume that the pulley and rope are massless and
move without friction, and neglect any tilting effects of the platform. Assume g = 10m/sec2. (a) What are the tensions in the ropes A, B, and C? (b) Draw isolation diagrams for the man and the platform and
draw a separate force diagram for each, showing all the forces acting
on them. Label each force and clearly indicate its direction. (C) What is the force of contact exerted on the man by the platform? 7—7 In an equalarm arrangement, a mass 51m) is balanced by the
masses 3/710 and 2mg, which are connected by a string over a pulley
of negligible mass and prevented from moving by the string A (see
the ﬁgure). Analyze what happens if the string A is suddenly severed,
e.g., by means ofa lighted match. 2/11., 7—8 A prisoner in jail decides to escape by sliding to freedom down
a rope provided by an accomplice. He attaches the top end of the
rope to a hook outside his window; the bottom end of the rope hangs
clear of the ground. The rope has a mass of 10 kg, and the prisoner
has a mass of 70 kg. The hook can stand a pull of 600 N without
giving way. If the prisoner‘s window is 15 m above the ground, what
is the least velocity with which he can reach the ground, starting from
rest at the top end of the rope? 7—9 (a) Suppose that a uniform rope of length L, resting on a fric
tionless horizontal surface, is accelerated along the direction of its
length by means ofa force F pulling it at one end. Derive an expression
for the tension 7‘ in the rope as a function of position along its length.
How is the expression for T changed if the rope is accelerated vertically
in a constant gravitational ﬁeld? (b) A mass M is accelerated by the rope in part (21). Assuming
the mass of the rope to be m, calculate the tension for the horizontal
and vertical cases. 7—10 In 1931 F. Kirchner performed an experiment to determine the
chargetomass ratio, e/m, for electrons. An electron gun (see the
ﬁgure) produced a beam of electrons that passed through two “gates,”
each gate consisting of a pair of parallel plates with the upper plates
connected to an alternating voltage source. Electrons could pass B, and C?
and the platform and
lg all the forces acting
ts direction. 1 on the man by the 10 is balanced by the
a string over a pulley
by the string A (see
4 is suddenly severed, 2111., 1g to freedom down the top end of the
ld of the rope hangs
1g, and the prisoner
ll of 600 N without
re the ground, what
ound, starting from ., resting on a fric—
he direction of its
Derive an expression
)n along its length.
:celerated vertically art (a). Assuming
for the horizontal t to determine the
tron gun (see the
:mgh two “gates,”
1 the upper plates
:trons could pass 237 straight through a gate only if the voltage on the upper plate were
momentarily zerc. With the gates separated by a distance I equal to
50.35 cm, and with a gate voltage varying sinusoidally at a frequency
f equal to 2.449 X '107 Hz (1 Hz = 1 cycle/sec), Kirchner found that
electrons could pass completely undeﬂected through both gates when
the initial accelerating voltage (V0) was set at 1735 V. Under these
conditions the ﬂighttime between the gates corresponded to one
halfcycle of the alternating voltage. (a) What was the electron speed, deduced directly from land f? (b) What value of e/m is implied by the data? (c) Were corrections due to special relativity signiﬁcant?
[For Kirchner’s original paper, see Ann. Physik, 8, 975 (1931).] 7—11 A certain loaded car is known to have its center of gravity half
way between the front and rear axles. It is found that the drive wheels
(at the rear) start slipping when the car is driven up a 20° incline.
How far back must the load (weighing a quarter the weight of the
empty car) be shifted for the car to get up a 25° slope '3 (The distance
between the axles is 10ft.) 7—12 A child sleds down a snowy hillside, starting from rest. The
hill has a 15° slope, with a long stretch of level ﬁeld at the foot. The
child starts 50ft up the slope and continues for 100 ft on the level
ﬁeld before coming to a complete stop. Find the coefﬁcient of friction
between the sled and the snow, assuming that it is constant throughout
the ride. Neglect air resistance. 7—13 A beam of electrons from an electron gun passes between two
parallel plates, 3 mm apart and 2 cm long. After leaving the plates
the electrons travel to form a spot on a ﬂuorescent screen 25cm
farther on. It is desired to get the spot to deﬂect vertically through
3 cm when 100 V are applied to the deﬂector plates. What must be
the accelerating voltage V0 on the electron gun? (Show ﬁrst, in
general, that if the linear displacement caused by the deﬂector plates
can be neglected, the required voltage is given by V0 = V(/D/2 Yd),
where Y is the linear displacement of the spot on the screen. The
notation is that used on p. 197.) 7—14 A ball of mass m is attached to one end of a string of length I.
It is known that the string will break if pulled with a force equal to
nine times the weight of the ball. The ball, supported by a frictionless
table, is made to travel a horizontal circular path, the other end of the
string being attached to a ﬁxed point 0. What is the largest number
of revolutions per unit time that the mass can make without breaking the string ? 7—15 A mass of 100g is attached to one end ofa very light rigid rod
20 cm long. The other end of the rod is attached to the shaft of a
motor so that the rod and the mass are caused to rotate in a vertical
circle with a constant angular velocity of 7 rad/sec. (a) Draw a force diagram showing all the forces acting on the
mass for an arbitrary angle 00f the rod'to the downward vertical. (b) What are the magnitude and the direction of the force exerted
by the rod on the mass when'the rod points in a horizontal direction,
i.e., at 0 = 90°? 7—16 You are ﬂying along in your Sopwith Camel at 60 mph and
2000ft altitude in the vicinity of Saint Michel when suddenly you
notice that the Red Baron is just 300 ft behind you ﬂying at 90 mph.
Recalling from captured medical data that the Red Baron can with
stand only 4 g's of acceleration before blacking but, whereas you can
withstand 5 g’s, you decide on the following plan. You dive straight
down at full power, then level out by ﬂying in a circular arc that comes
out horizontally just above the ground. Assume that your speed is
constant after you start to pull out and that the acceleration you
experience in the arc is 5 g's. Since you know that the Red Baron
will follow you, you are assured he will black out and crash. Assuming
that both planes dive with 2 g’s acceleration from the same initial
point (but with initial speeds given above), to what altitude must you
descend so that the Red Baron, in trying to follow your subsequent
arc, must either crash or black out? Assuming that the Red Baron
is a poor shot and must get within 100 ft of your plane to shoot you
down, will your plan succeed? After starting down you recall reading
that the wings of your plane will fall off if you exceed 300 mph. ls ‘
your plan sound in view of this limitation on your plane? 7—17 A curve of 300 m radius on a level road is banked for a speed
of 25 m/sec (z 55 mph) so that the force exerted on a car by the road
is normal to the surface at this speed. (a) What is the angle of bank? (b) The friction between tires and road can provide a maximum
tangential force equal to 0.4 of the force normal to the road surface.
What is the highest speed at which the car can take this curve without
skidding? 7—18 A large mass M hangs (stationary) at the end of a string that l of a string of length I.
:d with a force equal to
pported by a frictionless
ath, the other end of the
at is the largest number
make without breaking of a very light rigid rod
ached to the shaft of a
:d to rotate in a vertical
l/sec. the forces acting on the
: downward vertical.
:tion of the force exerted
n a horizontal direction, Camel at 60 mph and
be] when suddenly you
.d you ﬂying at 90 mph.
Ie Red Baron can with
ig out, whereas you can
plan. You dive straight
1 circular arc that comes
.ume that your speed is
at the acceleration you
ow that the Red Baron
ut and crash. Assuming
I from the same initial
what altitude must you
follow your subsequent
ing that the Red Baron
(our plane to shoot you down you recall reading ou exceed 300 mph. Is
your plane? 1 is banked for a speed
ted on a car by the road :an provide a maximum
nal to the road surface.
I take this curve without the end of a string that 239 passes through a smooth tube to a small mass m that whirls around
in a circular path of radius [sin 0, where l is the length of the string
from m to the top end of the tube (see the ﬁgure). Write down the
dynamical equations that apply to each mass and show that m must
complete one orbit in a time of 21r(lm/gM)”2. Consider whether
there is any restriction on the value of the angle 0 in this motion. 7—19 A model rocket rests on a frictionless horizontal surface and is
joined by a string of length l to a ﬁxed point so that the rocket moves
in a horizontal circular path of radius [. The string will break if its
tension exceeds a value T. The rocket engine provides a thrust F of
constant magnitude along the rocket’s direction of motion. The
rocket has a mass m that does not decrease appreciably with time. (a) Starting from rest at t = 0, at what later time I; is the rocket
traveling so fast that the string breaks? Ignore air resistance. (b) What'Was 'the magnitude of the rocket’s instantaneous net
acceleration at time t1/2? Obtain the answer in terms of F, T, and m. (c) What distance does the rocket travel between the time ti
when the string breaks and the time 2!. ? The rocket engine continues to operate after the string breaks. 7—20 It has been suggested thatgthe biggest nuclear accelerator we
are likely to make will be an evacuated pipe running around the
earth’s equator. The strength of the earth‘s magnetic ﬁeld at the
equator is about 0.3 G or 3 x .105 MKS units (N—sec/Cm). With
what speed would an atom of lead (at. wt. 207), singly ionized (i.e.,
carrying one elementary charge), have to move around such an orbit
so that the magnetic force provided the correct centripetal acceleration?
(e = 1.6 X 10—19 C.) Through what voltage would a singly ionized
lead atom have to be accelerated to give it this correct orbital speed? 7—21 A trick cyclist rides his bike around a “wall of death" in the
form of a vertical cylinder (see the ﬁgure). The maximum frictional
force parallel to the surface of the cylinder is equal to a fraction 1:. of
the normal force exerted on the bike by the wall. (a) At what speed must the cyclist go to avoid slipping down? go
'—————r
I
I /,
Axis of '
cylinder $13
3 .L (b) At what angle (go) to the horizontal must he be inclined? (c) If u z 0.6 (typical of rubber tires on dry roads) and the
radius of the cylinder is 5 m, at what minimum speed must the cyclist
ride, and what angle does he make with the horizontal? 7—22 The following expression gives the resistive force exerted on a
sphere of radius r moving at speed 0 through air. It is valid over a
very wide range of speeds. W) = 3.1 x 10% + 0.87%2 where R is in N, r in m, and u in mf/sec. Consider water drops falling '
under their own weight and reaching a terminal speed. (a) For what range of values of small r is the terminal speed
determined within 1% by the ﬁrst term alone in the expression for
R(u)? (b) For what range of values. of larger r is the terminal speed
determined within 1% by the second term alone? (c) Calculate the terminal speed of a raindrop of radius 2 mm.
If there were no air resistance, from what height would it fall from
rest before reaching this speed? 7—23 An experiment is performed with the Millikan oil—drop ap
paratus. The plates are 8 mm apart. The experiment is done with
oil droplets of density 896 kg/m“. The droplets are timed between
two horizontal lines that are 2.58 mm apart. With the plates un~
charged, a droplet is observed to take 23.6 sec to fall from one line to
the other. When the upper plate is made 1100 V positive with respect
to the lower, the droplet rises and takes 22.0 sec to cover the same
distance. Assume that the resistive force is 3.1 X 10T4rv (MKS units). (a) What is the radius of the droplet? (b) What is its net charge, measured as a number of elementary
charges? (e = 1.6 x 10190) (c) What voltage would hold the droplet stationary? [Use the
precise value of the charge deduced in part (b)]. 7—24 Two solid plastic spheres of the same material but of different
radii, R and 2R, are used in a Millikan experiment. The spheres carry
equal charges q. The larger sphere is observed to reach terminal
speeds as follows: (1) plates uncharged: terminal speed no (down
ward), and (2) plates charged: terminal speed = u; (upward). As ll 7suming that the resistive force on a sphere of radius r at speed 0 is c150, ﬁnd, in terms of Do and ur, the corresponding terminal speeds
for the smaller sphere. 7—25 Analyze in retrospect the legendary Galilean experiment that
took place at the leaning tower of Pisa. Imagine such an experiment
done with two iron spheres, of radii 2 and 10 cm, respectively, dropped
simultaneously from a height of about 15 m. Make calculations to Lontal must he be inclined? r tires on dry roads) and the linimum speed must the cyclist
the horizontal? te resistive force exerted on a
hrough air. It is valid over a Consider water drops falling
erminal speed.
small r is the terminal speed
n alone in the expression for .arger r is the terminal speed
rn alone? f a raindrop of radius 2 mm.
fiat height would it fall from "t the Millikan oildrop ap
The experiment is done with
: droplets are timed between
apart. With the plates un
.6 sec to fall from one line to
: 1100 V positive with respect
5 22.0 sec to cover the same
s 3.1 x 10—4ru (MKS units).
:t? d as a number of elementary troplet stationary? [Use the rt (b)]. me material but of different
periment. The spheres carry
observed to reach terminal
terminal speed = D() (down
speed = U} (upward). As
:re of radius r at Speed 0 is
rresponding terminal speeds y Galilean experiment that
Imagine such an experiment
10 cm, respectively, dropped
Sm. Make calculations to 11p: 241 determine, approximately, the difference in the times at which they
hit the ground. Do you think this could be detected without special
measuring devices? (Density of iron z 7500 kg/m“.) 7—26 Estimate the terminal Speed of fall (in air) of an airtilled toy
balloon, with a diameter of 30 cm and a mass (not counting the air
inside) of about 0.5 g. About how long would it take for the balloon
to come to within a few percentof this terminal speed? Try making
some real observations of balloons inﬂated to different sizes. 7—27 A spring that obeys Hooke’s law in both extension and com pression is extended by 10cm when a mass of 2 kg is hung from it. (a) What is the spring constant k? (b) The spring and the 2kg mass are placed on a smooth table.
The mass is pulled so as to extend the spring by 5 cm and is then
released at I = 0. What is the equation of the ensuing motion ? (c) If, instead of being released from rest, the mass were started
off at x = 5 cm with a speed of 1 m/sec in the direction of increasing
x, what would be the equation of motion? 7—28 When the mass is doubled in diagram (a), the end of the spring
descends an additional distance Ii. What is the frequency of oscillation
for the arrangement in diagram (b)? All individual springs shown
are identical. 7—29 Any object, partially or wholly submerged in a ﬂuid, is buoyed
up by a force equal to the weight of the displaced ﬂuid. A uniform
cylinder of density p and length l is ﬂoating with its axis vertical, in a
ﬂuid of density pm. What is the frequency of smallamplitude vertical
oscillations of the cylinder ? 7—30 (a) A small bead of mass m is attached to the midpoint of a
string (itself of negligible mass). The string is of length L and is under
constant tension T. Find the frequency of the SHM that the mass
describes when given a slight transverse displacement. (b) Find the frequency in the case where the mass is attached
at a distance D from one end instead of the midpoint. 7~3l A block rests on a tabletop that is undergoingsimple harmonic
motion of amplitude A and period T. (a) If the oscillation is vertical, what is the maximum value of A
that will allow the block to remain always in contact with the table? (b) If the oscillatior. is horizontal, and the coefficient of friction between block and tabletop is [1, what is the maximum value of A that
will allow the block to remain on the surface without slipping? 7—32 The springs of a car of mass 1200 kg give it a period when empty
of 0.5 sec for small vertical oscillations. (at) How far does the car sink down when a driver and three
passengers, each of mass 75 kg, get into the car? 4352;; q.. (b) The car with its passengers is traveling along a horizontal
road when it suddenly runs onto a piece of new road surface, raised
2in. above the old surface. Assume that this suddenly raises the
wheels and the bottom ends of the springs through 2 in. before the
body of the car begins to move upward. In the ensuing rebound, are
the passengers thrown clear of their seats? Consider the maximum
acceleration of the resulting simple harmonic motion. ed not in terms
I "curvature of
:t and mathe
:ometries.
‘avitation gives
)unds for pre~
than practical.
ions there is a
This is in the
try. The phe
s‘ distinctly el—
.ses in its own
erent direction
this precession
'itury) can be
3 other planets
rere remains a
ldS of are per
n theory—for
ide Mercury's
other facts of
1’s theory, on
c parameters,
exactly with
:nce ofa very
nce than the
.y in which a
)recess is dis
ttions of the
the inverse
:d his theory,
r led to false
)ry, however.
ing term was
of the planet
try, with its irth is actually
JOUS change in
.luinoxes — see PROBLEMS 8—1 Given a knowledge of Kepler’s third law as it applies to the
solar system, together with the knowledge that the disk of the sun
subtends an angle of about §° at the earth, deduce the period of a
hypothetical planet in a circular orbit that skims the surface of the sun. 8—2 It is well known that the gap between the four inner planets and
the ﬁve outer planets is occupied by the asteroid belt instead of by a
tenth planet. This asteroid belt extends over a range of orbital radii
from about 2.5 to 3.0 AU. Calculate the corresponding range of
periods, expressed as multiples of the earth’s year. 8—3 It is proposed to put up an earth satellite in a circular orbit with
a period of 2 hr.
(a) How high above the earth's surface would it have to be?
(b) If its orbit were in the plane of the earth’s equator and in
the same direction as the earth’s rotation, for how long would it be
continuously visible from a given place on the equator at sea level? 8—4 A satellite is to be placed in synchronous circular orbit around
the planet Jupiter to study the famous “red spot” in Jupiter’s lower
atmosphere. How high above the surface of Jupiter will the satellite
be? The rotation period of Jupiter is 9.9 hr, its mass M; is about
320 times the earth’s mass, and its radius RJ is about 11 times that
of the earth. You may ﬁnd it convenient to calculate ﬁrst the gravita
tional acceleration g; at Jupiter’s surface as a multiple of g, using the
above values of M] and RJ, and then use a relationship analogous to
that developed in the text for earth satellites [Eq. (8—16) or (847)]. 8—5 A satellite is to be placed in a circular orbit 10km above the
surface of the moon. What must be its orbital speed and what is the
period of revolution? 8—6 A satellite is to be placed in synchronous circular earth orbit.
The satellite’s power supply is expected to last 10 years. If the maxi
mum aceeptable eastward or westward drift in the longitude of the
satellite durin its lifetime is 10°, what is the margin of error in the
radius of its 0%3it? 8—7 The sprihgs found in retractable ballpoint pens have a relaxed
length of about 3cm and a spring constant of perhaps 0.05 N/mm.
Imagine that two lead spheres, each of 10,000 kg, are placed on a
frictionless surface so that one of these springs just ﬁts, in its un
compressed state, between their nearest points. (a) How much would the spring be compressed by the mutual
gravitational attraction of the two spheres? The density of lead is
about 11,000 kg/m”. I . ,.r’\_/_Z? (b) Let the system be rotated in the horizontal plane. At what
frequency of rotation would the presence of the spring become ir
relevant to the separation of the masses? 8—8 During the eighteenth century, an ingenious attempt to ﬁnd the
mass of the earth was made by the British Astronomer Royal, Nevil
Maskelynef He observed the extent to which a plumbline was pulled
out of true by the gravitational attraction of a mountain. The ﬁgure
illustrates the principle of the method. The change of direction of the
plumbline was measured between the two sides of the mountain. (This was done by sighting on stars.) After allowance had been made
for the change in direction of the local vertical because of the curvature
of the earth, the residual angular difference a was given by 2FM/FE,
where :bFM is the horizontal force on the plumbbob due to the moun
tain, and FE = GMEm/Ryz. (m is the mass of the plumb~bob.) The value of a is about 10 seconds of arc for measurements on
opposite sides of the base ofa mountain about 2000 m high. Suppose
that the mountain can be approximated by a cone of rock (of density
2.5 times that of water) whose radius at the base is equal to the height
and whose mass can be considered to be concentrated at the center
of the base. Deduce an approximate value of the earth’s mass from
these ﬁgures. (The true answer is about 6 X 1024 kg.) Compare the
gravitational deflection a to the change of direction associated with
the earth‘s curvature in this experiment. 8—9 Imagine that in a certain region of the ocean ﬂoor there is a
roughly cone—shaped mound of granite 250 m high and 1 km in diam
eter. The surrounding ﬂoor is relatively ﬂat for tens of kilometers in
all directions. The ocean depth in the region is 5 km and the density
of the granite is 3000 kg/m”. Could the mound’s presence be de
tected by a surface vessel equipped with a gravity meter that can
detect a change in g of 0.1 mgal?. (Hint: Assume that the ﬁeld produced by the mound at the
surface can be approximated by the ﬁeld of a mass point of the same
total mass located at the level of the surrounding ﬂoor. Note that in
calculating the change in g you must keep in mind that the mound
has displaced its own volume of water. The density of water, even at such depths, can be taken as about equal to its surface value of about
1000 kg/m“. horizontal plane. At what
of the spring become ir ;enious attempt to ﬁnd the
Astronomer Royal, Nevil
ch a plumbline was pulled
)f a mountain. The ﬁgure
change of direction of the
0 sides of the mountain. 0/2
\; ——>l allowance had been made
al because of the curvature
a was given by ZFM/Fg,
imbbob due to the moun
; of the plumb—bob.)
’ arc for measurements on
)ut 2000 m high. Suppose
a cone of rock (of density
base is equal to the height
:oncentrated at the center
of the earth’s mass from
< 1024 kg.) Compare the
direction associated with he ocean ﬂoor there is a
11 high and 1 km in diam
L for tens of kilometers in
n is 5 km and the density
mound’s presence be de
1 gravity meter that can :d by the mound at the
a mass point of the same
riding ﬂoor. Note that in
in mind that the mound
density of water, even at
its surface value of about 8~10 Show that the period of a particle that moves in a circular orbit
close to the surface of a sphere depends only on G and the mean
density of the sphere. Deduce what this period would be for any
sphere having a mean density equal to the density of water. (Jupiter
almost corresponds to this case.) 811 Calculate the mean density of the sun, given a knowledge of G,
the length of the earth’s year, and the fact that the sun’s diameter
subtends an angle of about 0.55° at the earth. 812 An astronaut who can lift 50 kg on earth is exploring a planetoid
(roughly spherical) of 10 km diameter and density 3500 kg/m3. (a) How large a rock can he pick up from the planetoid’s surface,
assuming that he ﬁnds a wellplaced handle? (b) The astronaut observes a rock falling from a cliff. The
rock’s radius is only 1 m and as it approaches the surface its velocity
is l m/sec. Should he try to catch it? (This is obviously a fanciful
problem. One would not expect a planetoid to have cliﬂ‘s or loose
rocks, even if an astronaut were to get there in the ﬁrst place.) 8—13 It is pointed out in the text that a person can properly be termed
“weightless” when he is in a satellite circling the earth. The moon is a
satellite, yet it is noted in many discussions that we would weigh % of
our normal weight there. Is there a contradiction here? 8—14 A dedicated scientist performs the following experiment. After
installing a huge spring at the bottom of a 20storyhigh elevator shaft,
he takes the elevator to the top, positions himself on a bathroom scale
inside the airtight car with a stopwatch and with pad and pencil to
record the scale reading, and directs an assistant to cut the car’s
support cable at t = O. Presuming that the scientist survives the
ﬁrst encounter with the spring, sketch a graph of his measured weight
versus time from t = 0 up to the beginning of the second bounce.
(Note: Twenty stories is ample distance for the elevator to aCQuire
terminal velocity.) 8—15 A planet of mass M and a single satellite of mass M/10 revolve
in circular orbits about their stationary center of mass, being held
together by their gravitational attraction. The distance between their
centers is D. (a) What is the period of this orbital motion? (b) What fraction of the total kinetic energy resides in the
satellite?
(Ignore any spin of planet and satellite about their own axes.) 816 We have considered the problem of the moon’s orbit around
the earth as if the earth’s center represented a ﬁxed point about which
the motion takes place. In fact, however, the earth and the moon
revolve about their common center of mass. ' (a) Calculate the position of the center of mass, given that the
earth’s mass is 81 times that of the moon and that the distance be—
tween their centers is 60 earth radii. (b) How much longer would the month be if the moon were of
negligible mass compared to the earth '? 8—17 The sun appears to be moving at a speed of about 250 km/sec
in a circular orbit of radius about 25,000 lightyears around the center
of our galaxy. (One lightyear 2 10H5 m.) The earth takes 1 year to
describe an almost circular orbit of radius about 1.5 X 10” m around
the sun. What do these facts imply about the total mass responsible
for keeping the sun in its orbit? Obtain this mass as a multiple of the
sun’s mass M. (Note that you do not need to introduce the numerical
value ’of G torobtain the answer.) 8—18 (A good problem for discussion.) In 1747 Georges Louis Lesage
explained the inversesquare law of gravitation by postulating that
vast numbers of invisible particles were ﬂying through space in all
directions at high speeds. Objects like the sun and planets block these
particles, leading to a shadowing effect that has the same quantitative
result as a gravitational attraction. Consider the arguments for and
against this theory. (Suggestion: First consider a theory in which opaque objects
block the particles completely. This proposal is fairly easy to refute.
Next consider a theory in which the attenuation of the particles by
objects is incomplete or even very small. This theory is much harder
to dismiss.) 8—19 The continuous output of energy by the sun corresponds (through
Einstein‘s relation E = Meg) to a steady decrease in its mass M, at
the rate of about 4 X 106 tons/sec. This implies a progressive in
crease in the orbital periods of the planets, because for an orbit of a
given radius we have T~ M‘“2 [cf Eq. (8—23)]. A precise analysis of the effect must take into account the fact
that as M decreases the orbital radius itself increases the planets
gradually spiral away from the sun. However, one can get an order
ofmagnitude estimate of the size of the effect, albeit a little bit on the
low side, by assuming that r remains constant. (See Problem 13~21
for a more rigorous treatment.) Using the simplifying assumption of constant r, estimate the
approximate increase in the length of the year resulting from the sun‘s decrease in mass over the time span of accurate astronomical ob
servations—about 2500 years. 8—20 It is mentioned at the end of the chapter how Einstein’s theory
of gravitationreads to a small correction term on top of the basic
Newtonian to tee of gravitation. For a planet of mass m, traveling at
speed 1: in a e rcular orbit of radius r, the gravitational force becomes of mass, given that the
d that the distance be be if the moon were of i of about 250 km/sec
'ears around the center
e earth takes 1 year to
1.5 X 10" m around
total mass responsible
lSS as a multiple of the
(troduce the numerical Georges Louis Lesage
1 by postulating that
through space in all
1d planets block these
the same quantitative
Ie arguments for and 'hich opaque objects
fairly easy to refute.
1 of the particles by
Ieory is much harder :orresponds (through
se in its mass M, at
es a progressive in
ise for an orbit of a no account the fact
:reases———the planets
e can get an order
:it a little bit on the
:See Problem 13—21 int r, estimate the
[ting from the sun’s
: astronomical ob w Einstein’s theory
n top of the basic
lass m, traveling at
mal force becomes 305 ark / in effect the following: 2
p: 61"?"(1 +693)
r C” where c is the speed of light. (Correction terms of the order of 03/02
are typical of relativistic effects.) (a) Show that, if the period under a pure Newtonian force
GMm/r2 is denoted by To, the modiﬁed period T is given approxi
mately by 0 9
1211"?"
T_ To<1 — 237,02) (Treat the relativistic correction as representing, in effect, a small
fractional increase in the value of G, and use the value of u corre
sponding to the Newtonian orbit.) (b) Hence show that, in each revolution, a planet in a circular
orbit would travel through an angle greater by about 241r3r3/c3Tu‘3
than under the pure Newtonian force, and that this is also expressible
as 67rGM/cgr, where M is the mass of the sun. (c) Apply these results to the planet Mercury and verify that
the accumulated advance in angle amounts to about 43 seconds of
are per century. This corresponds to what is called the precession
of its orbit. ...
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 Fall '07
 Harris
 Force, Friction, Mass

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