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Unformatted text preview: nated by turbulence, however small the ratio B/A may be. The
same consideration guarantees that at sufﬁciently low speeds the
resistance will be dominated by the viscous term, directly propor
tional to u [see Fig. 5—9(b)]. CONCLUDING REMARKS PROBLEMS In this chapter we have given a brief account of the three major
types of physical interactions and have indicated the general
areas in which they are dominant. To recapitulate: Nuclear
forces are signiﬁcant only for nuclear distances, the gravitational
force is important only if objects of astronomical scale are in
volved, and nearly everything else ultimately depends on electro
magnetic interactions. The study of physics is essentially the
attempt to understand these interactions and all their con
sequences. In mechanics we have, for the most part, the more
modest goal of taking the forces as given and considering various
dynamical situations in which they enter. We shall, however, be
discussing two classic cases—gravitation and alphaparticle
scattering—in which Newtonian mechanics provided the key to
the basic laws of force. The present chapter has provided a kind
of preview, because it summarizes the state of our current knowl
edge without entering into any detailed discussion of how we
have come to know it. The real work lies ahead! 5—] At what distance from the earth, on the line from the earth to
the sun, do the gravitational forces exerted on a mass by the earth
and the sun become equal and opposite? Compare the result with
the radius of the moon’s orbit around the earth. 5—2 By what angle, in seconds of arc, will a plumbline be pulled out
of its normal vertical direction by the gravitational attraction of a
10ton truck that parks 20 ft away? Do you think that this effect
could be detected? 53 In a Cavendish—type apparatus (see the ﬁgure) the large spheres
are each 2 kg, the small spheres each 20 g. The length of the'arm con
necting the small spheres is 20 cm, and the distance between the centers
of a small sphere and the'big sphere close to it is 5 cm. The torsion
constant of the suspending ﬁber is 5 X 10‘8 mN/rad. The angular
deﬂection of the suspended system is deduced from the displacement ver small the ratio B/A may be. The
tees that at sufﬁciently low speeds the
i by the viscous term, directly propor . en a brief account of the three major
ons and have indicated the general
ominant. To recapitulate: Nuclear
)r nuclear distances, the gravitational
abjects of astronomical scale are in
I g else ultimately depends on electro—
: study of physics is essentially the
:se interactions and all their con
e have, for the most part, the more
rces as given and considering various
:h they enter. We shall, however, be
res—gravitation and alphaparticle
nian mechanics provided the key to
present chapter has provided a kind
irizes the state of our current knowl—
any detailed discussion of how we
real work lies ahead! 1e earth, on the line from the earth to
'orces exerted on a mass by the earth
1 opposite? Compare the result with
tround the earth. . of arc, will a plumbline be pulled out
I by the gravitational attraction of a
way? Do you think that this effect ratus (see the ﬁgure) the large spheres
each 20 g. The length of the arm con
n, and the distance between the centers
phere close to it is 5 cm. The torsion
' is 5 X 10‘8 mN/rad. The angular
em is deduced from the displacement an“; aw.
"Am 155 Mk .Ztli'i’lii L \20 g of a reﬂected Spot of light on a scale 5 m away. (Remember that the
change in direction of the reﬂected light is twice the angle through
which the mirror is turned.) It is observed that when the large spheres are moved from their initial
positions to equivalent positions on the other side (dashed lines) the
mean position of the spot of light is shifted by 8 cm. (a) Deduce the value of G according to these data, ignoring the
effect on each small sphere of the force due to the more distant of the
larger Spheres. _ (b) Estimate the percentage correction on the result of (a) re
quired to allow for the effect of the more distant spheres. 5—4 The original Cavendish experiment was done with a largesize
apparatus, as is natural if one wants to make the gravitational forces
and torques as big as possible. However, this requires a strong, stiff
wire to support the suspended masses. Much later (1895) C. V. Boys
made a miniature apparatus, using thin ﬁbers of fused quartz for the
suspensions. It is an interesting exercise to see how the attainable
sensitivity of the apparatus depends on its size. Imagine two versions
of the Cavendish apparatus, A and B, both using solid lead spheres,
in which the radii and separations of all the masses in B, together with
the length of the torsion ﬁber, are scaled down by a certain factor L
with respect to A. We then design for maximum sensitivity in each
apparatus by using the thinnest possible torsion ﬁber that will take
the weight of the suspended masses without breaking. Now for a
torsion ﬁber of given material and of circular cross section, the maxi
mum supportable load is proportional to (P, where d is its diameter,
and its torsion constant is proportional to 114/], where l is its length.
Using this information, compare the maximum angular deﬂections
obtainable with the two different sizes of apparatus. (Remember, the
lengths of the torsion ﬁbers also differ by the scaling factor L.) 55 The radius of the hydrogen atom according to the original Bohr theory is 0.5 A. (a) What is the Coulomb force between the proton and the
electron at this distance? What is the gravitational force? (b) How far apart must the proton and electron be for the
Coulomb force to be equal to the value that the gravitational attrac
tion has at 0.5 A? What familiar astronomical distance is this com parable to? 5—6 Suppose electrons could be added to earth and moon until the
Coulomb repulsion thus developed was of just the size to balance the
gravitational attraction. What would be the smallest total mass of
electrons that would achieve this? 5—7 For a person living at 45° latitude, what is the approximate
fractional difference, during the day, between the maximum and
minimum gravitational forces due to the moon—the change resulting
from the fact that the earth’s rotation causes the person’s distance
from the moon to vary? What is a manifestation of this kind of force effect in nature? 58 You know that the Coulomb force and the gravitational force
both obey an inversesquare law. Suppose that it were put to you
that the origin of the gravitational force is a minute difference be
tween the natural unit of positive charge, as carried by a proton, and
the natural unit of negative charge, as carried by an electron. Thus
“neutral” matter, containing equal numbers of protons and electrons, would not be quite neutral in fact. (a) What fractional difference between the positive and negative
elementary charges would lead to “gravitational” forces of the right
magnitude between lumps of ordinary “neutral” matter? How could
such a difference be looked for by laboratory experiments? (b) Is the theory tenable? 5—9 (a) The text (p. 148) quotes a value of the nuclear force for two
nucleons close together but also suggests that to describe the nuclear
interactions in terms of forces is not very practical. Can you suggest
any way in which a nuclear force as such could be measured? (b) According to one of the earliest and simplest theoretical
descriptions of the nuclear interaction (by H. Yukawa) the force of
attraction between two nucleons at large separation would be given by m) = _ A I" where the distance ro is about lO—lsm and the constant A is about
10‘11 Nm. At about what separation between a proton and a neutron
would the nuclear force be equal to the gravitational force between these two particles ? 510 Can you think of any systems or processes in which gravita— mb force between the proton and the
at is the gravitational force? st the proton and electron be for the
> the value that the gravitational attrac.
iiliar astronomical distance is this com be added to earth and moon until the oped was of just the size to balance the
it would be the smallest total mass of
his? 45° latitude, what is the approximate
the day, between the maximum and
due to the moon—the change resulting
i rotation causes the person’s distance
L is a manifestation of this kind of force lomb force and the gravitational force
aw. Suppose that_it were put to you
tional force is a minute difference be
ive charge, as carried by a proton, and
large, as carried by an electron. Thus
qual numbers of protons and electrons,
act. ‘ence between the positive and negative
! to “gravitational” forces of the right
rdinary “neutral” matter? How could
by laboratory experiments ? l lets a value of the nuclear force for two
0 suggests that to describe the nuclear
.s not very practical. Can you suggest
:e as such could be measured ‘1 the earliest and simplest theoretical
traction (by H. Yukawa) the force of
‘. at large separation would be given by 10*15 m and the constant A is about
.ration between a proton and a neutron
ml to the gravitational force between stems or processes in which gravita mus. a tional, electromagnetic, and nuclear forces all play an important role? 5—11 As mentioned in the text, the attractive (longrange) part of the
force between neutral molecules varies as l/r7. For a number of
molecules, the order of magnitude of this van der Waals force is well
represented by the equation mm x —1o‘7“/r7
where F yw is in newtons and r in meters. Compare the magnitude of
the van der Waals force with the Coulomb force between two ele
mentary charges [Eq. (5—2), with q, = q2 = e = 1.6 x 10"” C]:
(a) For r = 4 A. (This is a distance about equal to the diameter
of a molecule of oxygen or nitrogen and hence barely exceeding the
closest approach of the centers of two such molecules in a collision.)
(b) For a value of r correSponding to the mean distance between
molecules in a gas at STP. ' 5—12 One of the seemingly weakest forms of contact force is the
surface tension of a liquid ﬁlm. One of the seemingly strongest is the
tensile force of a stretched metal wire. However, when expressed in
terms of a force between individual atoms in contact, they do not
look so different. Use the following data to evaluate them in these
terms: (a) If a water ﬁlm is formed between a rectangular wire frame,
3 in. wide, and a freely sliding transverse wire (see the ﬁgure), it takes
the weight of about 1 g to prevent the ﬁlm from contracting. This
contractile force can be ascribed to the contact of the atoms lying
within a monomolecular layer along each side of the ﬁlm. Supposing
that the molecules are 3 A across and closely packed, calculate the
force per molecule. (b) A copper wire of 0.025in. diameter was found to break
when a weight of about 10 kg was hung from its lower end. First,
calculate this breaking force in tons per square inch. If the fracture is
assumed to involve the rupturing of the contacts between the atoms
on the upper and lower sides of a horizontal section right across the
wire, calculate the force per atom, assuming an atomic diameter of
about 3 A. 513 A timehonored trick method for approximately locating the
midpoint of a long uniform rod or bar is to support it horizontally
at any two arbitrary points on one’s index ﬁngers and then move the
ﬁngers together. (Of course, just ﬁnding its balance point on one
ﬁnger alone works very well, too!) Explain the workings of the trick
method, using your knowledge of the basic principles of static equi—
librium and a property of frictional forces: that they have a maximum
value equal to a constant [I (the coefﬁcient of friction) times the com
ponent of force normal to the surface of contact between two objects. 5—14 (a) A string in tension is in contact with a circular rod (radius r)
over an arc sintending a small angle A0 (see the ﬁgure). Show that
the force with which the string presses radially inward on the pulley
(and hence the normal force AN With which the pulley pushes on the
string) is equal to TAB. I (b) Hence show that the normal force per unit length is equal
to T/r. This is a sort of pressure which, for a given value of T, gets
bigger as r decreases. (This helps to explain why, when a string is
tightly tied around a package, it cuts into the package most deeply
as it passes around corners, where r is least.) (c) If the contact is not perfectly smooth, the values of the
tension at the two ends of the arc can differ by a certain amount AT
before slipping occurs. The value of AT is equal to uAN, where u is
the coefﬁcient of friction between string and rod. Deduce from this
the exponential relation m9) = Toe” where To is the tension applied at one end of an arbitrary are (0) of
string and T(0) is the tension at~the other end. (d) The above result expresses the possibility of withstanding a
large tension T in a rope by wrapping the rope around a cylinder, a
phenomenon that has been exploited since time immemorial by sailors.
Suppose, for example, that the value ofp in the contact between a rope
and a bollard on a dock is 0.2. For To = 100 lb, calculate the values
of T corresponding to one, two, three and four complete turns of rope
around the bollard.
(It is interesting to note that T is proportional to To. This allows
sailors to produce a big pull or not, at will, by having a rope passing
around a continuously rotating motordriven drum. The arrangement
can be described as a force ampliﬁer.) 5—15 In a very delicate torsionbalance experiment, such as the
Cavendish experiment, the stray forces due to the ﬂuid friction of slow
air currents pushing on the suspended system may be quite signiﬁcant.
To make this quantitative, consider the gravity torsionbalance ex
periment described in Problem 5—3. For suspended spheres of the
size stated (r z 0.8 cm), the force due to a ﬂow of air of speed 0 is
given approximately by the formula [Eg. (5—4)] R (newtons) = 2.5 X 10—6!) + 5 X 10—5172 where v is in m/sec. Calculate the value of u that would cause a force
due to air currents that equaled the gravitational force exerted on the
sphere in this experiment (i.e., the force exerted on a 20g sphere by a
2kg sphere with their centers 5 cm apart). (Hint: Do not bother to )ntact With 8 Circular rod (radius r) i V I solve a quadratic equation for v. Just ﬁnd the values of v for which gle A0 (see the figure) ShOW that 5' the contributions to R, taken separately, would equal the gravitational
35565 radially inward 0“ the puuey .v j   force. The smaller of the two values ofv so obtained is clearly already
'th WhiCh the puney pUShes 0“ the  enough to spoil the experiment.) mal force per unit length is equal
which, for a given value of T, gets
to explain why, when a string is
uts into the package most deeply
' is least.) rfectly smooth, the values of the
:an differ by a certain amount AT
)f AT is equal to uAN, where [.l. is
tring and rod. Deduce from this one end of an arbitrary arc (0) of
other end.
; the possibility of withstanding a
ing the rope around a cylinder, a
: since time immemorial by sailors.
of u in the contact between a rope
To = 100 lb, calculate the values
e and four complete turns of rope proportional to To. This allows
at will, by having a rope passing
)r—driven drum. The arrangement ~) alance experiment, such as the
es due to the ﬂuid friction of slow
d system may be quite signiﬁcant.
r the gravity torsionbalance ex For suspended spheres of the
lue to a ﬂow of air of speed 0 is Big. (54)] 5 x 10502 alue of u that would cause a force
gravitational force exerted on the
rce exerted on a 20g sphere by a
apart). (Hint: Do not bother to 159 §_32'<>i‘3§em3 ,ments gathered themselves together into
n jumped up onto the table, this would
‘nature doesn’t act like that. Yet a
ividual atomic encounters at every stage
>e perfectly timereversible.
vith a puzzle: Newton’s law implies that
cal behavior of an individual particle is
when one takes a system of very large
parently the behavior ceases to be time
n of this mystery is found in the detailed
.nyparticle systems—the subject known
As long as we are dealing, as we shall
of only a few particles, the problems
:rsal do not arise and we shall not con :ome apparent to you during the course
: foundation of classical mechanics, as
: second law, is a complex and in many
The precise content of the law is still a
ly three centuries after Newton stated
n a ﬁne discussion entitled “The Origin
; Laws of Motion,” one author (Brian
Newton’s second law of motion? What
llS law? Is it a deﬁnition of force? Of
cal preposition relating force, mass, and
gues that it is something of all of these: an’s second law is actually used. In some
tably true that Newton’s second law is used
'orce. How else, for example, can we mea
gravitational forces? But it is also un
at Newton’s second law is sometimes used
nass. Consider, for example, the use of the
And in yet other ﬁelds, where force, mass,
: all easily and independently measurable,
v of motion functions as an empirical cor
se three quantities. Consider, for example, :ssays, Beyond the Edge of Certain! y (Robert G.
Englewood Cliffs, N.J., 1965. PROBLEMS 181 the application of Newton’s second law in ballistics and rock
etry . .. To suppose that Newton’s second law of motion, or
any law for that matter, must have a unique role that we can
describe generally and call the logical status is an unfounded
and unjustiﬁable supposition. Since force and mass are both abstract concepts and not
objective realities, we might conceive of a description of nature
in which we dispensed with both of them. But, as one physicist
(D. H. Frisch) has remarked, “Whatever we think about ultimate
reality it is convenient to follow Newton and split the description
of our observations into ‘forces,’ which are what make masses
accelerate, and ‘masses,’ which are what forces make accelerate.
This would be just tautology were it not that the observed phe
nomena can best be classiﬁed as the result of dlﬂerent forces
acting on the same’set of masses.” Ellis spells out this same idea
in more detail: Now there are, in fact, many and various procedures by which
the magnitudes of the individual forces acting on a given system
may be determined—electrostatic forces by charge and distance
measurements, elastic forces by measurement of strain, magnetic
forces by current and distance determinations, gravitational
forces by mass and distance measurements, and so on. And it is
an empirical fact that when all such force measurements are
made and the magnitude of the resultant force determined, then
the rate of change of momentum of the system under considera
tion is found to be proportional to the magnitude of this re
sultant force. And so it is that we obtain an immensely fruitful and accurate
description of a very large part of our whole experience of
objects in motion, through the simple and compact statement
of Newton’s second law.1 6—1 Make a graphical analysis of the data represented by the strobe
scopic photograph of Fig. 6—2(a) to test whether this is indeed ac
celeration under a constant force. 1For further discussion of these questions, the American Journal of Physics
is a perennial source. See, for example, the following articles: I... Eisenbud,
“On the Classical Laws of Motion," Am. J. Phys, 26, 144 ( 1958); N. Austern,
“Presentation of Newtonian Mechanics,” Am. J. Phys., 29, 617 (196]);
R. Weinstock, “Laws of Classical Motion: What's F? What‘s m? What's
0?", Am. J. Phys., 29, 698 (1961). ’5' .., i. .,»..t
t); §.?i;>§;f:§£;:: 6—2 A cabin cruiser of mass 15 metric tons drifts in toward a dock
at a Speed of 0.3 m/sec after its engines have been cut. (A metric ton
is 103 kg.) A man on the dock is able to touch the boat when it is l m
from the dock, and thereafter he pushes on it with a force of 700 N to
try to stop it. Can he bring the boat to rest before it touches the dock? 6—3 (a) A man of mass 80 kg jumps down to a concrete patio from
a window ledge only 0.5 m above the ground. He neglects to bend
his knees on landing, so that his motion is arrested in a distance of
about 2 cm. With what average force does this jar his bone structure? (b) If the man jumps from a ledge 1.5 m above the ground but
bends his knees so that his center of gravity descends an additional
distance 11 after his feet touch the ground, what must h be so that the
average force exerted on him by the ground is only three times his
normal weight? 6—4 An object of mass 2 kg is acted upon by the following combina
tion of forces in the xy plane: 5 N at 0 = 0, 10 N at 0 = 7r/4, and
20 N at 0 — 47r/3. The direction 0 = 0 corresponds to the +x direc
tion. At! = 0 the object is at the pointx = —6mandy = 3 m and has
velocity components v1 = 2m/sec and 1),, = 4m/sec. Find the
object’s velocity and position at t = 2 sec. 6~5 The graphs shown give information regarding the motion in the
xy plane of three different particles. In diagrams (a) and (b) the small y Note!
Parabolic \u (a) t
. x I
0 (Vertical lines 0
.v equally spaced) .v
Direction of motion
X ()
(C) I )f mass 15 metric tons drifts in toward a dock
: after its engines have been cut. (A metric ton
he dock is able to touch the boat when it is l m
reafter he pushes on it with a force of 700 N to
ring the boat to rest before it touches the dock? ss 80 kg jumps down to a concrete patio from
.5 m above the ground. He neglects to bend
0 that his motion is arrested in a distance of
average force does this jar his bone structure?
nps from a ledge 1.5 m above the ground but
his center of gravity descends an additional
touch the ground, what must I: be so that the
n him by the ground is only three times his 2 kg is acted upon by the following combina
plane: 5N at 0 = 0,10N at 0 = 7r/4, and
direction 6 = 0 corresponds to the +x direc
sat thepointx = n6mandy == 3 m and has = 2m/sec and Dy = 4m/sec. Find the
tion at t = 2 sec. give information regarding the motion in the.
t particles. In diagrams (a) and (b) the small Note!
abolic \‘u,
L
.r 0 r
l
. . .V
Direction
31‘ motion
\
X 0 ’ (C) dots indicate the positions at equal intervals of time. For each case,
write equations that describe the force components F,n and Fy. 6—6 An observer ﬁrst measures the velocity of an approaching object
to be 10—2 m/sec and then, lsec later, to be 2 X 10‘2 m/sec. No
intermediate readings are possible because the observer’s instruments
take a full second to determine a velocity. If the object has a mass
of 5 g, what conclusions can the observer make about (a) The size of the force that had been acting? (b) The impulse supplied by the force? (c) The work done by the force? 6—7 A particle of mass 2 kg oscillates along the x axis according to
the equation . 1r‘
—— 0.2 sm (5! — E) where x is in meters and t in seconds.
(a) What is the force acting on the particle at t = 0?
(b) What is the maximum force that acts on the particle? 6—8 A car of mass 103 kg is traveling at 28 m/sec (a little over
60 mph) along a horizontal straight road when the driver suddenly
sees a fallen tree blocking the road 100 m ahead. The driver applies
the brakes as soon as his reaction time (0.75 sec) allows and comes to
rest 9 m short of the tree. (a) Assuming constant deceleration caused by the brakes, what
is the decelerating force? What fraction is it of the weight of the car
(take g = 9.8 m/sec2)? (b) If the car had been on a downward grade of sin"(715) with
the brakes supplying the same decelerating force as before, with what
speed would the car have hit the tree? 6—9 A particle of mass m follows a path in the xy plane that is de
scribed by the following equations: x = A(at —— sin at)
y = A(l — cosa!) (a) Sketch this path.
(b) Find the timedependent force vector that causes this motion.
Can you suggest a way of producing such a situation in practice? 6'10 A piece of string of length l, which can support a maximum
tension T, is used to whirl a particle of mass m in a circular path.
What is the maximum Speed with which the particle may be whirled
if the circle is (a) horizontal; (b) vertical? ...
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