Scientific American Supplement, Vol. XXI., No. 531, March 6, 1886 by Various
V >>
Various >> Scientific American Supplement, Vol. XXI., No. 531, March 6, 1886
Pages:
1 | 2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10
"But how about the Indians?" I asked. "Why," he replied, "we never had
any trouble from the Indians. I knew we wouldn't have. Men who supposed
I was such a fool as to go about this undertaking before that was all
settled didn't know me. No Indian ever harmed that line. The Indians are
the best friends we have got. You see, we taught the Indians the Great
Spirit was in that line; and what was more, we proved it to them. It
was, by all odds, the greatest medicine they ever saw. They fairly
worshiped it. No Indian ever dared to do it harm."
"But," he added, "there was one thing I didn't count on. The border
ruffians in Missouri are as bad as anybody ever feared the Indians might
be. They have given us so much trouble that we are now building a line
around that State, through Iowa and Nebraska. We are obliged to do it."
This opened another phase of the subject. The telegraph line to the
Pacific had a value beyond that which could be expressed in money. It
was perhaps the strongest of all the ties which bound California so
securely to the Union, in the dark days of its struggle for existence.
The secession element in Missouri recognized the importance of the line
in this respect, and were persistent in their efforts to destroy it. We
have seen by what means their purpose was thwarted.
I have always felt that, among the countless evidences of the ordering
of Providence by which the war for the preservation of the Union was
signalized, not the least striking was the raising up of this remarkable
man, to accomplish alone, and in the very nick of time, a work which at
once became of such national importance.
This is the man who has crowned his useful career, and shown again his
eminently practical character and wise foresight, by the endowment of
this College, which cannot fail to be a perennial source of benefit to
the country whose interests he has done so much to promote, and which
his remarkable sagacity and energy contributed so much to preserve.
We have an excellent rule, followed by all successful designers of
machinery, which is, to make provision for the extreme case, for the
most severe test to which, under normal conditions, and so far as
practicable under abnormal conditions also, the machinery can be
subjected. Then, of course, any demands upon it which are less than the
extreme demand are not likely to give trouble. I shall apply this
principle in addressing you to-day. In what I have to say, I shall speak
directly to the youngest and least advanced minds among my auditors. If
I am successful in making an exposition of my subject which shall be
plain to them, then it is evident that I need not concern myself about
being understood by the higher class men and the professors.
The subject to which your attention is now invited is
THE PRINCIPLES AND METHODS OF BALANCING FORCES DEVELOPED IN MOVING
BODIES.
This is a subject with which every one who expects to be concerned with
machinery, either as designer or constructor, ought to be familiar. The
principles which underlie it are very simple, but in order to be of use,
these need to be thoroughly understood. If they have once been mastered,
made familiar, incorporated into your intellectual being, so as to be
readily and naturally applied to every case as it arises, then you
occupy a high vantage ground. In this particular, at least, you will not
go about your work uncertainly, trying first this method and then that
one, or leaving errors to be disclosed when too late to remedy them. On
the contrary, you will make, first your calculations and then your
plans, with the certainty that the result will be precisely what you
intend.
Moreover, when you read discussions on any branch of this subject, you
will not receive these into unprepared minds, just as apt to admit error
as truth, and possessing no test by which to distinguish the one from
the other; but you will be able to form intelligent judgments with
respect to them. You will discover at once whether or not the writers
are anchored to the sure holding ground of sound principles.
It is to be observed that I do not speak of balancing bodies, but of
balancing forces. Forces are the realities with which, as mechanical
engineers, you will have directly to deal, all through your lives. The
present discussion is limited also to those forces which are developed
in moving bodies, or by the motion of bodies. This limitation excludes
the force of gravity, which acts on all bodies alike, whether at rest or
in motion. It is, indeed, often desirable to neutralize the effect of
gravity on machinery. The methods of doing this are, however, obvious,
and I shall not further refer to them.
Two very different forces, or manifestations of force, are developed by
the motion of bodies. These are
MOMENTUM AND CENTRIFUGAL FORCE.
The first of these forces is exerted by every moving body, whatever the
nature of the path in which it is moving, and always in the direction of
its motion. The latter force is exerted only by bodies whose path is a
circle, or a curve of some form, about a central body or point, to which
it is held, and this force is always at right angles with the direction
of motion of the body.
Respecting momentum, I wish only to call your attention to a single
fact, which will become of importance in the course of our discussion.
Experiments on falling bodies, as well as all experience, show that the
velocity of every moving body is the product of two factors, which must
combine to produce it. Those factors are force and distance. In order to
impart motion to the body, force must act through distance. These two
factors may be combined in any proportions whatever. The velocity
imparted to the body will vary as the square root of their product.
Thus, in the case of any given body,
Let force 1, acting through distance 1, impart velocity 1.
Then " 1, " " " 4, will " " 2, or
" 2, " " " 2, " " " 2, or
" 4, " " " 1, " " " 2;
And " 1, " " " 9, " " " 3, or
" 3, " " " 3, " " " 3, or
" 9, " " " 1, " " " 3.
This table might be continued indefinitely. The product of the force
into the distance will always vary as the square of the final velocity
imparted. To arrest a given velocity, the same force, acting through the
same distance, or the same product of force into distance, is required
that was required to impart the velocity.
The fundamental truth which I now wish to impress upon your minds is
that in order to impart velocity to a body, to develop the energy which
is possessed by a body in motion, force must act through distance.
Distance is a factor as essential as force. Infinite force could not
impart to a body the least velocity, could not develop the least energy,
without acting through distance.
This exposition of the nature of momentum is sufficient for my present
purpose. I shall have occasion to apply it later on, and to describe the
methods of balancing this force, in those cases in which it becomes
necessary or desirable to do so. At present I will proceed to consider
the second of the forces, or manifestations of force, which are
developed in moving bodies--_centrifugal force_.
This force presents its claims to attention in all bodies which revolve
about fixed centers, and sometimes these claims are presented with a
good deal of urgency. At the same time, there is probably no subject,
about which the ideas of men generally are more vague and confused. This
confusion is directly due to the vague manner in which the subject of
centrifugal force is treated, even by our best writers. As would then
naturally be expected, the definitions of it commonly found in our
handbooks are generally indefinite, or misleading, or even absolutely
untrue.
Before we can intelligently consider the principles and methods of
balancing this force, we must get a correct conception of the nature of
the force itself. What, then, is centrifugal force? It is an extremely
simple thing; a very ordinary amount of mechanical intelligence is
sufficient to enable one to form a correct and clear idea of it. This
fact renders it all the more surprising that such inaccurate and
confused language should be employed in its definition. Respecting
writers, also, who use language with precision, and who are profound
masters of this subject, it must be said that, if it had been their
purpose to shroud centrifugal force in mystery, they could hardly have
accomplished this purpose more effectually than they have done, to minds
by whom it was not already well understood.
Let us suppose a body to be moving in a circular path, around a center
to which it is firmly held; and let us, moreover, suppose the impelling
force, by which the body was put in motion, to have ceased; and, also,
that the body encounters no resistance to its motion. It is then, by our
supposition, moving in its circular path with a uniform velocity,
neither accelerated nor retarded. Under these conditions, what is the
force which is being exerted on this body? Clearly, there is only one
such force, and that is, the force which holds it to the center, and
compels it, in its uniform motion, to maintain a fixed distance from
this center. This is what is termed centripetal force. It is obvious,
that the centripetal force, which holds this revolving body _to_ the
center, is the only force which is being exerted upon it.
Where, then, is the centrifugal force? Why, the fact is, there is not
any such thing. In the dynamical sense of the term "force," the sense in
which this term is always understood in ordinary speech, as something
tending to produce motion, and the direction of which determines the
direction in which motion of a body must take place, there is, I repeat,
no such thing as centrifugal force.
There is, however, another sense in which the term "force" is employed,
which, in distinction from the above, is termed a statical sense. This
"statical force" is the force by the exertion of which a body keeps
still. It is the force of inertia--the resistance which all matter
opposes to a dynamical force exerted to put it in motion. This is the
sense in which the term "force" is employed in the expression
"centrifugal force." Is that all? you ask. Yes; that is all.
I must explain to you how it is that a revolving body exerts this
resistance to being put in motion, when all the while it _is_ in motion,
with, according to our above supposition, a uniform velocity. The first
law of motion, so far as we now have occasion to employ it, is that a
body, when put in motion, moves in a straight line. This a moving body
always does, unless it is acted on by some force, other than its
impelling force, which deflects it, or turns it aside, from its direct
line of motion. A familiar example of this deflecting force is afforded
by the force of gravity, as it acts on a projectile. The projectile,
discharged at any angle of elevation, would move on in a straight line
forever, but, first, it is constantly retarded by the resistance of the
atmosphere, and, second, it is constantly drawn downward, or made to
fall, by the attraction of the earth; and so instead of a straight line
it describes a curve, known as the trajectory.
Now a revolving body, also, has the same tendency to move in a straight
line. It would do so, if it were not continually deflected from this
line. Another force is constantly exerted upon it, compelling it, at
every successive point of its path, to leave the direct line of motion,
and move on a line which is everywhere equally distant from the center
to which it is held. If at any point the revolving body could get free,
and sometimes it does get free, it would move straight on, in a line
tangent to the circle at the point of its liberation. But if it cannot
get free, it is compelled to leave each new tangential direction, as
soon as it has taken it.
This is illustrated in the above figure. The body, A, is supposed to be
revolving in the direction indicated by the arrow, in the circle, A B F
G, around the center, O, to which it is held by the cord, O A. At the
point, A, it is moving in the tangential direction, A D. It would
continue to move in this direction, did not the cord, O A, compel it to
move in the arc, A C. Should this cord break at the point, A, the body
would move; straight on toward D, with whatever velocity it had.
You perceive now what centrifugal force is. This body is moving in the
direction, A D. The centripetal force, exerted through the cord, O A,
pulls it aside from this direction of motion. The body resists this
deflection, and this resistance is its centrifugal force.
[Illustration: Fig. 1]
Centrifugal force is, then, properly defined to be the disposition of a
revolving body to move in a straight line, and the resistance which such
a body opposes to being drawn aside from a straight line of motion. The
force which draws the revolving body continually to the center, or the
deflecting force, is called the centripetal force, and, aside from the
impelling and retarding forces which act in the direction of its motion,
the centripetal force is, dynamically speaking, the only force which is
exerted on the body.
It is true, the resistance of the body furnishes the measure of the
centripetal force. That is, the centripetal force must be exerted in a
degree sufficient to overcome this resistance, if the body is to move in
the circular path. In this respect, however, this case does not differ
from every other case of the exertion of force. Force is always exerted
to overcome resistance: otherwise it could not be exerted. And the
resistance always furnishes the exact measure of the force. I wish to
make it entirely clear, that in the dynamical sense of the term "force,"
there is no such thing as centrifugal force. The dynamical force, that
which produces motion, is the centripetal force, drawing the body
continually from the tangential direction, toward the center; and what
is termed centrifugal force is merely the resistance which the body
opposes to this deflection, _precisely like any other resistance to a
force_.
The centripetal force is exerted on the radial line, as on the line, A
O, Fig. 1, at right angles with the direction in which the body is
moving; and draws it directly toward the center. It is, therefore,
necessary that the resistance to this force shall also be exerted on the
same line, in the opposite direction, or directly from the center. But
this resistance has not the least power or tendency to produce motion in
the direction in which it is exerted, any more than any other resistance
has.
We have been supposing a body to be firmly held to the center, so as to
be compelled to revolve about it in a fixed path. But the bond which
holds it to the center may be elastic, and in that case, if the
centrifugal force is sufficient, the body will be drawn from the center,
stretching the elastic bond. It may be asked if this does not show
centrifugal force to be a force tending to produce motion from the
center. This question is answered by describing the action which really
takes place. The revolving body is now imperfectly deflected. The bond
is not strong enough to compel it to leave its direct line of motion,
and so it advances a certain distance along this tangential line. This
advance brings the body into a larger circle, and by this enlargement of
the circle, assuming the rate of revolution to be maintained, its
centrifugal force is proportionately increased. The deflecting power
exerted by the elastic bond is also increased by its elongation. If this
increase of deflecting force is no greater than the increase of
centrifugal force, then the body will continue on in its direct path;
and when the limit of its elasticity is reached, the deflecting bond
will be broken. If, however, the strength of the deflecting bond is
increased by its elongation in a more rapid ratio than the centrifugal
force is increased by the enlargement of the circle, then a point will
be reached in which the centripetal force will be sufficient to compel
the body to move again in the circular path.
Sometimes the centripetal force is weak, and opportunity is afforded to
observe this action, and see its character exhibited. A common example
of weak centripetal force is the adhesion of water to the face of a
revolving grindstone. Here we see the deflecting force to become
insufficient to compel the drops of water longer to leave their direct
paths, and so these do not longer leave their direct paths, but move on
in those paths, with the velocity they have at the instant of leaving
the stone, flying off on tangential lines.
If, however, a fluid be poured on the side of the revolving wheel near
the axis, it will move out to the rim on radial lines, as may be
observed on car wheels universally. The radial lines of black oil on
these wheels look very much as if centrifugal force actually did produce
motion, or had at least a very decided tendency to produce motion, in
the radial direction. This interesting action calls for explanation. In
this action the oil moves outward gradually, or by inconceivably minute
steps. Its adhesion being overcome in the least possible degree, it
moves in the same degree tangentially. In so doing it comes in contact
with a point of the surface which has a motion more rapid than its own.
Its inertia has now to be overcome, in the same degree in which it had
overcome the adhesion. Motion in the radial direction is the result of
these two actions, namely, leaving the first point of contact
tangentially and receiving an acceleration of its motion, so that this
shall be equal to that of the second point of contact. When we think
about the matter a little closely, we see that at the rim of the wheel
the oil has perhaps ten times the velocity of revolution which it had on
leaving the journal, and that the mystery to be explained really is, How
did it get that velocity, moving out on a radial line? Why was it not
left behind at the very first? Solely by reason of its forward
tangential motion. That is the answer.
When writers who understand the subject talk about the centripetal and
centrifugal forces being different names for the same force, and about
equal action and reaction, and employ other confusing expressions, just
remember that all they really mean is to express the universal relation
between force and resistance. The expression "centrifugal force" is
itself so misleading, that it becomes especially important that the real
nature of this so-called force, or the sense in which the term "force"
is used in this expression, should be fully explained.[1] This force is
now seen to be merely the tendency of a revolving body to move in a
straight line, and the resistance which it opposes to being drawn aside
from that line. Simple enough! But when we come to consider this action
carefully, it is wonderful how much we find to be contained in what
appears so simple. Let us see.
[Footnote 1: I was led to study this subject in looking to see what had
become of my first permanent investment, a small venture, made about
thirty-five years ago, in the "Sawyer and Gwynne static pressure
engine." This was the high-sounding name of the Keely motor of that day,
an imposition made possible by the confused ideas prevalent on this very
subject of centrifugal force.]
FIRST.--I have called your attention to the fact that the direction in
which the revolving body is deflected from the tangential line of motion
is toward the center, on the radial line, which forms a right angle with
the tangent on which the body is moving. The first question that
presents itself is this: What is the measure or amount of this
deflection? The answer is, this measure or amount is the versed sine of
the angle through which the body moves.
Now, I suspect that some of you--some of those whom I am directly
addressing--may not know what the versed sine of an angle is; so I must
tell you. We will refer again to Fig. 1. In this figure, O A is one
radius of the circle in which the body A is revolving. O C is another
radius of this circle. These two radii include between them the angle A
O C. This angle is subtended by the arc A C. If from the point O we let
fall the line C E perpendicular to the radius O A, this line will divide
the radius O A into two parts, O E and E A. Now we have the three
interior lines, or the three lines within the circle, which are
fundamental in trigonometry. C E is the sine, O E is the cosine, and E A
is the versed sine of the angle A O C. Respecting these three lines
there are many things to be observed. I will call your attention to the
following only:
_First_.--Their length is always less than the radius. The radius is
expressed by 1, or unity. So, these lines being less than unity, their
length is always expressed by decimals, which mean equal to such a
proportion of the radius.
_Second_.--The cosine and the versed sine are together equal to the
radius, so that the versed sine is always 1, less the cosine.
_Third_.--If I diminish the angle A O C, by moving the radius O C toward
O A, the sine C E diminishes rapidly, and the versed sine E A also
diminishes, but more slowly, while the cosine O E increases. This you
will see represented in the smaller angles shown in Fig. 2. If, finally,
I make O C to coincide with O A, the angle is obliterated, the sine and
the versed sine have both disappeared, and the cosine has become the
radius.
_Fourth_.--If, on the contrary, I enlarge the angle A O C by moving the
radius O C toward O B, then the sine and the versed sine both increase,
and the cosine diminishes; and if, finally, I make O C coincide with O
B, then the cosine has disappeared, the sine has become the radius O B,
and the versed sine has become the radius O A, thus forming the two
sides inclosing the right angle A O B. The study of this explanation
will make you familiar with these important lines. The sine and the
cosine I shall have occasion to employ in the latter part of my lecture.
Now you know what the versed sine of an angle is, and are able to
observe in Fig. 1 that the versed sine A E, of the angle A O C,
represents in a general way the distance that the body A will be
deflected from the tangent A D toward the center O while describing the
arc A C.
The same law of deflection is shown, in smaller angles, in Fig. 2. In
this figure, also, you observe in each of the angles A O B and A O C
that the deflection, from the tangential direction toward the center, of
a body moving in the arc A C is represented by the versed sine of the
angle. The tangent to the arc at A, from which this deflection is
measured, is omitted in this figure to avoid confusion. It is shown
sufficiently in Fig. 1. The angles in Fig. 2 are still pretty large
angles, being 12 deg. and 24 deg. respectively. These large angles are used for
convenience of illustration; but it should be explained that this law
does not really hold in them, as is evident, because the arc is longer
than the tangent to which it would be connected by a line parallel with
the versed sine. The law is absolutely true only when the tangent and
arc coincide, and approximately so for exceedingly small angles.
[Illustration: Fig. 2]
In reality, however, we have only to do with the case in which the arc
and the tangent do coincide, and in which the law that the deflection is
_equal to_ the versed sine of the angle is absolutely true. Here, in
observing this most familiar thing, we are, at a single step, taken to
that which is utterly beyond our comprehension. The angles we have to
consider disappear, not only from our sight, but even from our
conception. As in every other case when we push a physical investigation
to its limit, so here also, we find our power of thought transcended,
and ourselves in the presence of the infinite.
We can discuss very small angles. We talk familiarly about the angle
which is subtended by 1" of arc. On Fig. 2, a short line is drawn near
to the radius O A'. The distance between O A' and this short line is 1 deg.
of the arc A' B'. If we divide this distance by 3,600, we get 1" of arc.
The upper line of the Table of versed sines given below is the versed
sine of 1" of arc. It takes 1,296,000 of these angles to fill a circular
space. These are a great many angles, but they do not make a circle.
They make a polygon. If the radius of the circumscribed circle of this
polygon is 1,296,000 feet, which is nearly 213 geographical miles, each
one of its sides will be a straight line, 6.283 feet long. On the
surface of the earth, at the equator, each side of this polygon would be
one-sixtieth of a geographical mile, or 101.46 feet. On the orbit of the
moon, at its mean distance from the earth, each of these straight sides
would be about 6,000 feet long.
The best we are able to do is to conceive of a polygon having an
infinite number of sides, and so an infinite number of angles, the
versed sines of which are infinitely small, and having, also, an
infinite number of tangential directions, in which the body can
successively move. Still, we have not reached the circle. We never can
reach the circle. When you swing a sling around your head, and feel the
uniform stress exerted on your hand through the cord, you are made aware
of an action which is entirely beyond the grasp of our minds and the
reach of our analysis.
Pages:
1 | 2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10