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° and 24° 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°
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.
So always in practical operation that law is absolutely true which we
observe to be approximated to more and more nearly as we consider
smaller and smaller angles, that the versed sine of the angle is the
measure of its deflection from the straight line of motion, or the
measure of its fall toward the center, which takes place at every point
in the motion of a revolving body.
Then, assuming the absolute truth of this law of deflection, we find
ourselves able to explain all the phenomena of centrifugal force, and to
compute its amount correctly in all cases.
We have now advanced two steps. We have learned _the direction_ and _the
measure_ of the deflection, which a revolving body continually suffers,
and its resistance to which is termed centrifugal force. The direction
is toward the center, and the measure is the versed sine of the angle.
SECOND.--We next come to consider what are known as the laws of
centrifugal force. These laws are four in number. They are, that the
amount of centrifugal force exerted by a revolving body varies in four
ways.
_First_.--Directly as the weight of the body.
_Second_.--In a given circle of revolution, as the square of the speed
or of the number of revolutions per minute; which two expressions in
this case mean the same thing.
_Third_.--With a given number of revolutions per minute, or a given
angular velocity[1] _directly_ as the radius of the circle; and
_Fourth_.--With a given actual velocity, or speed in feet per minute,
_inversely_ as the radius of the circle.
[Footnote 1: A revolving body is said to have the same angular velocity,
when it sweeps through equal angles in equal times. Its actual velocity
varies directly as the radius of the circle in which it is revolving.]
Of course there is a reason for these laws. You are not to learn them by
rote, or to accept them on any authority. You are taught not to accept
any rule or formula on authority, but to demand the reason for it--to
give yourselves no rest until you know the why and wherefore, and
comprehend these fully. This is education, not cramming the mind with
mere facts and rules to be memorized, but drawing out the mental powers
into activity, strengthening them by use and exercise, and forming the
habit, and at the same time developing the power, of penetrating to the
reason of things.
In this way only, you will be able to meet the requirement of a great
educator, who said: "I do not care to be told what a young man knows,
but what he can _do_." I wish here to add my grain to the weight of
instruction which you receive, line upon line, precept on precept, on
this subject.
The reason for these laws of centrifugal force is an extremely simple
one. The first law, that this force varies directly as the weight of the
body, is of course obvious. We need not refer to this law any further.
The second, third, and fourth laws merely express the relative rates at
which a revolving body is deflected from the tangential direction of
motion, in each of the three cases described, and which cases embrace
all possible conditions.
These three rates of deflection are exhibited in Fig. 2. An examination
of this figure will give you a clear understanding of them. Let us first
suppose a body to be revolving about the point, O, as a center, in a
circle of which A B C is an arc, and with a velocity which will carry it
from A to B in one second of time. Then in this time the body is
deflected from the tangential direction a distance equal to A D, the
versed sine of the angle A O B. Now let us suppose the velocity of this
body to be doubled in the same circle. In one second of time it moves
from A to C, and is deflected from the tangential direction of motion a
distance equal to A E, the versed sine of the angle, A O C. But A E is
four times A D. Here we see in a given circle of revolution the
deflection varying as the square of the speed. The slight error already
pointed out in these large angles is disregarded.
The following table will show, by comparison of the versed sines of very
small angles, the deflection in a given circle varying as the square of
the speed, when we penetrate to them, so nearly that the error is not
disclosed at the fifteenth place of decimals.
The versed sine of 1" is 0.000,000,000,011,752
" " " " 2" is 0.000,000,000,047,008
" " " " 3" is 0.000,000,000,105,768
" " " " 4" is 0.000,000,000,188,032
" " " " 5" is 0.000,000,000,293,805
" " " " 6" is 0.000,000,000,423,072
" " " " 7" is 0.000,000,000,575,848
" " " " 8" is 0.000,000,000,752,128
" " " " 9" is 0.000,000,000,951,912
" " " " 10" is 0.000,000,001,175,222
" " " " 100" is 0.000,000,117,522,250
You observe the deflection for 10" of arc is 100 times as great, and for
100" of arc is 10,000 times as great as it is for 1" of arc. So far as
is shown by the 15th place of decimals, the versed sine varies as the
square of the angle; or, in a given circle, the deflection, and so the
centrifugal force, of a revolving body varies as the square of the
speed.
The reason for the third law is equally apparent on inspection of Fig.
2. It is obvious, that in the case of bodies making the same number of
revolutions in different circles, the deflection must vary directly as
the diameter of the circle, because for any given angle the versed sine
varies directly as the radius. Thus radius O A' is twice radius O A, and
so the versed sine of the arc A' B' is twice the versed sine of the arc
A B. Here, while the angular velocity is the same, the actual velocity
is doubled by increase in the diameter of the circle, and so the
deflection is doubled. This exhibits the general law, that with a given
angular velocity the centrifugal force varies directly as the radius or
diameter of the circle.
We come now to the reason for the fourth law, that, with a given actual
velocity, the centrifugal force varies _inversely_ as the diameter of
the circle. If any of you ever revolved a weight at the end of a cord
with some velocity, and let the cord wind up, suppose around your hand,
without doing anything to accelerate the motion, then, while the circle
of revolution was growing smaller, the actual velocity continuing nearly
uniform, you have felt the continually increasing stress, and have
observed the increasing angular velocity, the two obviously increasing
in the same ratio. That is the operation or action which the fourth law
of centrifugal force expresses. An examination of this same figure (Fig.
2) will show you at once the reason for it in the increasing deflection
which the body suffers, as its circle of revolution is contracted. If we
take the velocity A' B', double the velocity A B, and transfer it to the
smaller circle, we have the velocity A C. But the deflection has been
increasing as we have reduced the circle, and now with one half the
radius it is twice as great. It has increased in the same ratio in which
the angular velocity has increased. Thus we see the simple and necessary
nature of these laws. They merely express the different rates of
deflection of a revolving body in these different cases.
THIRD.--We have a coefficient of centrifugal force, by which we are
enabled to compute the amount of this resistance of a revolving body to
deflection from a direct line of motion in all cases. This is that
coefficient. The centrifugal force of a body making _one_ revolution per
minute, in a circle of _one_ foot radius, is 0.000341 of the weight of
the body.
According to the above laws, we have only to multiply this coefficient
by the square of the number of revolutions made by the body per minute,
and this product by the radius of the circle in feet, or in decimals of
a foot, and we have the centrifugal force, in terms of the weight of the
body. Multiplying this by the weight of the body in pounds, we have the
centrifugal force in pounds.
Of course you want to know how this coefficient has been found out, and
how you can be sure it is correct. I will tell you a very simple way.
There are also mathematical methods of ascertaining this coefficient,
which your professors, if you ask them, will let you dig out for
yourselves. The way I am going to tell you I found out for myself, and
that, I assure you, is the only way to learn anything, so that it will
stick; and the more trouble the search gives you, the darker the way
seems, and the greater the degree of perseverance that is demanded, the
more you will appreciate the truth when you have found it, and the more
complete and permanent your possession of it will be.
The explanation of this method may be a little more abstruse than the
explanations already given, but it is very simple and elegant when you
see it, and I fancy I can make it quite clear. I shall have to preface
it by the explanation of two simple laws. The first of these is, that a
body acted on by a constant force, so as to have its motion uniformly
accelerated, suppose in a straight line, moves through distances which
increase as the square of the time that the accelerating force continues
to be exerted.
The necessary nature of this law, or rather the action of which this law
is the expression, is shown in Fig. 3.
[Illustration: Fig. 3]
Let the distances A B, B C, C D, and D E in this figure represent four
successive seconds of time. They may just as well be conceived to
represent any other equal units, however small. Seconds are taken only
for convenience. At the commencement of the first second, let a body
start from a state of rest at A, under the action of a constant force,
sufficient to move it in one second through a distance of one foot. This
distance also is taken only for convenience. At the end of this second,
the body will have acquired a velocity of two feet per second. This is
obvious because, in order to move through one foot in this second, the
body must have had during the second an average velocity of one foot per
second. But at the commencement of the second it had no velocity. Its
motion increased uniformly. Therefore, at the termination of the second
its velocity must have reached two feet per second. Let the triangle A B
F represent this accelerated motion, and the distance, of one foot,
moved through during the first second, and let the line B F represent
the velocity of two feet per second, acquired by the body at the end of
it. Now let us imagine the action of the accelerating force suddenly to
cease, and the body to move on merely with the velocity it has acquired.
During the next second it will move through two feet, as represented by
the square B F C I. But in fact, the action of the accelerating force
does not cease. This force continues to be exerted, and produces on the
body during the next second the same effect that it did during the first
second, causing it to move through an additional foot of distance,
represented by the triangle F I G, and to have its velocity accelerated
two additional feet per second, as represented by the line I G. So in
two seconds the body has moved through four feet. We may follow the
operation of this law as far as we choose. The figure shows it during
four seconds, or any other unit, of time, and also for any unit of
distance. Thus:
Time 1 Distance 1
" 2 " 4
" 3 " 9
" 4 " 16
So it is obvious that the distance moved through by a body whose motion
is uniformly accelerated increases as the square of the time.
But, you are asking, what has all this to do with a revolving body? As
soon as your minds can be started from a state of rest, you will
perceive that it has everything to do with a revolving body. The
centripetal force, which acts upon a revolving body to draw it to the
center, is a constant force, and under it the revolving body must move
or be deflected through distances which increase as the squares of the
times, just as any body must do when acted on by a constant force. To
prove that a revolving body obeys this law, I have only to draw your
attention to Fig. 2. Let the equal arcs, A B and B C, in this figure
represent now equal times, as they will do in case of a body revolving
in this circle with a uniform velocity. The versed sines of the angles,
A O B and A O C, show that in the time, A C, the revolving body was
deflected four times as far from the tangent to the circle at A as it
was in the time, A B. So the deflection increased as the square of the
time. If on the table already given, we take the seconds of arc to
represent equal times, we see the versed sine, or the amount of
deflection of a revolving body, to increase, in these minute angles,
absolutely so far as appears up to the fifteenth place of decimals, as
the square of the time.
The standard from which all computations are made of the distances
passed through in given times by bodies whose motion is uniformly
accelerated, and from which the velocity acquired is computed when the
accelerating force is known, and the force is found when the velocity
acquired or the rate of acceleration is known, is the velocity of a body
falling to the earth. It has been established by experiment, that in
this latitude near the level of the sea, a falling body in one second
falls through a distance of 16.083 feet, and acquires a velocity of
32.166 feet per second; or, rather, that it would do so if it did not
meet the resistance of the atmosphere. In the case of a falling body,
its weight furnishes, first, the inertia, or the resistance to motion,
that has to be overcome, and affords the measure of this resistance,
and, second, it furnishes the measure of the attraction of the earth, or
the force exerted to overcome its resistance. Here, as in all possible
cases, the force and the resistance are identical with each other. The
above is, therefore, found in this way to be the rate at which the
motion of any body will be accelerated when it is acted on by a constant
force equal to its weight, and encounters no resistance.
It follows that a revolving body, when moving uniformly in any circle at
a speed at which its deflection from a straight line of motion is such
that in one second this would amount to 16.083 feet, requires the
exertion of a centripetal force equal to its weight to produce such
deflection. The deflection varying as the square of the time, in 0.01 of
a second this deflection will be through a distance of 0.0016083 of a
foot.
Now, at what speed must a body revolve, in a circle of one foot radius,
in order that in 0.01 of one second of time its deflection from a
tangential direction shall be 0.0016083 of a foot? This decimal is the
versed sine of the arc of 3°15', or of 3.25°. This angle is so small
that the departure from the law that the deflection is equal to the
versed sine of the angle is too slight to appear in our computation.
Therefore, the arc of 3.25° is the arc of a circle of one foot radius
through which a body must revolve in 0.01 of a second of time, in order
that the centripetal force, and so the centrifugal force, shall be equal
to its weight. At this rate of revolution, in one second the body will
revolve through 325°, which is at the rate of 54.166 revolutions per
minute.
Now there remains only one question more to be answered. If at 54.166
revolutions per minute the centrifugal force of a body is equal to its
weight, what will its centrifugal force be at one revolution per minute
in the same circle?
To answer this question we have to employ the other extremely simple
law, which I said I must explain to you. It is this: The acceleration
and the force vary in a constant ratio with each other. Thus, let force
1 produce acceleration 1, then force 1 applied again will produce
acceleration 1 again, or, in other words, force 2 will produce
acceleration 2, and so on. This being so, and the amount of the
deflection varying as the squares of the speeds in the two cases, the
centrifugal force of a body making one revolution per minute in a circle
of
1²
one foot radius will be ---------- = 0.000341
54.166²
--the coefficient of centrifugal force.
There is another mode of making this computation, which is rather neater
and more expeditious than the above. A body making one revolution per
minute in a circle of one foot radius will in one second revolve through
an arc of 6°. The versed sine of this arc of 6° is 0.0054781046 of a
foot. This is, therefore, the distance through which a body revolving at
this rate will be deflected in one second. If it were acted on by a
force equal to its weight, it would be deflected through the distance of
16.083 feet in the same time. What is the deflecting force actually
exerted upon it? Of
0.0054781046
course, it is ------------.
16.083
This division gives 0.000341 of its weight as such deflecting force, the
same as before.
In taking the versed sine of 6°, a minute error is involved, though not
one large enough to change the last figure in the above quotient. The
law of uniform acceleration does not quite hold when we come to an angle
so large as 6°. If closer accuracy is demanded, we can attain it, by
taking the versed sine for 1°, and multiplying this by 6². This gives as
a product 0.0054829728, which is a little larger than the versed sine of
6°.
I hope I have now kept my promise, and made it clear how the coefficient
of centrifugal force may be found in this simple way.
We have now learned several things about centrifugal force. Let me
recapitulate. We have learned:
1st. The real nature of centrifugal force. That in the dynamical sense
of the term force, this is not a force at all: that it is not capable of
producing motion, that the force which is really exerted on a revolving
body is the centripetal force, and what we are taught to call
centrifugal force is nothing but the resistance which a revolving body
opposes to this force, precisely like any other resistance.
2d. The direction of the deflection, to which the centrifugal force is
the resistance, which is straight to the center.
3d. The measure of this deflection; the versed sine of the angle.
4th. The reason of the laws of centrifugal force; that these laws merely
express the relative amount of the deflection, and so the amount of the
force required to produce the deflection, and of the resistance of the
revolving body to it, in all different cases.
5th. That the deflection of a revolving body presents a case analogous
to that of uniformly accelerated motion, under the action of a constant
force, similar to that which is presented by falling bodies;[1] and
finally,
6th. How to find the coefficient, by which the amount of centrifugal
force exerted in any case may be computed.
[Footnote 1: A body revolving with a uniform velocity in a horizontal
plane would present the only case of uniformly accelerated motion that
is possible to be realized under actual conditions.]