Scientific American Supplement, Vol. XXI., No. 531, March 6, 1886 by Various

V >> Various >> Scientific American Supplement, Vol. XXI., No. 531, March 6, 1886

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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 deg.15', or of 3.25 deg.. 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 deg. 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 deg., 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

1 squared
one foot radius will be ---------- = 0.000341
54.166 squared

--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 deg.. The versed sine of this arc of 6 deg. 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 deg., 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 deg.. If closer accuracy is demanded, we can attain it, by
taking the versed sine for 1 deg., and multiplying this by 6 squared. This gives as
a product 0.0054829728, which is a little larger than the versed sine of
6 deg..

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.]

I now pass to some other features.

_First_.--You will observe that, relatively to the center, a revolving
body, at any point in its revolution, is at rest. That is, it has no
motion, either from or toward the center, except that which is produced
by the action of the centripetal force. It has, therefore, this identity
also with a falling body, that it starts from a state of rest. This
brings us to a far more comprehensive definition of centrifugal force.
This is the resistance which a body opposes to being put in motion, at
any velocity acquired in any time, from a state of rest. Thus
centrifugal force reveals to us the measure of the inertia of matter.
This inertia may be demonstrated and exhibited by means of apparatus
constructed on this principle quite as accurately as it can be in any
other way.

_Second_.--You will also observe the fact, that motion must be imparted
to a body gradually. As distance, _through_ which force can act, is
necessary to the impartation of velocity, so also time, _during_ which
force can act, is necessary to the same result. We do not know how
motion from a state of rest begins, any more than we know how a polygon
becomes a circle. But we do know that infinite force cannot impart
absolutely instantaneous motion to even the smallest body, or to a body
capable of opposing the least resistance. Time being an essential
element or factor in the impartation of velocity, if this factor be
omitted, the least resistance becomes infinite.

We have a practical illustration of this truth in the explosion of
nitro-glycerine. If a small portion of this compound be exploded on the
surface of a granite bowlder, in the open air, the bowlder will be rent
into fragments. The explanation of this phenomenon common among the
laborers who are the most numerous witnesses of it, which you have
doubtless often heard, and which is accepted by ignorant minds without
further thought, is that the action of nitro-glycerine is downward. We
know that such an idea is absurd.

The explosive force must be exerted in all directions equally. The real
explanation is, that the explosive action of nitro-glycerine is so
nearly instantaneous, that the resistance of the atmosphere is very
nearly equal to that of the rock; at any rate, is sufficient to cause
the rock to be broken up. The rock yields to the force very nearly as
readily as the atmosphere does.

_Third_. An interesting solution is presented here of what is to many an
astronomical puzzle. When I was younger than I am now, I was greatly
troubled to understand how it could be that if the moon was always
falling to the earth, as the astronomers assured us it was, it should
never reach it, nor have its falling velocity accelerated. In popular
treatises on astronomy, such for example as that of Professor Newcomb,
this is explained by a diagram in which the tangential line is carried
out as in Fig. 1, and by showing that in falling from the point A to the
earth as a center, through distances increasing as the square of the
time, the moon, having the tangential velocity that it has, could never
get nearer to the earth than the circle in which it revolves around it.
This is all very true, and very unsatisfactory. We know that this long
tangential line has nothing to do with the motion of the moon, and while
we are compelled to assent to the demonstration, we want something
better. To my mind the better and more satisfactory explanation is found
in the fact that the moon is forever commencing to fall, and is
continually beginning to fall in a new direction. A revolving body, as
we have seen, never gets past that point, which is entirely beyond our
sight and our comprehension, of beginning to fall, before the direction
of its fall is changed. So, under the attraction of the earth, the moon
is forever leaving a new tangential direction of motion at the same
rate, without acceleration.

(_To be continued_.)

* * * * *

COMPRESSED AIR POWER SCHEMES.

By J. STURGEON, Engineer of the Birmingham Compressed Air Power Company.

In the article on "Gas, Air, and Water Power" in the _Journal_ for Dec.
8 last, you state that you await with some curiosity my reply to certain
points in reference to the compressed air power schemes alluded to in
that article. I now, therefore, take the liberty of submitting to you
the arguments on my side of the question (which are substantially the
same as those I am submitting to Mr. Hewson, the Borough Engineer of
Leeds). The details and estimates for the Leeds scheme are not yet in a
forward enough state to enable me to give them at present; but the whole
case is sufficiently worked out for Birmingham to enable a fair
deduction to be made therefrom as regards the utility of the system in
other towns. In Birmingham, progress has been delayed owing to
difficulties in procuring a site for the works, and other matters of
detail. We have, however, recently succeeded in obtaining a suitable
place, and making arrangements for railway siding, water supply, etc.;
and we hope to be in a position to start early in the present year.

I inclose (1) a tabulated summary of the estimates for Birmingham
divided into stages of 3,000 gross indicated horse power at a time; (2)
a statement showing the cost to consumers in terms of indicated horse
power and in different modes, more or less economical, of applying the
air power in the consumers' engines; (3) a tracing showing the method of
laying the mains; (4) a tracing showing the method of collecting the
meter records at the central station, by means of electric apparatus,
and ascertaining the exact amount of leakage. A short description of the
two latter would be as well.

TABLE I.--_Showing the Progressive Development of the Compressed Air
System in stages of 3000 Indicated Horse Power (gross) at a Time, and
the Profits at each Stage_

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