G. P. (George Payn) Quackenbos.

A natural philosphy: embracing the most recent discoveries in the various branches of physics .. online

. (page 10 of 42)
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vers of the first kind may be combined
into Compound Levers.

213. In compound levers, equilibrium is established when
the power, multiplied by the first arms of all the levers, is
equal to the weight multiplied by the last arms of all the
levers.

Fig. 96.

F





A COMPOUND LEVEB.

Thus, in Fig. 9G, which represents a compound lever formed of three sim-
ple ones, let the long arm of each lever be three times the length of its short
arm ; then 1 pound at P will balance 27 pounds at W, because
1 pound X3X3X3 27 pounds X 1 X 1 X 1.

214. LEVERS OF THE SECOND KIND. In levers of the
second kind, the relative position is

must he sit to preserve the balance ? 211. What is meant by a bent lever ? How are
the arms of a bent lever estimated ? Give some familiar examples of bent levers.
212. How may simple levers of the first kind be combined? 213. When is equilib-
rium established in a compound lever ? Illustrate this with Fig. 95. 214. In levers




100 MECHANICS.

POWER WEIGHT FULCRUM OR FULCRUM WEIGHT POWER.

Fig. 97. Fig. 97 shows how the crow-bar may

be used as a lever of the second kind.
The power is applied at the handle ; the
fulcrum is at the other end, and the
weight to be moved is between them.

215. The nearer the weight
is to the fulcrum the greater
the advantage gained, and con-
sequently the greater the space
that P will have to pass through

in moving W a given distance. This principle is stated in

the following

Law. "With levers of the second Idnd, intensity of force

is gained, and time is lost, in proportion as the distance

between the power and the fulcrum exceeds the distance

between the weight and the fulcrum.

Thus, in Fig. 97, if the distance P F be five times as great as W F, a pres-
sure of 10 pounds at P will counterbalance a weight of 50 pounds at W, and
move any thing under 50 pounds ; while, for every inch that W is moved, P
will have to move five inches in the same direction.

Fig. 98. p 216. Practical Applications. The

common chipping-knife, used by apothe-
caries, and represented in Fig. 98, is a
familiar illustration of levers of the sec-
ond kind. The knife is fastened at one
end, F, which thus becomes the fulcrum ; the hand is ap-
plied, as the power, at the other end, P ; and the substance
to be cut is the resistance, or weight, between them. Nut-
crackers and lemon-squeezers work on the same principle,
and are levers of the second kind.

A door turned on its hinges, and an oar used in rowing,
are also examples of this kind of lever. In the former case,
the hinge is the fulcrum ; the hand applied at the knob is
the power ; and the weight of the door, which may be re-

of the second kind, what is the relative position of the three important points ? How
may the crow-bar be used as a lever of the second kind ? 215. What is the law of levers
of the second kind ? Apply this in Fig. 97. 216. What familiar articles will serve as
Illustrations of levers of the second kind ? Show how a door turned on its hinges is




LEVERS OF THE THIRD KIND. 101

gardcd as concentrated in its centre of gravity somewhere
between the two, is the resistance. In the latter case, the
point at which the oar enters the water is the fulcrum ; the
rower's hand is the power ; and the weight of the boat,
acting at the row-lock, is the resistance. According to the
law laid down in 215, the further from the row-lock we
grasp the oar, the more easily we overcome the resistance
and produce motion.

217. Two persons carrying a weight suspended from a stick between them,
use a double lever of the second kind. Power is applied at each end, and
each end in turn becomes the fulcrum to the other, the weight resting on
some intermediate point. The relation of the power at one end to the weight
is governed by the same law as that of the power at the other end ; and there-
fore the weight, to be divided equally, must be suspended from the middle of
the stick. If it is not so suspended, the man who is nearer the weight car-
ries more than the other in proportion as he is nearer.

Thus, let a 12-pound weight, W, be suspended Fig. 99.

from a bar three feet long, at a distance of one foot -A , B

from A and two feet from B. Then A will carry A

two-thirds of the weight, and B one-third. On

this principle, when it is desired that one of the horses harnessed to a car-
riage should draw more than the other, it is necessary only to make the arm
of the vvhiffle-tree to which he is attached proportionally shorter.

Fig. 100 shows how a weight may be equally p Fig. loo

distributed between three persons. B, being i> / E
twice as far from E as D is, bears one-third of the



weight, W ; while A and C, at the extremities ^ W

of the equal-armed lever ADC, bear equal portions of the remaining two-
thirds, or one-third each.

218. LEVERS OF THE THIRD KIND. In le- Fig. 101.
vers of the third kind, the relative position is

FULCRUM POWER WEIGHT OR WEIGHT POWER FULCRUM.

The forceps, represented in Fig. 101, is a lever of the
third kind. The two sides unite at one end to form the ful-
crum ; the article to be grasped is the weight ; and the fin-
gers, applied between the two, constitute the power.

219. Levers of the third kind, unlike those
before described, involve a mechanical disad-

a lever of the second kind. Show how an oar acts as a levor of the second kind.
217. When two persons carry a weight suspended from a stick between them, what
kind of a lever do they use ? Where is the fulcrum ? To be equally divided, where
must the weight be suspended? If the weight does not hang from the middle of the




f02



MECHANICS.



vantage ; that is, to produce equilibrium, the power must
always be greater than the weight.

Law. With levers of the third kind, intensity of force
is lost, and time is gained, in proportion as the distance
from the weight to the fulcrum exceeds the distance from
the power to the fulcrum.

Thus, in Fig. 101, if FW be three times as great as FP, it will require a
power of three pounds at P to counterbalance a resistance of one pound at
W. Levers of this class, therefore, are never used when great power is re-
quired, but only when a slight resistance is to be overcome with great ra-
pidity.

220. Practical Applications. The sugar-tongs, which
resembles in shape the forceps above described, is a familiar
example of the third kind of lever. So is the fire-tongs ;
and hence the difficulty of raising heavy pieces of coal with
this instrument, particularly when the hand is applied near
the rivet or fulcrum.

The sheep-shears is another lever of the third kind, admirably adapted to
the work it performs ; because the wool, being flexible, has to be cut rapidly,
while it does not require any great degree of force.

A door becomes a lever of the third kind when one attempts to move it by
pushing at the edge near the hinges. The mechanical disadvantage is shown
by the great strength required to move it when the power is there applied.
So, when a painter attempts to raise a ladder lying on the ground with its
bottom against a wall, by lifting the top and walking under it grasping round
after round in succession, he experiences great difficulty as he approaches
the bottom, because the ladder, when he passes its centre of gravity, becomes
a lever of the third kind.

Fig. 102. ^2 Nature uses levers of the

third kind in the bones of
animals. The fore-arm of a
man, represented in Fig.
102,will serve as an example.




HUMAN ARM AND HAND.



stick, -which man will carry the more ? Illustrate this with Fig. 99. How may ono
of the horses harnessed to a carriage be made to draw more than the other? How
may a weight be equally distributed between three persons ? 218. In levers of the
third kind, what is the position of the three important points? What instrument is
an example of the third kind of levers ? *219. To produce equilibrium in the third kind
of levers, what is necessary? State the law for levers of the third kind. Illustrate
this with Fig. 101. 22!). What common articles are levers of the third kind ? What
is said of the sheep-shears ? When docs a door become a lever of the third kind ?



THE WHEEL AND AXLE.



103



The fulcrum, F, is at the elbow-joint ; the biceps muscle, descending from
the upper part of the arm and inserted near the elbow at P, operates as the
power ; while the weight, W, rests on the hand. If the distance F W be 15
times as great as F P, it will take a power of 15 pounds at P to counterbal-
ance one pound at \V ; and when the arm is extended, the disadvantage is
still greater, in consequence of the muscle's not acting perpendicularly to the
bone, but obliquely.

This accounts for the difficulty of holding out a heavy weight at arm's
length. In proportion as power is lost, however, quickness of motion is
gained ; a very slight contraction of the muscle moves the hand through a
comparatively large space with great rapidity. Here, as in all the works of
creation, the wisdom of Providence is shown in exactly adapting the part to
the purpose for which it is designed. With so many external agents at his
command, man does not need any great strength of his own ; quickness of
motion is much more necessary to him, and this the structure of his arm
ensures.

Tfao Wheel and Axle.

221. The Wheel and Axle is the second of the simple
mechanical powers. It consists of a Wheel attached to a
cylinder, or Axle, in such a way that when set in motion
they revolve around the same axis.

222. In the simplest form of the wheel and axle, the
power is applied to a rope passing round the wheel, while



the weight is attached to another
rope passing round the axle.

This form of the machine is shown in Fig. 103.
C D is a frame ; B is the wheel ; A is the axle,
attached to the frame at its extremities E and F
by gudgeons, or iron pins, on which it turns. P
is the power, and W is the weight.

223. The wheel and axle is simply
a revolving lever of the first kind.
One application of the lever can not
move a body any great distance ; but,
by means of the wheel and axle, the
action of the lever is continued unin-



Fig. 103.




T1IE WHEEL AND AXLE.



Under what circumstances does a ladder become a lever of the third kind ? In what
does Nature use levers of the third kind? Show, by Fig. 102, how the fore-arm i.s a
lever, and point out the relation between power and weight. How is the wisdom of
Providence shown, in making the arm such a lever? 221. What is the second sim-
ple mi'chanical power ? Of what does the Wheel and Axle consist ? 222. In the s'm-
\lest form of this machine, how is the power applied, and how the weight ? Ilius-



104 MECHANICS.

terruptedly. This machine has therefore been called the
perpetual or endless lever.

224. The wheel and axle must turn round their common axis in the same
time. In each revolution, a length of rope equal to the wheel's circumfer-
ence is pulled down from the wheel, while only as much rope is wound round
the axle as is equal to the axle's circumference. There is, therefore, a loss
of time, greater or less according as the circumference of the wheel exceeds
that of the axle ; but, by the law of Mechanics already stated, there must be
a corresponding gain of power.

Viewing the wheel and axle as a lever of the first kind, we have the cir-
cumference of the wheel for the long arm, and that of the axle for the short
arm. If the diameters of the wheel and the axle are given instead of their
circumferences, they may be taken for the two arms ; and so with the radii,
if they are given. In practice, an allowance of 10 per cent, of the weight
must be made for the stiffness of the ropes and the friction of the gudgeons.
From these principles is deduced the following law :

225. LAW OF THE WHEEL AND AXLE. With the icheel
and axle, intensity of force is gained, and time is lost, in
proportion as the circumference of the wheel exceeds that of
the axle.

Thus, in Fig. 103, if the circumference of the wheel B is five feet and that
of the axle A one foot, a power of 40 pounds at P will counterbalance a weight
of 200 pounds at W, and of course lift any thing under 200 pounds.

226. DIFFERENT FOEMS. The wheel and axle is exten-
sively used, and assumes a variety of forms.

Fig. 104. Instead of having a rope attached to it, th<

wheel is often provided with projecting pins, a
shown in Fig. 104, to which the hand is directly
applied. This form of the machine is used in
the pilot-houses of steamboats for moving the
rudder. In calculating the advantage in this
case, instead of the circumference of the wheel
we must take the circumference of the circle
described by the point to which the hand is ap-
plied.

A still more common form, much used in drawing water from wells and
loaded buckets from mines, is shown in Fig. 105. Instead of a wheel, we

trate this with Fig. 103. 223. What has the wheel nnd axle been called, and why?
22 i. Explain the operation of the wheel and axle, and show how great the loss of time
and gain of power will be. Viewing the wheel and axle as a lever, what is the lung
arm? What is the short arm ? What, besides the circumference, maybe taken as
the arms of the lever ? "What allowance must be made in practice ? 225. State the law
of the wheel and axle. Illustrate this law with Fig. 103. 226. Describe the form of




CAPSTAN AND WINDLASS.



105



Fig. 105.



FI& 106.





have here a Winch, or handle, attached to the axle.
In this case, to calculate the advantage gained, we
must compare the circle described by the extrem-
ity of the handle (shown in the Figure by a dotted
line) with the circumference of the axle.

Fig. 106 shows a
third form of the
wheel and ^axle.
Here the axle A is

vertical, instead of horizontal. A bar insert-
ed in its head, at the extremity of which the
hand is applied, takes the place of the wheel.
If the circumference of A is 3 feet and the
circle described by P is 12 feet, a power of 1
pound at P will counterbalance a weight of 4
pounds at W.

227. The Capstan. The Capstan (see Fig. 107) is a fa-
miliar example of this form of the' wheel and axle. It is
used by sailors for warping vessels up to a

dock, raising anchors, &c. ; and consists of a
massive piece of timber, round which a rope
passes. This is surmounted by a circular head, // / 1^^3\ \\
perforated with holes, into which, when the in-
strument is to be used, strong bars, called
handspikes, are inserted. Several men may work at each
handspike, pushing it before them as they walk round the
capstan. The handspikes act on the principle of the lever.
The longer they are, therefore, the more easily the men
overcome the resistance, but the further they have to walk
in doing it.

228. The Windlass. This is a similar form of the wheel
and axle, used on shipboard for various purposes.

The windlass is not vertical, like the capstan, but horizontal or parallel
to the deck. It is a round piece of timber, supported at each end, and per-
forated with rows of holes. Pushing against handspikes inserted in these

the wheel and axle used in the pilot-houses of steamboats. In calculating the advan-
tage in this case, what must we substitute for the circumference of the wheel? De-
scribe the form of the machine used in drawing water from wells. How is the ad-
vantage ascertained in this case? Describe a third form of the wheel and axle,
exhibited in Fig. 106. 227. What machine is a familiar example of this third form?
For what is the Capstan used? Of what does it consist? How is it worked? How do
the handspikes act ? 223. "What similar instrument is often substituted for the cap-




THE CAPSTAN.



106



MECHANICS.



holes, the boatmen turn the barrel of the windlass halfway over. It is held
there by a suitable apparatus, till the handspikes are removed and put in a
new row of holes, when the process is repeated. The windlass acts on the
same principle as the capstan, but is less convenient, on account of the man-
ner in which the force is applied, and the necessity of removing the hand-
spikes to new holes from time to time.

229. Wheels enter largely into machinery. The modes
of connecting them will be considered hereafter.

The Pulley.

230. The Pulley is the third of the simple mechanical
Fig. los. powers. It consists of a wheel with a

grooved circumference, over which a rope
passes, and , an axis or pin, round which
the wheel may be made to turn. The
ends of the axis are fixed in a frame
called a bloc7c.

Fig. 103 gives a view of the pulley. A represents
the block, B the axis, and C the wheel. Round the
groove in the wheel passes a rope, at one end of
which the power acts, while the weight is attached
to the other.

231. IVIXDS OF PULLEY. Pulleys are of two kinds,
Fixed and Movable.

232. Fixed Pulleys. A Fixed Pulley is
one that has a fixed block.

- Fig. 109 represents a fixed pulley. The block is at-
tached to a projecting beam. P is the power, and W the
weight. For every inch that P descends, W ascends the
same distance. There is, therefore, no loss of time, and no
gain in intensity of force. One pound at P will just coun-
terbalance one pound at "W.

233. In this rule, as well as all the others pertaining
to the Mechanical Powers, it must be remembered that
friction is not taken into account. In the case of the pul-
ley, in consequence of the stiffness of the rope and the
friction of the pin, an allowance of 20 per cent, of the
weight, and often more, must be made in practice.




THE PULLEY.



Fig. 109.




FIXED m.LEY.



stan? How docs the Windlass differ from the capstan? Ofwin.tdoes the windlass
consist ? How is it worked ? What makes it less convenient than the capstan ?
229. What is said of wheels? 230. What is the third simple mechanical power? Of



FIXED PULLEYS.



107



Fig. 110.



234. Though no power is gained with the fixed pulley,
it is frequently used to change the direction of motion.
The sailor, instead of climbing the mast to hoist his sails,
stands on deck, and by pulling on a rope attached to a
pulley raises them with far less difficulty. With equal ad-
vantage the builder uses a fixed pulley in raising huge
blocks of stone or marble, and the porter in hoist-
ing heavy boxes to the lofts of a warehouse.

235. With two fixed pulleys, horizontal motion
may be changed into vertical ; horses are thus en-
abled to hoist weights, as shown in Fig. 84.

236. Fig. 110 shows how a person may raise
himself from the ground, or let himself down from
a height, by means of a fixed pulley. In lofty
buildings an apparatus of this kind is sometimes
rigged near a window, to furnish means of escape
in case of fire.

237. Movable Pulleys. A Movable Pulley
is one that has a movable block.

Fig. Ill represents a movable pulley. A is the wheel.
One end of the rope is fastened to a support at D, while
the power is applied to the other at P.

238. To raise the weight a given distance with the
movable pulley, the hand must be raised twice that dis-
Fig. 112. tance. Time, therefore, being lost in the

proportion of 2 to 1, the intensity of the
force is doubled. A power of one pound
at P will counterbalance two pounds at
W, and raise anything under two pounds.

239. A movable pulley is
seldom used alone. It is gen- UOVABLE PULLEY -
erally combined with a fixed pulley, as shown
in Fig. 112. No additional power is thus




Fig. 111.





what does the Pulley consist ? What is the Block ? Point out the parts of the pul-
ley in Fig. 103. 231. How many kinds of pulleys are there? 232. What is a Fixed
Pulley? Point out the parts in the Figure. What is the gain with this pulley?
2:):"!. What allowance must be made for friction in the case of the pulley ? 234. If no
power is gained by the use of the fixed pulley, why is it' used? Give examplos.
2:5. How may horizontal motion be changed into vertical ? 236. What does Fig. 110



108



MECHANICS.



Pig. 113. gained ; on the contrary, there is a loss, the
friction of two pulleys being double that of one.
But this loss is more than counterbalanced by
the greater convenience of pulling downward.

240. When a high degree of force is required, several mov-
able and fixed pulleys may be combined, as represented in
Fig. 113. A and B are fixed pulleys; C and D are movable
ones, from the block of which the weight W is suspended.
One end of the rope is attached to the lower extremity of the
fixed block, F ; to the other end the power is applied, after the
rope has passed in succession over each of the four pulleys.

To move W an inch with this combination, each length of
rope must be shortened an inch, and therefore P must move
as many inches as there are lengths of rope. Since there are
two lengths of rope for each movable pulley, we may lay down
the following law :

241. Law of Movable Pulleys. With mov-
able pulleys, a power will balance a iceight as many times
greater tJian itself as twice the number of movable pulleys
employed.

In Fig. 113, a power of 1 pound will balance a weight of 4 pounds. If
three movable pulleys were used, 1 pound at P would balance 6 pounds at W ;
if four were used, 8 pounds, &c. Friction, however, nullifies much of this
gain.

242. White's Pulley. To lessen the friction, when a
number of pulleys are required, the wheels are made to
turn on the same axis. This is effected by having but one
block for all the upper pulleys, and one for the lower ;
grooves being cut in each, to take the place of separate
wheels. The friction in each block is thus reduced to that
of a single wheel. This system is called, from its inventor,
White's Pulley.

Fig. 114 gives a front and a side view of White's Pulley. A is the fixed

show ? For what is an apparatus of this kind sometimes used ? 237. What is a Mov-
able Pulley? Describe it with Fig. 111. 233. To move a weight a given distance
with a movable pnlley, how far must the power travel ? What, then, is the law of
this machine 9 239. With what is a movable pulley generally combined? What is
gained by this combination ? 240. Describe the combination of movable and fixed pul-
leys represented in Fig. 113. 241. What is the law of movable pulleys? Apply this
law in the case of the pulley represented in Fig. 113. By what i* much of this gain
nullified ? 242. When a nmnber of pulleys are required, how is the friction lessened ?



MOVABLE PULLEYS.



109



Fig. 114.



block, with grooves of different sizes representing the separate wheels. B is

the movable block, similarly prepared. A single rope is used, which is

fastened at one end to the smallest fixed

p'Uley, and acted on by the power at the

other. Here again, if friction is left out of

account, the power will counterbalance a

weight as many times greater than itself as

twice the number of movable pulleys. In

Fig. 114 there are six movable pulleys ;

consequently, with a pressure of 1 pound

nt P, equilibrium will be established when

W is twice six, or 12, pounds.

243. Fig. 115 shows another

system of movable pulleys, each

of which has a separate rope of

its own attached at one end to a

fixed support.

Fig. 115. To raise the low-

est pulley, A, and the
weight suspended
from it one inch, two
inches of its rope
must be pulled up.
This is done by pull-
ing up twice 2, or 4, inches of B's rope ; and this, i
turn, by pulling up twice 4, or 8, inches of C's rope-
P, therefore, must descend 8 inches, to raise W one inch.
If there were four movable pulleys, P would have to de-
scend 16 inches to raise W one inch ; if 5, 32 inches, and
so on, P's distance doubling for each new pulley add-
ed. Hence, with this combination, the power balances a
weifjJtt as many times greater than itself as 2 raised to the
power denoted by the number of movable pulleys.

244. The pulley is so cheap and conve'
nient that it is much used in its simple forms. In com-
plicated systems, more than half the advantage is lost by
friction and the stiffness of the ropes ; and consequently
such systems are used only when immense weights are to
be raised.





WHITE'S PTTLLEY.



What pulloy is constructed on this principle? Describe White's Pulley. With
White's Pulley, what is the gain ? 243. Describe the system of pulleys represented in
Fig. 115. Explain its operation. What is the gain with this system ? 244. What is



110



MECHANICS.




The Inclined Plane.

245. The Inclined Plane is the fourth of the simple me-
chanical powers. It is a plane surface, inclined to the ho-
rizon at any angle. Every road not perfectly level is an
inclined plane.

Fig. 116. A D, in Fig. 116, is an inclined plane,

of which AC is the length, AB the



Online LibraryG. P. (George Payn) QuackenbosA natural philosphy: embracing the most recent discoveries in the various branches of physics .. → online text (page 10 of 42)