would have to draw obliquely downward, as well as forward, aiid
thus expend part of his strength in drawing against the ground.*
* In descending a steep hill, the wheels of a carriage are often locked (as
it is called), that is, fastened in such a manner as to prevent their turning;
and thus the rolling is converted into the sliding friction, and the vehicle
descends more safely.
Castors are put on the legs of tables and other articles of furniture to
facilitate the moving of them ; and thus the sliding is converted into tha
86 NATURAL PHILOSOPHY.
310. PRACTICAL EXAMPLES OF POWER APPLIED TO THE \A IIREL AND AJL.
Questions for Solution.
(1.) With a wheel 5 feet in diameter and a power of 6 pounds, whaJ
nust be the diameter of the axle to support 3 cwt. 1 Ans. 1.2 in.
(2.) How large must be the diameter of the wheel to support with 10
lb~ a weight of 5 cwt. on an axle inches in diameter 1 Ans. 81.5ft.
(3.) A wheel has a diameter of 4 feet, an axle of 6 inches. What power
must be applied tf the wheel to balance 2 cwt. on the axle ? Ans. 25 Ib.
(4.) There is a connexion of cogged wheels having 6 leaves on the pinion
and 36 cogs on the wheel. What is the proportion of the power to the
weight in equilibrium * Ans. As 1 to 6.
(5.) Suppose a lever of six feet inserted in a capstan 2 feet in diameter,
and six men whose united strength is represented by i of a ton at the capstan,
how heavy an anchor can they draw up, allowing the loss of of their powe?
from friction } Ans. 2 T.
(6.) What must be the proportion of the axle to the wheel, to sustain a
weight 30 cwt. with a power of 3 cwt. 1 Ans. As 1 to 10.
(7.) The weight is to the power in the proportion of six to one. What
must bo the proportion of the wheel to the axle 1 Ans. 6 to 1.
(8.) The power is represented by 10, the axle by 2. How can you repre-
sert the wheel and axle 1 An*. 10 : weight:: 2 : wheel.
(9.) The weight is expressed by 15, the power by 3. What will repre-
sent the wheel and axle 7 Ans. 5 and 1.
(10.) The axle is represented by 16, the power by 4. Required the pro
portion of the wheel and axle. Ans. 4: weight:: 16: wheel.
(11.) What is the weight of an anchor requiring 6 men to weigh it, by
means of a capstan 2 feet in diameter, with a lever 8 feet long, 2 feet of ita
length being inserted in the capstan ; supposing the power of each man to
be represented by 2 cwt., and a loss of the power by friction? Ans. b&cwt.
(12.) A stone weighing 2 tons is to be raised by a windlass with spoked
2 feet in length, projecting from an axle 9 inches in diameter. How many
men must be employed, supposing each man's power equal to 2 cwt., and the re-
bistance increased j by friction ? Ana, 5 men.
What is a 311. THE PULLEY. The Pulley is a small
Pulley 7 wheel turning on an axis, with a string or rope
in a groove running around it.
How many kinds There ftre twQ kindg Q f pu lleys the
of pulleys are ?
there i fixed and the movable. The fixed pulley
is a pulley that has no other motion than a revolution on
its axis, and it is used only for changing the direction of
Explain 312. Fig. 46 represents a fixed pulley. P is a
"# small wheel turning on its axia, with a string running
round it in a groove. W is a weight to be raised, F is the force
t>r power applied. It is evident that, by pulling the string at
F, the weight must rise just as much as the string is drawn
THE MECHANICAL FOWU128. S7
down. As, therefore, the velocity of the weight and the m%. 46
power is precisely the same, it is manifest that they
balance each other, and that no mechanical advantage
is gained.* But this pulley is very useful for changing
the direction of motion. If, for instance, we wish to
raise a weight to the top of a high building, it can be done
with tbe assistance of a fixed pulley, by a man standing
below. A curtain, or a sail, also, can be raised by means of a
fixed pulley, without ascending with it, by drawing down a string
running over the pulley.
On what prin- 313. The fixed pulley operates on the same
tiple does the principle as a lever of the first kind with equal
fixed pulley act? armgj where the f u]crum being in the centre
of gravity, the power and the weight are equally distant from it.
and no mechanical advantage is gained.
314. The movable pulley differs from
How does the *
movable pulley the fixed pulley by being attached to Fig. 47.
differ from the tne we ight ; it therefore rises and
falls with the weight.
Explain 315. Fig. 47 represents a movable pull ny,
Fig. 47. w ith the weight W attached to it by a hook
below. One end of the rope is fastened at F ; and, as
the power P draws the weight upwards, the pulley
rises with the weight. Now, in order to raise the
weight one inch, it is evident that both sides of the string
* Although the fixed pulley gives no direct mechanical advantage, a
man may advantageously use his own strength by the use of it. Thus, if
he seat himself on a chair suspended from one end of a rope passing over a
fixed pulley, he may draw himself up by the other end of the rope by exert-
ing a force equal only to one-half of his own weight. One half of his weight
is supported by the chair and the other half by his hands, and the effect is
tne same as if he drew only one half of himself at a time; for, the rope being
doubled across the pulley, two feet of the rope must pass Ihrough his hands
before he can raise himself one foot. In this manner laborers and others
frequently descend into wells, and from the upper floors of stores, by meaue
of a rope passing over a Cxei' wheel ( f pulley.
88 NATURAL PHILOSOPHY.
must be shortened ; in order to do which, the power P rnunt
pass over two inches. As the velocity of the power is double
that of the weight, it follows that a power of one pound will bal-
ance a weight on the movable pulley of two pounds. 1 *
What is the ad- 316. The power gained by the use of pul-
vantage gained , . t . , , , . , . .
in the use of the ^ e J s 1S ascertained by multiplying the num-
movable pulley ? ber of movable pulleys by 2.f
317. A weight of 72 pounds may be balanced by a power of 9
pounds with four pulleys, by a power of 18 pounds with two pul-
leys, or by a power of 3G pounds with one pulley. But in each
case the space passed over by the power must be double the space
passed over by the weight, multiplied by the number of movable
pulleys. That is, to raise the weight one foot, with one pulley, the
power must pa^s over two feet, with two pulleys four feet, with
four pulleys eight feet.
Explain 318. Fig. 48 represents a sy.stem of fixed and
8 ' movable pulleys. In the block F there
are four fixed pulleys, and in the block M there
are four movable pulleys, all turning on their com-
mon axis, and rising and falling with the weight
W. The movable pulleys are connected with the
fixed ones by a string attached to the hook H,
passing over the alternate grooves of the pulleys
in each block, forming eight cords, and terminating
at the power P. Now, to raise the weight one foot,
it is evident that each of the eight cords must be
Thus, it is seen that pulleys act on the same principle with the lever
and the wheel and axle, the deficiency of the strength of the power being
compensated by superior velocity. Now, as we cannot increase our natural
strength, but can increase the velocity of motion, it is evident that we are
enabled, by pulleys, and other mechanical powers, to reduce the resistance
or weight of any body to the level of our strength.
f This rule applies only to the movable pulleys in the same block, or
when the parts of the rope which sustains the weight are parallel to each
other. The mechanical advantage, however, which the pulley seems to possess
in theory, is considerably diminished in practice by the stiffness of the ropes
aud the fricMon of the wheels and blocks. When the parts of the cord,
also, are not parallel, the pulley becomes less efficacious ; and when the
parts of the cord which supports the weight very widely depart from par-
allelism, the pulley becomes wholly useless. There are certain arrange-
ments of the Jord aud the pulley by which the effective power of tb
THE MECHANICAL POWEKS. 89
shortened one foot, and, consequently, that the power I* must
descend eight times that distance. The power, therefore, must
pass over eight times the distance that the weight moves.
319. The movable pulley, as well as the fixed, acts on the same
principle with the lever, the deficiency of the strength of the
power with the movable pulley being compensated by its superior
On what princi- 32 - The fixed P ulle J acts on the principle of
pie is the mov- a lever with equal arms. [See No. 313.] The
structed? ^ ' mova kl e P u ^ e y> on * ne contrary, by giving a
superior velocity to the power, operates like a
lever with unequal arms.
321. Practical use of Pulleys. Pulleys are used to raise goods
into warehouses, and in ships, &c., to draw up the sails. Both kind?
of pulleys are in these causes advantageously applied : for the sails
are raised up to the xuasts by the sailors on deck by means of the
fixed pulleys, while the labor is facilitated by the mechanical power
of the movable ones.
322, Both fixed and movable pulleys are constructed in a great
variety of forms, but the principle on which all kinds are con-
structed is the same. What is generally called a tackle and fall ',
or a block and tackle, is nothing more than a pulley. Pulleys have
likewise lately been attached to the harness of a horse, to enable
the driver to govern the animal with less exertion of strength
323. It may be observed, in relation to the Me-
What law ap- . J ,
plies to all the chanical Powers in general, that power is always
Mechanical* gained at the expense of time and velocity ; thai
is, the same power which will raise one pound in
one minute will raise two pounds in two minutes, six pounds in
six minutes, sixty pomids in sixty minutes, <J-c. : and that the
same quantity of force used to raise two pounds one foot will
raise one, pound two feet, fyc. And, further, it may be stated
that the product of the weight multiplied by the velocity of the
weight will always be equal to the product of the power multi
plied by tke velocity of the power.
Siihey mny be augmented in a three-fold instead of a two-fold proportion.
u.t when such an advantage is secured, it must be by contriving to make
the power pass over three times the space of the weight.
* See Appendix.
'JO NATURAL PHILOSOPHY.
In what proper- Hence we have the following rule . The
tton is the power 7
to the weight P ower ls M **& same proportion to trie
when the mov- weight as the velocity of the weight is tn
able pulley is j7 , ., - 7
used i Me velocity of the power.*
324. PRACTICAL EXAMPLES OF APPLICATION OF THE PULLET.
Questions for Solution.
(1.) Suppose a power of 9 Ibs. applied to a set of 3 movable pulleys. Ai
lowing } loss for friction, what weight can be sustained by them 1 A. 36 Ib.
(2.) Six movable pulleys are attached to a weight of 1800 Ibs.; what
power will support them, allowing a loss of two-thirds of the power from
friction 1 Ans. 4,50 Ib.
(3.) Six men, with a block and tackle containing nine movable pulltys,
ars required to raise a sail. Suppose each man's strength to be represented
by two cwt. and two-thirds of the power lost by friction, what is the
weight of the sail, with its appendages 1 Ans. 72 cwt.
(4.) If a stone weighing 3 tons is to be raised by horse power to the wall
of a building in process of erection, by means of a derrick from which are
suspended 3 movable pulleys, how many horses must be employed, sup-
posing each horse capable of drawing as much as eight men, each of whom
can lift 2 cwt., making an allowance of two-thirds for friction 1 Ans. 1. '
(5.) A block contains 5 movable pulleys, connected with a beam contain-
ing 5 fixed pulleys. A weight of half a ton is to be raised. Allowing a loss
of two- thirds for friction, what power must be applied to raise it 1 A. 3 cwt.
(7.) The power is 3, the weight is 27; how many pulleys murft be usea,
if friction requires an allowance of two-thirds 1 Ana. 27,
(8.) Friction one-third of the power, power 6, weight 72, how many pul-
leys 1 Ans. IS.
(9.) Weight 84, friction nothing, pulleys, 3 fixed, 3 movable ; required
the power. Ans. 14
(10.) Power 12, friction 8, four pulleys, two of them fixed ; required the
weight. Ans. 16.
(11.) Six movable and six fixed pulleys. The weight is raised 3 feet.
How far has the power moved '! Ans. 36/t
,12.) The power has moved 12 feet ; how far has the weight moved un-
der two pulleys, one fixed, the other movable 1 Ans. &ft.
(13.) The weight, suspended from a fixed pulley, has moved 6 feet. How
far has the power moved 1 Ans. 6ft.
(14.) The power has moved 2-0 feet under a fixed pulley ; how far haa
the weight moved * Ans. 20ft.
What is the In- 325. THE INCLINED PLANE. The In-
dined Plane? c j ine(i Pl ane consists of a hard plain surface,
inclined to the horizon.
326. The principle on which the inclined plane acts as a me-
"rmnical power is simply the fact that it supports part of the weight.
[f a body be placed on a horizontal plane, its whole weight will be
* The stiffness of the cords and the friction of the blocks frequentlj
require large deduction to be made from the effective power of pulleys
TMu loss thus occasioned will sometimes amount to two-thirds of the p'.vrcr
THE MECHANICAL POWERS. 91
supported , but, if the plane be elevated at one end, by degrees, it
will support less of the weight in proportion to the elevation, until
the plane becomes at right angles to the horizon, when it will sup-
port no part of the weight, and the body will fall perpendicularly.
^27. A body, in ascending or descending an inclined plane, wiL
ha ye a greater space to traverse than if it should rise or fall per-
pendicularly. The time, therefore, of its ascent or descent will be
longer, and thus it will oppose less resistance, and thus, also, a less
force will be required to cause its ascent. Hence, we see that the
fundamental principle of Mechanics, " What is gained in power is
lost in time," applies to the Inclined Plane as well as to the Me-
chanical Powers that have already been described.
What is the ad- 328. The advantage gained by the use of
vantage gained , ,. , ,
by the use of the tne inclined plane is in proportion as the
inclined plane ? length of the plane exceeds its perpen-
Fig. 49 represents an inclined plane. C A its height, C B
its length, and W a weight which is to be K 49
moved on it. If the length C B be four
times the height C A then a power of one
pound at G will balance a weight of four
pounds on the inclined plane C B.
329. The greater the inclination of the plane, the greater must
be its perpendicular height, compared with its length ; and, of
course, the greater must be the power to elevate a weight along its
330. Instances of the application of the inclined plane are very
common. Sloping planks or pieces of timber leading into a cellar,
and on which casks are rolled up and down ; a plank or board with
one end elevated on a step, for the convenience of trundling wheel-
barrows, or rolling barrels into a store, &c., are inclined planes.
331. Chisels and other cutting instruments, which are cham-
fered, or sloped only on one side, are constructed on the principle
of the inclined plane.*
332. Roads which are not level may be considered as inclined
planes, and the inclination of the road is estimated by the height
corresponding to some proposed length. To raise a load up an
inclined plane requires a power sufficient to carry it along the
whole distance of the length of the base, and then to lift it up to
* Chisels for cutting wood should have their edges at an angle of about
30 ; for cutting ; ron from 50 J to 60, and for cutting brass at about 80 or
90. Tools urged by pressure may be sharper than those which, like the
wedge, are diiven by percussion
93 NATUKAL PHILOSOPHY.
the elevation ; but in the inclined plane a feebler force will accom-
plish the desired object, because the resistance is spread equally
over, the whole distance.*
What is the 333. THE WEDGE. The Wedge consists
ff;|7 7 *
of two inclined planes united at their bases.
What is the ad- 334. The advantage gained by the wedge
ls in Proportion, as its length exceeds the
thickness between the converging sides.
In what pro-
portion is the It follows that the power of the wedge ia in pio-
poiver of the portion to its sharpness.
835. Fig. 50 represents a wedge. The line a b
represents the base of each of the inclined planes
of which it is composed, and at which they are
336. The wedge is a very important mechanical power, used to
split rocks, timber, &c., which could not be effected by any othei
337. Axes, hatchets, knives, and all other cutting instruments,
chamfered, or sloped on both sides, are constructed on the principle
of the wedge; also pins, needles, nails, and all piercing instru-
On what does 338. The effective power of the wedge depends
the on friction 5 for tf there were no Diction, the
wedge depend ? wedge would fly back after every stroke.
* Mention has already been made of the sagacity of animals in a former
page [see No. 54], and a sort of intuitive knowledge which they appear
to possess of philosophical principles. In ascending a steep hill, a common
dray-horse will drag his load from side to side, as if he were conscious that
be thus made the plane longer in proportion to its height, and thereby
made his load the lighter.
t The wedge is an instrument of exceedingly effective power, and la
frequently used in presses for extracting the juice of seeds, fruits, <fco. It
Is used especially in the oil mill, by which the oil is extracted from seeds.
The seeds are placed in hair bags, between planes of hard wood, which are
pressed together by wedges. The pressure thus exerted is so intense that
the seeds, after the extraction of the -oil, are converted into masses as hard
and compact as the most dense woods.
Wedges are used also in the launching of vessels, and also for restoring
buildings to the perpendicular which have been inclined by the sinking of
THE MECHANICAL POWERS. 93
339. The wedge derives much of its efficiency from the force of
percussion, which in its nature is so different from continued force,
such as the pressure of weights, the force of springs, &c., that it
would be difficult to submit it to numerical calculation ; and, there-
fore, we cannot properly represent the proportion which a blow
bears to the weight.
What: 'Ufa 340. THE SCREW. The Screw is an in-
Screw? clined plane wound around a cylinder, thus
producing a circular inclined plane, forming what is called
the threads of the screw.
341. Cut a piece of paper in the shape of an inclined plane, as
represented by Fig. 49, and, beginning with the end represented
by the height C A, in that Figure, wind it around a pencil, or a
round ruler. The edge of the paper will be a circular inclined plane,
and will represent the threads of the screw. The distance between
any two threads on the same side of the rule will represent the per-
pendicular height of the inclined plane that extends once around the
cylinder, and the advantage gained in the use of the screw (when
used without a lever) will be the same as in the inclined plane ;
namely, as the length of the plane exceeds the perpendicular height.
But the screw is seldom used alone. A lever is generally attached
to the screw, and it is with this attachment the screw will now be
342. The Screw is generally accompanied
What appendage J
generally attends by an appendage called the nut, which consists
the Screw ? O f a concave cylinder or block, with a hollow
spiral cavity cut so as to correspond exactly with the threads of
the screw. When thus fitted together, the screw and the nut
form two inclined planes, the one resting on the other.
343. Sometimes the screw is movable and
Is the screw, or
the nut mov- the nut is stationary, and sometimes the screw
able ? is stationary and the nut is movable.
344. At every revolution the screw or the nut advances or
retreats through a *Dace equal to the distance between the threads
of the screw.
In what manner 345. The power applied to a screw gener-
% e plied e to W the al l v Describes a circle around the screw,
screw move? perpendicular to the direction in which
the screw or nut moves.
\Tn<A is the advan- 346 The advantage gained by the
* gained by the screw is in proportion as the circumfer-
ence described by the power exceeds the
distance between the threads of the screw.
What is meant by 347. The cylinder with its threads is called
the Convex and the Convex Screw, and the nut is called the
Concave Screw? /, a m , ,
Concave bcrew. The lever is sometimes at-
tached to the screw, and sometimes to the nut.
Explain 348. Fig. 51 represents a fixed screw
*&' ' S, with a movable nut N, to which is
attached the lever L. By turning the lever in one
direction the nut descends, and by turning it in the
opposite direction the nut ascends, at every revo-
lution of the lever, through a space equal to the dis-
tance between the threads of the screw ; to accomplish which, the
hand or power applied to the end of the lever L will describe a
circle around the sorew S, of which the radius is L S. The
power thus passes over a space represented by the circumfer-
ence of this circle, and the advantage gained is in the same pro-
portion as the space exceeds the distance between each threa
of the screw
Explain 349. Fig. 52 represents a movable
I ^ P " screw, with a nut fixed in a frame, and
consequently immovable. As the lever L is
turned, the screw ascends or descends at every
revolution of the lever through a space equal to
the distance between the threads of the screw, and
the advantage gained is in the same proportion as in the case of
the movable nut in Fig. 51.
350. It will thus be seen that, although the screw is usually con-
sidered distinctly as a mechanical power, it is in fact a compound
power, consisting of two circular inclined planes, moved by a lever.
351. The power of the screw being estimated by the distance
between the threads, it follows that the closer the threads are
toother, the greater will be the power, but the slower will be the
motion produced : lor. every revolution of the lever advances the
screw 01 the nut only through a space as great as the distance of the
threads from each other.
352 The screw is applied to presses and engines of all kinds
where great power is to be applied, without percussion, through
small distances It is used in bookbinders' presses, in oider and
wme presses, in raising buildings. It is also used for
coining, and for punching square or circular holes
through thick plates of metal. When used for this
purpose, the lever passes through the head of the
screw and terminates at both ends with heavy
balls or weights, the momentum of which adds to
the force of the screw, and invests it with immense
353. HUNTER'S SCREW. The ingenious contrivance known by
the name of Hunter's Screw consists of two screws of different
threads playing one within the other ; and such will be the effect, that
while one is advancing forward the other will retreat, and the resist-
ance will be urged forward through a distance equal only to the
difference between the threads of the two screws. An indefinite
increase in the power is thus obtained, without diminishing the
thread of the screw.*
* From what has been stated with regard to the Mechanical Powers, it
appears that by their aid a man is enabled to perform works to which hif
unassisted natural strength is wholly inadequate. But the power of all
machines is limited by the strength of the materials of which they are com-
posed. Iron, which is the strongest of all substances, will not resist a strain
beyond a certain limit. Its cohesive attraction may be destroyed, acd it
can withstand no resistance which is stronger than its cohesive attraction.
Besides the strength of the materials, it is necessary, also, to consider the
time which is expended in the application of mechanical assistance. Archim-
edes is said to have boasted to Hiero, King of Syracuse, that, if he would