used, unless absolutely necessary, as is the case in raising a ladder in
order to place it against a wall; the man who raises it cannot place his
hands on the upper part of the ladder, the power, therefore, is
necessarily placed much nearer to the fulcrum than to the weight.
_Caroline._ Yes, the hands are the power, the ground the fulcrum, and
the upper part of the ladder the weight.
_Mrs. B._ Nature employs this kind of lever in the structure of the
human frame. In lifting a weight with the hand, the lower part of the
arm becomes a lever of the third kind; the elbow is the fulcrum, the
muscles of the fleshy part of the arm, the power; and as these are
nearer to the elbow than to the hand, it is necessary that their power
should exceed the weight to be raised.
_Emily._ Is it not surprising that nature should have furnished us with
such disadvantageous levers?
_Mrs. B._ The disadvantage, in respect to power, is more than
counterbalanced by the convenience resulting from this structure of the
arm; and it is that no doubt which is best adapted to enable it to
perform its various functions.
There is one rule which applies to every lever, which is this: In order
to produce an equilibrium, the power must bear the same proportion to
the weight, as the length of the shorter arm does to that of the longer;
as was shown by Emily with the weights of 1 _lb._ and of 3 _lb._ Fig. 3.
plate 4.
We have dwelt so long on the lever, that we must reserve the examination
of the other mechanical powers, to our next interview.
Questions
1. (Pg. 54) How many mechanical powers are there, and what are they
named?
2. (Pg. 54) What is a mechanical power defined to be?
3. (Pg. 54) What four particulars must be observed?
4. (Pg. 54) Upon what will the velocities depend?
5. (Pg. 55) What is a lever?
6. (Pg. 55) Give a familiar example.
7. (Pg. 55) When and why do the scales balance each other, and where is
their centre of gravity? (fig. 1. plate 4.)
8. (Pg. 55) Why would they not balance with unequal weights?
9. (Pg. 55) Were the fulcrum removed from the middle of the beam what
would result?
10. (Pg. 55) What do we mean by the arms of a lever?
11. (Pg. 56) How may a pair of scales be false, and yet appear to be
true?
12. (Pg. 56) If the fulcrum be removed from the centre of gravity, how
may the equilibrium be restored?
13. (Pg. 56) How is this exemplified by fig. 3. plate 4?
14. (Pg. 56) What proportion must the weights bear to the lengths of the
arms?
15. (Pg. 57) On what principle do we weigh with a pair of steelyards,
and what will be the difference in the motion of the extremities of such
a lever?
16. (Pg. 58) How is this exemplified by fig. 4. plate 4?
17. (Pg. 58) What line is described by the ends of a lever? fig. 4.
plate 4.
18. (Pg. 58) How many kinds are there; and in the first how is the
fulcrum situated?
19. (Pg. 58) When may the fulcrum be so situated that this lever is not
a mechanical power, and why?
20. (Pg. 59) What is represented by fig. 5. plate 4?
21. (Pg. 59) Give a familiar example of the use of a lever of the first
kind.
22. (Pg. 59) In what instruments are two such levers combined?
23. (Pg. 59) How may two horses of unequal strength, be advantageously
coupled in a carriage?
24. (Pg. 60) Describe a lever of the second kind. (Fig. 6. plate 4.)
25. (Pg. 60) What is represented in fig. 7. plate 4, and in what
proportion does this lever gain power?
26. (Pg. 60) What is said respecting a door?
27. (Pg. 60) Describe a lever of the third kind.
28. (Pg. 60) In what instance do we use this?
29. (Pg. 61) What remarks are made on its employment in the limbs of
animals?
30. (Pg. 61) What are the conditions of equilibrium in every lever?
CONVERSATION V.
CONTINUED.
ON THE MECHANICAL POWERS.
OF THE PULLEY. OF THE WHEEL AND AXLE. OF THE INCLINED PLANE. OF THE
WEDGE. OF THE SCREW.
MRS. B.
The pulley is the second mechanical power we are to examine. You both, I
suppose, have seen a pulley?
_Caroline._ Yes, frequently: it is a circular, and flat piece of wood or
metal, with a string which runs in a groove round it: by means of which,
a weight may be pulled up; thus pulleys are used for drawing up
curtains.
_Mrs. B._ Yes; but in that instance the pulleys are fixed; that is, they
retain their places, and merely turn round on their axis; these do not
increase the power to raise the weights, as you will perceive by this
figure. (plate 5. fig. 1.) Observe that the fixed pulley is on the same
principle as the lever of a pair of scales, in which the fulcrum F being
in the centre of gravity, the power P and the weight W, are equally
distant from it, and no advantage is gained.
_Emily._ Certainly; if P represents the power employed to raise the
weight W, the power must be greater than the weight in order to move it.
But of what use then is a fixed pulley in mechanics?
_Mrs. B._ Although it does not increase the power, it is frequently
useful for altering its direction. A single fixed pulley enables us to
draw a curtain up, by pulling the string connected with it downwards;
and we should be at a loss to accomplish this simple operation without
its assistance.
_Caroline._ There would certainly be some difficulty in ascending to the
head of the curtain, in order to draw it up. Indeed I now recollect
having seen workmen raise weights to a considerable height by means of a
fixed pulley, which saved them the trouble of going up themselves.
_Mrs. B._ The next figure represents a pulley which is not fixed; (fig.
2.) and thus situated, you will perceive that it affords us mechanical
assistance.
A is a moveable pulley; that is, one which is attached to the weight to
be raised, and which consequently moves up or down with it. There is
also a fixed pulley D, which is only of use to change the direction of
the power P. Now it is evident that the velocity of the power, will be
double that of the weight W; for if the rope be pulled at P, until the
pulley A ascends with the weight to the fixed pulley D, then both parts
of the rope, C and B, must pass over the fixed pulley, and consequently
the hand at P, will have descended through a space equal to those two
parts; but the weight will have ascended only one half of that distance.
_Caroline._ That I understand: if P drew the string but one inch, the
weight would be raised only half an inch, because it would shorten the
strings B and C half an inch each, and consequently the pulley with the
weight attached to it, can be raised only half an inch.
_Emily._ But I do not yet understand the advantage of moveable pulleys;
they seem to me to increase rather than diminish the difficulty of
raising weights, since you must draw the string double the length that
you raise the weight; whilst with a single pulley, or without any
pulley, the weight is raised as much as the string is shortened.
_Mrs. B._ The advantage of a moveable pulley consists in dividing the
difficulty; we must, it is true, draw twice the length of the string,
but then only half the strength is required that would be necessary to
raise the weight without the assistance of a moveable pulley.
_Emily._ So that the difficulty is overcome in the same manner as it
would be, by dividing the weight into two equal parts, and raising them
successively.
_Mrs. B._ Exactly. You must observe, that with a moveable pulley the
velocity of the power, is double that of the weight; since the power P
(fig. 2.) moves two inches whilst the weight W moves one inch; therefore
the power need not be more than half the weight, to make their momentums
equal.
_Caroline._ Pulleys act then on the same principle as the lever; the
deficiency of weight in the power, being compensated by its superior
velocity, so as to make their momentums equal.
_Mrs. B._ You will find, that all gain of power in mechanics is founded
on the same principle.
_Emily._ But may it not be objected to pulleys, that a longer time is
required to raise a weight by their aid, than without it? for what you
gain in power, you lose in time.
_Mrs. B._ That, my dear, is the fundamental law in mechanics: it is the
case with the lever, as well as the pulley; and you will find it to be
so with all the other mechanical powers.
_Caroline._ I do not see any advantage in the mechanical powers then, if
what we gain by them in one way, is lost in another.
_Mrs. B._ Since we are not able to increase our natural strength is not
any instrument of obvious utility, by means of which we may reduce the
resistance or weight of any body, to the level of that strength? This
the mechanical powers enable us to accomplish. It is true, as you
observe, that it requires a sacrifice of time to attain this end, but
you must be sensible how very advantageously it is exchanged for power.
If one man by his natural strength could raise one hundred pounds only,
it would require five such men to raise five hundred pounds; and if one
man performs this by the help of a suitable engine, there is then no
actual loss of time; as he does the work of five men, although he is
five times as long in its accomplishment.
You can now understand, that the greater the number of moveable pulleys
connected by a string, the more easily the weight is raised; as the
difficulty is divided amongst the number of strings, or rather of parts
into which the string is divided, by the pulleys. Two, or more pulleys
thus connected, form what is called a tackle, or system of pulleys.
(fig. 3.) You may have seen them suspended from cranes to raise goods
into warehouses.
_Emily._ When there are two moveable pulleys, as in the figure you have
shown to us, (fig. 3.) there must also be two fixed pulleys, for the
purpose of changing the direction of the string, and then the weight is
supported by four strings, and of course, each must bear only one fourth
part of the weight.
_Mrs. B._ You are perfectly correct, and the rule for estimating the
power gained by a system of pulleys, is to count the number of strings
by which the weight is supported; or, which amounts to the same thing,
to multiply the number of moveable pulleys by two.
In shipping, the advantages of both an increase of power, and a change
of direction, by means of pulleys, are of essential importance: for the
sails are raised up the masts by the sailors on deck, from the change of
direction which the pulley effects, and the labour is facilitated by the
mechanical power of a combination of pulleys.
[Illustration: PLATE V.]
_Emily._ But the pulleys on ship-board do not appear to me to be united
in the manner you have shown us.
_Mrs. B._ They are, I believe, generally connected as described in
figure 4, both for nautical, and a variety of other purposes; but in
whatever manner pulleys are connected by a single string, the mechanical
power is the same.
The third mechanical power, is the wheel and axle. Let us suppose (plate
6. fig. 5) the weight W, to be a bucket of water in a well, which we
raise by winding round the axle the rope, to which it is attached; if
this be done without a wheel to turn the axle, no mechanical assistance
is received. The axle without a wheel is as impotent as a single fixed
pulley, or a lever, whose fulcrum is in the centre: but add the wheel to
the axle, and you will immediately find the bucket is raised with much
less difficulty. The velocity of the circumference of the wheel is as
much greater than that of the axle, as it is further from the centre of
motion; for the wheel describes a great circle in the same space of time
that the axle describes a small one, therefore the power is increased in
the same proportion as the circumference of the wheel is greater than
that of the axle. If the velocity of the wheel is twelve times greater
than that of the axle, a power twelve times less than the weight of the
bucket, would balance it; and a small increase would raise it.
_Emily._ The axle acts the part of the shorter arm of the lever, the
wheel that of the longer arm.
_Caroline._ In raising water, there is commonly, I believe, instead of a
wheel attached to the axle, only a crooked handle, which answers the
purpose of winding the rope round the axle, and thus raising the bucket.
_Mrs. B._ In this manner (fig. 6;) now if you observe the dotted circle
which the handle describes in winding up the rope, you will perceive
that the branch of the handle A, which is united to the axle, represents
the spoke of a wheel, and answers the purpose of an entire wheel; the
other branch B affords no mechanical aid, merely serving as a handle to
turn the wheel.
Wheels are a very essential part of most machines; they are employed in
various ways; but, when fixed to the axle, their mechanical power is
always the same: that is, as the circumference of the wheel exceeds that
of the axle, so much will the energy of the power be increased.
_Caroline._ Then the larger the wheel, in proportion to the axle, the
greater must be its effect?
_Mrs. B._ Certainly. If you have ever seen any considerable mills or
manufactures, you must have admired the immense wheel, the revolution of
which puts the whole of the machinery into motion; and though so great
an effect is produced by it, a horse or two has sufficient power to turn
it; sometimes a stream of water is used for that purpose, but of late
years, a steam-engine has been found both the most powerful and the most
convenient mode of turning the wheel.
_Caroline._ Do not the vanes of a windmill represent a wheel, Mrs. B.?
_Mrs. B._ Yes; and in this instance we have the advantage of a
gratuitous force, the wind, to turn the wheel. One of the great benefits
resulting from the use of machinery is, that it gives us a sort of
empire over the powers of nature, and enables us to make them perform
the labour which would otherwise fall to the lot of man. When a current
of wind, a stream of water, or the expansive force of steam, performs
our task, we have only to superintend and regulate their operations.
The fourth mechanical power is the inclined plane; this is generally
nothing more than a plank placed in a sloping direction, which is
frequently used to facilitate the raising of weights, to a small height,
such as the rolling of hogsheads or barrels into a warehouse. It is not
difficult to understand, that a weight may much more easily be rolled up
a slope than it can be raised the same height perpendicularly. But in
this, as well as the other mechanical powers, the facility is purchased
by a loss of time (fig. 7;) for the weight, instead of moving directly
from A to C, must move from B to C, and as the length of the plane is to
its height, so much is the resistance of the weight diminished.
_Emily._ Yes; for the resistance, instead of being confined to the short
line A C, is spread over the long line B C.
_Mrs. B._ The wedge, which is the next mechanical power, is usually
viewed as composed of two inclined planes (fig. 8:) you may have seen
wood-cutters use it to cleave wood. The resistance consists in the
cohesive attraction of the wood, or any other body which the wedge is
employed to separate; the advantage gained by this power is differently
estimated by philosophers; but one thing is certain, its power is
increased, in proportion to the decrease of its thickness, compared with
its length. The wedge is a very powerful instrument, but it is always
driven forward by blows from a hammer, or some other body having
considerable momentum.
_Emily._ The wedge, then, is rather a compound than a distinct
mechanical power, since it is not propelled by simple pressure, or
weight, like the other powers.
_Mrs. B._ It is so. All cutting instruments are constructed upon the
principle of the inclined plane, or the wedge: those that have but one
edge sloped, like the chisel, may be referred to the inclined plane;
whilst the axe, the hatchet, and the knife, (when used to split asunder)
are used as wedges.
_Caroline._ But a knife cuts best when it is drawn across the substance
it is to divide. We use it thus in cutting meat, we do not chop it to
pieces.
_Mrs. B._ The reason of this is, that the edge of a knife is really a
very fine saw, and therefore acts best when used like that instrument.
The screw, which is the last mechanical power, is more complicated than
the others. You will see by this figure, (fig. 9.) that it is composed
of two parts, the screw and the nut. The screw S is a cylinder, with a
spiral protuberance coiled round it, called the thread; the nut N is
perforated to receive the screw, and the inside of the nut has a spiral
groove, made to fit the spiral thread of the screw.
_Caroline._ It is just like this little box, the lid of which screws on
the box as you have described; but what is this handle L which projects
from the nut?
_Mrs. B._ It is a lever, which is attached to the nut, without which the
screw is never used as a mechanical power. The power of the screw,
complicated as it appears, is referable to one of the most simple of the
mechanical powers; which of them do you think it is?
_Caroline._ In appearance, it most resembles the wheel and axle.
_Mrs. B._ The lever, it is true, has the effect of a wheel, as it is the
means by which you turn the nut, or sometimes the screw, round; but the
lever is not considered as composing a part of the screw, though it is
true, that it is necessarily attached to it.
_Emily._ The spiral thread of the screw resembles, I think, an inclined
plane: it is a sort of slope, by means of which the nut ascends more
easily than it would do if raised perpendicularly; and it serves to
support it when at rest.
_Mrs. B._ Very well: if you cut a slip of paper in the form of an
inclined plane, and wind it round your pencil, which will represent the
cylinder, you will find that it makes a spiral line, corresponding to
the spiral protuberance of the screw. (Fig. 10.)
_Emily._ Very true; the nut then ascends an inclined plane, but ascends
it in a spiral, instead of a straight line: the closer the threads of
the screw, the more easy the ascent: it is like having shallow, instead
of steep steps to ascend.
_Mrs. B._ Yes; excepting that the nut takes no steps, as it gradually
winds up or down; then observe, that the closer the threads of the
screw, the less is its ascent in turning round, and the greater is its
power; so that we return to the old principle,--what is saved in power
is lost in time.
_Emily._ Cannot the power of the screw be increased also, by lengthening
the lever attached to the nut?
_Mrs. B._ Certainly. The screw, with the addition of the lever, forms a
very powerful machine, employed either for compression or to raise heavy
weights. It is used by book-binders, to press the leaves of books
together; it is used also in cider and wine presses, in coining, and for
a variety of other purposes.
_Emily._ Pray, Mrs. B., by what rule do you estimate the power of the
screw?
_Mrs. B._ By measuring the circumference of the circle, which the end of
the lever would form in one whole revolution, and comparing this with
the distance from the centre of one thread of the screw, to that of its
next contiguous turn; for whilst the lever travels that whole distance,
the screw rises or falls only through the distance from one coil to
another.
_Caroline._ I think that I have sometimes seen the lever attached to the
screw, and not to the nut, as it is represented in the figure.
_Mrs. B._ This is frequently done, but it does not in any degree affect
the power of the instrument.
All machines are composed of one or more of these six mechanical powers
we have examined; I have but one more remark to make to you relative to
them, which is, that friction in a considerable degree diminishes their
force: allowance must therefore always be made for it, in the
construction of machinery.
_Caroline._ By friction, do you mean one part of the machine rubbing
against another part contiguous to it?
_Mrs. B._ Yes; friction is the resistance which bodies meet with in
rubbing against each other; there is no such thing as perfect smoothness
or evenness in nature; polished metals, though they wear that appearance
more than most other bodies, are far from really possessing it; and
their inequalities may frequently be perceived through a good magnifying
glass. When, therefore, the surfaces of the two bodies come in contact,
the prominent parts of the one, will often fall into the hollow parts of
the other, and occasion more or less resistance to motion.
_Caroline._ But if a machine is made of polished metal, as a watch for
instance, the friction must be very trifling?
_Mrs. B._ In proportion as the surfaces of bodies are well polished, the
friction is doubtless diminished; but it is always considerable, and it
is usually computed to destroy one-third of the power of a machine. Oil
or grease is used to lessen friction: it acts as a polish, by filling up
the cavities of the rubbing surfaces, and thus making them slide more
easily over each other.
_Caroline._ Is it for this reason that wheels are greased, and the locks
and hinges of doors oiled?
_Mrs. B._ Yes; in these instances the contact of the rubbing surfaces is
so close, and they are so constantly in use, that they require to be
frequently oiled, or a considerable degree of friction is produced.
There are two kinds of friction; the first is occasioned by the rubbing
of the surfaces of bodies against each other, the second, by the rolling
of a circular body; as that of a carriage wheel upon the ground: the
friction resulting from the first is much the most considerable, for
great force is required to enable the sliding body to overcome the
resistance which the asperities of the surfaces in contact oppose to its
motion, and it must be either lifted over, or break through them;
whilst, in the second kind of friction, the rough parts roll over each
other with comparative facility; hence it is, that wheels are often used
for the sole purpose of diminishing the resistance from friction.
_Emily._ This is one of the advantages of carriage wheels, is it not?
_Mrs. B._ Yes; and the larger the circumference of the wheel the more
readily it can overcome any considerable obstacles, such as stones, or
inequalities in the road. When, in descending a steep hill, we fasten
one of the wheels, we decrease the velocity of the carriage, by
increasing the friction.
_Caroline._ That is to say, by converting the rolling friction into the
rubbing friction. And when you had casters put to the legs of the table,
in order to move it more easily, you changed the rubbing into the
rolling friction.
_Mrs. B._ There is another circumstance which we have already noticed,
as diminishing the motion of bodies, and which greatly affects the
power of machines. This is the resistance of the medium, in which a
machine is worked. All fluids, whether elastic like air, or non-elastic
like water and other liquids, are called mediums; and their resistance
is proportioned to their density; for the more matter a body contains,
the greater the resistance it will oppose to the motion of another body
striking against it.
_Emily._ It would then be much more difficult to work a machine under
water than in the air?
_Mrs. B._ Certainly, if a machine could be worked in _vacuo_, and
without friction, it would not be impeded, but this is unattainable; a
considerable reduction of power must therefore be allowed for, from
friction and the resistance of the medium.
We shall here conclude our observations on the mechanical powers. At our
next meeting I shall endeavour to give you an explanation of the motion
of the heavenly bodies.
Questions
31. (Pg. 62) Describe a pulley, and its use.
32. (Pg. 62) What is meant by a fixed pulley and why is not power gained
by its employment? (fig. 1. plate 5.)
33. (Pg. 62) Of what use is the fixed pulley?
34. (Pg. 63) How is the power gained by a moveable pulley, explained by
means of fig. 2. plate 5?
35. (Pg. 63) What proportion must the power bear to the weight in fig.
2, that their momentums may be equal?
36. (Pg. 64) What is a fundamental law as respects power and time?
37. (Pg. 64) If to gain power we must lose time, what advantage do we
derive from the mechanical powers?
38. (Pg. 64) What name is given to two or more pulleys connected by one
string?
39. (Pg. 64) How do we estimate the power gained by a system of pulleys?
40. (Pg. 65) What is represented by fig. 5. plate 5?
41. (Pg. 65) How does the wheel operate in increasing power?
42. (Pg. 65) How is this compared with the lever?
43. (Pg. 65) How does a handle fixed to an axle, represent a wheel, fig.
6?
44. (Pg. 65) How could we increase the power in this instrument?
45. (Pg. 66) What other forces besides the power of men, do we employ to
move machines?
46. (Pg. 66) What will serve as an example of an inclined plane?