the centre of gravity will then change its situation, and descend to B,
as that will now be the point about which the parts of the less heavily
laden wagon will balance each other. Will the wagon now be upset?
_Caroline._ No, because a perpendicular line from that point falls
within the wheels at D, and is supported by them; and when the centre of
gravity is supported, the body will not fall.
_Emily._ Yet I should not much like to pass a wagon in that situation,
for, as you see, the point D is but just within the left wheel; if the
right wheel was raised, by merely passing over a stone, the point D
would be thrown on the outside of the left wheel, and the wagon would
upset.
_Caroline._ A wagon, or any carriage whatever, will then be most firmly
supported, when the centre of gravity falls exactly between the wheels;
and that is the case in a level road.
_Mrs. B._ The centre of gravity of the human body, is a point somewhere
in a line extending perpendicularly through the middle of it, and as
long as we stand upright, this point is supported by the feet; if you
lean on one side, you will find that you no longer stand firm. A
rope-dancer performs all his feats of agility, by dexterously supporting
his centre of gravity; whenever he finds that he is in danger of losing
his balance, he shifts the heavy pole which he holds in his hands, in
order to throw the weight towards the side that is deficient; and thus
by changing the situation of the centre of gravity, he restores his
equilibrium.
_Caroline._ When a stick is poised on the tip of the finger, is it not
by supporting its centre of gravity?
_Mrs. B._ Yes; and it is because the centre of gravity is not supported,
that spherical bodies roll down a slope. A sphere being perfectly round,
can touch the slope but by a single point, and that point cannot be
perpendicularly under the centre of gravity, and therefore cannot be
supported, as you will perceive by examining this figure. (fig. 5. plate
3.)
_Emily._ So it appears: yet I have seen a cylinder of wood roll up a
slope; how is that contrived?
_Mrs. B._ It is done by plugging or loading one side of the cylinder
with lead, as at B, (fig. 5. plate 3.) the body being no longer of a
uniform density, the centre of gravity is removed from the middle of the
body to some point in or near the lead, as that substance is much
heavier than wood; now you may observe that should this cylinder roll
down the plane, as it is here situated, the centre of gravity must rise,
which is impossible; the centre of gravity must always descend in
moving, and will descend by the nearest and readiest means, which will
be by forcing the cylinder up the slope, until the centre of gravity is
supported, and then it stops.
_Caroline._ The centre of gravity, therefore, is not always in the
middle of a body.
_Mrs. B._ No, that point we have called the centre of magnitude; when
the body is of an uniform density, and of a regular form, as a cube, or
sphere, the centres of gravity and of magnitude are in the same point;
but when one part of the body is composed of heavier materials than
another, the centre of gravity can no longer correspond with the centre
of magnitude. Thus you see the centre of gravity of this cylinder
plugged with lead, cannot be in the same spot as the centre of
magnitude.
_Emily._ Bodies, therefore, consisting but of one kind of substance, as
wood, stone, or lead, and whose densities are consequently uniform, must
stand more firmly, and be more difficult to overset, than bodies
composed of a variety of substances, of different densities, which may
throw the centre of gravity on one side.
_Mrs. B._ That depends upon the situation of the materials; if those
which are most dense, occupy the lower part, the stability will be
increased, as the centre of gravity will be near the base. But there is
another circumstance which more materially affects the firmness of their
position, and that is their form. Bodies that have a narrow base are
easily upset, for if they are a little inclined, their centre of gravity
is no longer supported, as you may perceive in fig. 6.
_Caroline._ I have often observed with what difficulty a person carries
a single pail of water; it is owing, I suppose, to the centre of gravity
being thrown on one side; and the opposite arm is stretched out to
endeavour to bring it back to its original situation; but a pail hanging
to each arm is carried with less difficulty, because they balance each
other, and the centre of gravity remains supported by the feet.
_Mrs. B._ Very well; I have but one more remark to make on the centre of
gravity, which is, that when two bodies are fastened together by an
inflexible rod, they are to be considered as forming but one body; if
the two bodies be of equal weight, the centre of gravity will be in the
middle of the line which unites them, (fig. 7.) but if one be heavier
than the other, the centre of gravity will be proportionally nearer the
heavy body than the light one. (fig. 8.) If you were to carry a rod or
pole with an equal weight fastened at each end of it, you would hold it
in the middle of the rod, in order that the weights should balance each
other; whilst if the weights were unequal, you would hold it nearest the
greater weight, to make them balance each other.
_Emily._ And in both cases we should support the centre of gravity; and
if one weight be very considerably larger than the other, the centre of
gravity will be thrown out of the rod into the heaviest weight. (fig.
9.)
_Mrs. B._ Undoubtedly.
Questions
1. (Pg. 46) If a body be struck by two equal forces in opposite
directions, what will be the result?
2. (Pg. 46) What is fig. 5. plate 2. intended to represent?
3. (Pg. 47) How would the ball move, and how would you represent the
direction of its motion?
4. (Pg. 47) What is supposed respecting the forces represented in fig.
6?
5. (Pg. 47) How would the body move if so impelled?
6. (Pg. 47) If the forces are unequal and not at right angles, how would
the body move, as illustrated by fig. 7?
7. (Pg. 47) How must a body be acted on, to produce motion in a curve,
and what example is given?
8. (Pg. 48) When is a body said to revolve in a plane, and what is meant
by the centre of motion?
9. (Pg. 48) What is intended by the axis of motion, and what are
examples?
10. (Pg. 48) What is the middle point of a body called?
11. (Pg. 48) What is said of the axis of motion, whilst the body is
revolving?
12. (Pg. 48) When a body revolves on an axis, do all its parts move with
equal velocity?
13. (Pg. 49) How is this explained by fig. 1. plate 3?
14. (Pg. 49) What are the two forces called which cause a body to move
in a curve; and what proportion do these two forces bear to each other
when a body revolves round a centre?
15. (Pg. 49) If the centripetal force were destroyed, how would a body
be carried by the centrifugal?
16. (Pg. 50) Explain what is meant by a _tangent_, as shown in fig. 2.
plate 3.
17. (Pg. 50) What forces impede a body thrown horizontally?
18. (Pg. 50) Give the reason why a body so projected, falls in a curve.
(fig. 3. plate 3.)
19. (Pg. 51) The curve in which it falls, is not a part of a true
circle: what is it denominated?
20. (Pg. 51) What is the _centre of gravity_ defined to be?
21. (Pg. 51) What results from supporting, or not supporting the centre
of gravity?
22. (Pg. 51) What is intended to be explained by fig. 4. plate 3?
23. (Pg. 51) What would be the effect of taking off the upper portion of
the load?
24. (Pg. 52) When will a carriage stand most firmly?
25. (Pg. 52) What is said of the centre of gravity of the human body,
and how does a rope dancer preserve his equilibrium?
26. (Pg. 52) Why cannot a sphere remain at rest on an inclined plane?
(fig. 5. plate 3.)
27. (Pg. 52) A cylinder of wood, may be made to rise to a small distance
up an inclined plane. How may this be effected? (fig. 5. plate 3.)
28. (Pg. 53) When do we find the centres of gravity, and of magnitude in
different points?
29. (Pg. 53) What influence will the density of the parts of a body
exert upon its stability?
30. (Pg. 53) What other circumstance materially affects the firmness of
position? (fig. 6. plate 3.)
31. (Pg. 53) Why is it more easy to carry a weight in each hand, than in
one only?
32. (Pg. 53) What is said respecting two bodies united by an inflexible
rod?
33. (Pg. 53) What is fig. 7, plate 3, intended to illustrate? What fig.
8; what fig. 9?
CONVERSATION V.
ON THE MECHANICAL POWERS.
OF THE POWER OF MACHINES. OF THE LEVER IN GENERAL. OF THE LEVER OF THE
FIRST KIND, HAVING THE FULCRUM BETWEEN THE POWER AND THE WEIGHT. OF THE
LEVER OF THE SECOND KIND, HAVING THE WEIGHT BETWEEN THE POWER AND THE
FULCRUM. OF THE LEVER OF THE THIRD KIND, HAVING THE POWER BETWEEN THE
FULCRUM AND THE WEIGHT.
MRS. B.
We may now proceed to examine the mechanical powers; they are six in
number: The _lever_, the _pulley_, the _wheel_ and _axle_, the _inclined
plane_, the _wedge_ and the _screw_; one or more of which enters into
the composition of every machine.
A mechanical power is an instrument by which the effect of a given force
is increased, whilst the force remains the same.
In order to understand the power of a machine, there are four things to
be considered. 1st. The power that acts: this consists in the effort of
men or horses, of weights, springs, steam, &c.
2dly. The resistance which is to be overcome by the power: this is
generally a weight to be moved. The power must always be superior to the
resistance, otherwise the machine could not be put in motion.
_Caroline._ If for instance the resistance of a carriage was greater
than the strength of the horses employed to draw it, they would not be
able to make it move.
_Mrs. B._ 3dly. We are to consider the support or prop, or as it is
termed in mechanics, the _fulcrum_; this you may recollect is the point
upon which the body turns when in motion; and lastly, the respective
velocities of the power, and of the resistance.
_Emily._ That must in general depend upon their respective distances
from the fulcrum, or from the axis of motion; as we observed in the
motion of the vanes of the windmill.
_Mrs. B._ We shall now examine the power of the lever. The _lever is an
inflexible rod or bar, moveable about a fulcrum, and having forces
applied to two or more points on it_. For instance, the steel rod to
which these scales are suspended is a lever, and the point in which it
is supported, the fulcrum, or centre of motion; now, can you tell me why
the two scales are in equilibrium?
_Caroline._ Being both empty, and of the same weight, they balance each
other.
_Emily._ Or, more correctly speaking, because the centre of gravity
common to both, is supported.
_Mrs. B._ Very well; and where is the centre of gravity of this pair of
scales? (fig. 1. plate 4.)
_Emily._ You have told us that when two bodies of equal weight were
fastened together, the centre of gravity was in the middle of the line
that connected them; the centre of gravity of the scales must therefore
be supported by the fulcrum F of the lever which unites the two scales,
and which is the centre of motion.
_Caroline._ But if the scales contained different weights, the centre of
gravity would no longer be in the fulcrum of the lever, but remove
towards that scale which contained the heaviest weight; and since that
point would no longer be supported, the heavy scale would descend, and
out-weigh the other.
_Mrs. B._ True; but tell me, can you imagine any mode by which bodies of
different weights can be made to balance each other, either in a pair of
scales, or simply suspended to the extremities of the lever? for the
scales are not an essential part of the machine; they have no mechanical
power, and are used merely for the convenience of containing the
substance to be weighed.
_Caroline._ What! make a light body balance a heavy one? I cannot
conceive that possible.
_Mrs. B._ The fulcrum of this pair of scales (fig. 2.) is moveable, you
see; I can take it off the beam, and fasten it on again in another part;
this part is now become the fulcrum, but it is no longer in the centre
of the lever.
_Caroline._ And the scales are no longer true; for that which hangs on
the longest side of the lever descends.
_Mrs. B._ The two parts of the lever divided by the fulcrum, are called
its arms; you should therefore say the longest arm, not the longest side
of the lever.
Your observation is true that the balance is now destroyed; but it will
answer the purpose of enabling you to comprehend the power of a lever,
when the fulcrum is not in the centre.
_Emily._ This would be an excellent contrivance for those who cheat in
the weight of their goods; by making the fulcrum a little on one side,
and placing the goods in the scale which is suspended to the longest arm
of the lever, they would appear to weigh more than they do in reality.
_Mrs. B._ You do not consider how easily the fraud would be detected;
for on the scales being emptied they would not hang in equilibrium. If
indeed the scale on the shorter arm was made heavier, so as to balance
that on the longer, they would appear to be true, whilst they were
really false.
_Emily._ True; I did not think of that circumstance. But I do not
understand why the longest arm of the lever should not be in equilibrium
with the other?
_Caroline._ It is because the momentum in the longest, is greater than
in the shortest arm; the centre of gravity, therefore, is no longer
supported.
_Mrs. B._ You are right, the fulcrum is no longer in the centre of
gravity; but if we can contrive to make the fulcrum in its present
situation become the centre of gravity, the scales will again balance
each other; for you recollect that the centre of gravity is that point
about which every part of the body is in equilibrium.
_Emily._ It has just occurred to me how this may be accomplished; put a
great weight into the scale suspended to the shortest arm of the lever,
and a smaller one into that suspended to the longest arm. Yes, I have
discovered it--look Mrs. B., the scale on the shortest arm will carry 3
lbs., and that on the longest arm only one, to restore the balance.
(fig. 3.)
_Mrs. B._ You see, therefore, that it is not so impracticable as you
imagined, to make a heavy body balance a light one; and this is in fact
the means by which you observed that an imposition in the weight of
goods might be effected, as a weight of ten or twelve ounces, might thus
be made to balance a pound of goods. If you measure both arms of the
lever, you will find that the length of the longer arm, is three times
that of the shorter; and that to produce an equilibrium, the weights
must bear the same proportion to each other, and that the greater
weight, must be on the shorter arm. Let us now take off the scales, that
we may consider the lever simply; and in this state you see that the
fulcrum is no longer the centre of gravity, because it has been removed
from the middle of the lever; but it is, and must ever be, the centre of
motion, as it is the only point which remains at rest, while the other
parts move about it.
[Illustration: PLATE IV.]
_Caroline._ The arms of the lever being different in length, it now
exactly resembles the steelyards, with which articles are so frequently
weighed.
_Mrs. B._ It may in fact be considered as a pair of steelyards, by which
the same power enables us to ascertain the weight of different articles,
by simply increasing the distance of the power from the fulcrum; you
know that the farther a body is from the axis of motion, the greater is
its velocity.
_Caroline._ That I remember, and understand perfectly.
_Mrs. B._ You comprehend then, that the extremity of the longest arm of
a lever, must move with greater velocity than that of the shortest arm,
and that its momentum is greater in proportion.
_Emily._ No doubt, because it is farthest from the centre of motion. And
pray, Mrs. B., when my brothers play at _see-saw_, is not the plank on
which they ride, a kind of lever?
_Mrs. B._ Certainly; the log of wood which supports it from the ground
is the fulcrum, and those who ride, represent the power and the
resistance at the ends of the lever. And have you not observed that when
those who ride are of equal weight, the plank must be supported in the
middle, to make the two arms equal; whilst if the persons differ in
weight, the plank must be drawn a little farther over the prop, to make
the arms unequal, and the lightest person, who may be supposed to
represent the power, must be placed at the extremity of the longest arm.
_Caroline._ That is always the case when I ride on a plank with my
youngest brother; I have observed also that the lightest person has the
best ride, as he moves both further and quicker; and I now understand
that it is because he is more distant from the centre of motion.
_Mrs. B._ The greater velocity with which your little brother moves,
renders his momentum equal to yours.
_Caroline._ Yes; I have the most weight, he the greatest velocity; so
that upon the whole our momentums are equal. But you said, Mrs. B., that
the power should be greater than the resistance, to put the machine in
motion; how then can the plank move if the momentums of the persons who
ride are equal?
_Mrs. B._ Because each person at his descent touches and pushes against
the ground with his feet; the reaction of which gives him an impulse
which produces the motion; this spring is requisite to destroy the
equilibrium of the power and the resistance, otherwise the plank would
not move. Did you ever observe that a lever describes the arc of a
circle in its motion?
_Emily._ No; it appears to me to rise and descend perpendicularly; at
least I always thought so.
_Mrs. B._ I believe I must make a sketch of you and your brother riding
on a plank, in order to convince you of your error. (fig. 4. plate 4.)
You may now observe that a lever can move only round the fulcrum, since
that is the centre of motion; it would be impossible for you to rise
perpendicularly, to the point A; or for your brother to descend in a
straight line, to the point B; you must in rising, and he in descending,
describe arcs of your respective circles. This drawing shows you also
how much superior his velocity must be to yours; for if you could swing
quite round, you would each complete your respective circles, in the
same time.
_Caroline._ My brother's circle being much the largest, he must
undoubtedly move the quickest.
_Mrs. B._ Now tell me, do you think that your brother could raise you as
easily without the aid of a lever?
_Caroline._ Oh no, he could not lift me off the ground.
_Mrs. B._ Then I think you require no further proof of the power of a
lever, since you see what it enables your brother to perform.
_Caroline._ I now understand what you meant by saying, that in
mechanics, velocity is opposed to weight, for it is my brother's
velocity which overcomes my weight.
_Mrs. B._ You may easily imagine, what enormous weights may be raised by
levers of this description, for the longer, when compared with the
other, that arm is to which the power is applied, the greater will be
the effect produced by it; because the greater is the velocity of the
power compared to that of the weight.
Levers are of three kinds; in the first the fulcrum is between the power
and the weight.
_Caroline._ This kind then comprehends the several levers you have
described.
_Mrs. B._ Yes, when in levers of the first kind, the fulcrum is equally
distant from the power and the weight, as in the balance, there will be
an equilibrium, when the power and the weight are equal to each other;
it is not then a mechanical power, for nothing can in this case be
gained by velocity; the two arms of the lever being equal, the velocity
of their extremities must be so likewise. The balance is therefore of no
assistance as a mechanical power, although it is extremely useful in
estimating the respective weights of bodies.
But when (fig. 5.) the fulcrum F of a lever is not equally distant from
the power and the weight, and the power P acts at the extremity of the
longest arm, it may be less than the weight W; its deficiency being
compensated by its superior velocity, as we observed in the _see-saw_.
_Emily._ Then when we want to lift a great weight, we must fasten it to
the shortest arm of a lever, and apply our strength to the longest arm?
_Mrs. B._ If the case will admit of your putting the end of the lever
under the resisting body, no fastening will be required; as you will
perceive, when a nail is drawn by means of a hammer, which, though bent,
is a lever of the first kind; the handle being the longest arm, the
point on which it rests, the fulcrum, and the distance from that to the
part which holds the nail, the short arm. But let me hear, Caroline,
whether you can explain the action of this instrument, which is composed
of two levers united in one common fulcrum.
_Caroline._ A pair of scissors!
_Mrs. B._ You are surprised; but if you examine their construction, you
will discover that it is the power of the lever, that assists us in
cutting with scissors.
_Caroline._ Yes; I now perceive that the point at which the two levers
are screwed together, is the fulcrum; the power of the fingers is
applied to the handles, and the article to be cut, is the resistance;
therefore, the longer the handles, and the shorter the points of the
scissors, the more easily you cut with them.
_Emily._ That I have often observed, for when I cut paste-board or any
hard substance, I always make use of that part of the scissors nearest
the screw or rivet, and I now understand why it increases the power of
cutting; but I confess that I never should have discovered scissors to
have been double levers; and pray are not snuffers levers of a similar
description?
_Mrs. B._ Yes, and most kinds of pincers; the great power of which
consists in the great relative length of the handles.
Did you ever notice the swingle-tree of a carriage to which the horses
are attached when drawing?
_Emily._ O yes; this is a lever of the first kind, but the fulcrum being
in the middle, the horses should draw with equal power, whatever may be
their strength.
_Mrs. B._ That is generally the case, but it is evident that by making
one arm longer than the other, it might be adapted to horses of unequal
strength.
_Caroline._ And of what nature are the other two kinds of levers?
_Mrs. B._ In levers of the second kind, the weight, instead of being at
one end, is situated between the power and the fulcrum, (fig. 6.)
_Caroline._ The weight and the fulcrum have here changed places; and
what advantage is gained by this kind of lever?
_Mrs. B._ In moving it, the velocity of the power must necessarily be
greater than that of the weight, as it is more distant from the centre
of the motion. Have you ever seen your brother move a snow-ball by means
of a strong stick, when it became too heavy for him to move without
assistance?
_Caroline._ Oh yes; and this was a lever of the second kind, (fig. 7.)
the end of the stick, which he thrusts under the ball, and which rests
on the ground, becomes the fulcrum; the ball is the weight to be moved,
and the power his hands, applied to the other end of the lever. In this
instance there is a great difference in the length of the arms of the
lever; for the weight is almost close to the fulcrum.
_Mrs. B._ And the advantage gained is proportional to this difference.
The most common example that we have of levers of the second kind, is in
the doors of our apartments.
_Emily._ The hinges represent the fulcrum, our hands the power applied
to the other end of the lever; but where is the weight to be moved?
_Mrs. B._ The door is the weight, which in this example occupies the
whole of the space between the power and the fulcrum. Nut crackers are
double levers of this kind: the hinge is the fulcrum, the nut the
resistance, and the hands the power.
In levers of the third kind (fig. 8.) the fulcrum is again at one
extremity, the weight or resistance at the other, and the power is
applied between the fulcrum and the resistance.
_Emily._ The fulcrum, the weight, or the power, then, each in its turn,
occupies some part of the lever between its extremities. But in this
third kind of lever, the weight being farther than the power from the
centre of motion, the difficulty of raising it seems increased rather
than diminished.