the theory of attraction, I should be tempted to laugh at him for such a
supposition. What insignificant little creatures we should be!
_Mrs. B._ If our consequence arose from the size of our bodies, we
should indeed be but pigmies, but remember that the mind of Newton was
not circumscribed by the dimensions of its envelope.
_Emily._ It is, however, fortunate that heat keeps the pores of matter
open and distended, and prevents the attraction of cohesion from
squeezing us into a nut-shell.
_Mrs. B._ Let us now return to the subject of reaction, on which we have
some further observations to make. It is because reaction is in its
direction opposite to action, that _reflected motion_ is produced. If
you throw a ball against the wall, it rebounds; this return of the ball
is owing to the reaction of the wall against which it struck, and is
called _reflected motion_.
_Emily._ And I now understand why balls filled with air rebound better
than those stuffed with bran or wool; air being most susceptible of
compression and most elastic, the reaction is more complete.
_Caroline._ I have observed that when I throw a ball straight against
the wall, it returns straight to my hand; but if I throw it obliquely
upwards, it rebounds still higher, and I catch it when it falls.
_Mrs. B._ You should not say straight, but perpendicularly against the
wall; for straight is a general term for lines in all directions which
are neither curved nor bent, and is therefore equally applicable to
oblique or perpendicular lines.
_Caroline._ I thought that perpendicularly meant either directly upwards
or downwards?
_Mrs. B._ In those directions lines are perpendicular to the earth. A
perpendicular line has always a reference to something towards which it
is perpendicular; that is to say, that it inclines neither to the one
side or the other, but makes an equal angle on every side. Do you
understand what an angle is?
_Caroline._ Yes, I believe so: it is the space contained between two
lines meeting in a point.
_Mrs. B._ Well then, let the line A B (plate 2. fig. 1.) represent the
floor of the room, and the line C D that in which you throw a ball
against it; the line C D, you will observe, forms two angles with the
line A B, and those two angles are equal.
_Emily._ How can the angles be equal, while the lines which compose them
are of unequal length?
_Mrs. B._ An angle is not measured by the length of the lines, but by
their opening, or the space between them.
_Emily._ Yet the longer the lines are, the greater is the opening
between them.
_Mrs. B._ Take a pair of compasses and draw a circle over these spaces,
making the angular point the centre.
_Emily._ To what extent must I open the compasses?
_Mrs. B._ You may draw the circle what size you please, provided that it
cuts the lines of the angles we are to measure. All circles, of whatever
dimensions, are supposed to be divided into 360 equal parts, called
degrees; the opening of an angle, being therefore a portion of a circle,
must contain a certain number of degrees: the larger the angle the
greater is the number of degrees, and two angles are said to be equal,
when they contain an equal number of degrees.
_Emily._ Now I understand it. As the dimension of an angle depends upon
the number of degrees contained between its lines, it is the opening,
and not the length of its lines, which determines the size of the angle.
_Mrs. B._ Very well: now that you have a clear idea of the dimensions of
angles, can you tell me how many degrees are contained in the two angles
formed by one line falling perpendicularly on another, as in the figure
I have just drawn?
_Emily._ You must allow me to put one foot of the compasses at the point
of the angles, and draw a circle round them, and then I think I shall be
able to answer your question: the two angles are together just equal to
half a circle, they contain therefore 90 degrees each; 90 degrees being
a quarter of 360.
_Mrs. B._ An angle of 90 degrees or one-fourth of a circle is called a
right angle, and when one line is perpendicular to another, and distant
from its ends, it forms, you see, (fig. 1.) a right angle on either
side. Angles containing more than 90 degrees are called obtuse angles,
(fig. 2.) and those containing less than 90 degrees are called acute
angles, (fig. 3.)
_Caroline._ The angles of this square table are right angles, but those
of the octagon table are obtuse angles; and the angles of sharp pointed
instruments are acute angles.
[Illustration: PLATE II.]
_Mrs. B._ Very well. To return now to your observation, that if a ball
is thrown obliquely against the wall, it will not rebound in the same
direction; tell me, have you ever played at billiards?
_Caroline._ Yes, frequently; and I have observed that when I push the
ball perpendicularly against the cushion, it returns in the same
direction; but when I send it obliquely to the cushion, it rebounds
obliquely, but on an opposite side; the ball in this latter case
describes an angle, the point of which is at the cushion. I have
observed too, that the more obliquely the ball is struck against the
cushion, the more obliquely it rebounds on the opposite side, so that a
billiard player can calculate with great accuracy in what direction it
will return.
_Mrs. B._ Very well. This figure (fig. 4. plate 2.) represents a
billiard table; now if you draw a line A B from the point where the ball
A strikes perpendicular to the cushion, you will find that it will
divide the angle which the ball describes into two parts, or two angles;
the one will show the obliquity of the direction of the ball in its
passage towards the cushion, the other its obliquity in its passage back
from the cushion. The first is called _the angle of incidence_, the
other _the angle of reflection_; and these angles are always equal, if
the bodies are perfectly elastic.
_Caroline._ This then is the reason why, when I throw a ball obliquely
against the wall, it rebounds in an opposite oblique direction, forming
equal angles of incidence and of reflection.
_Mrs. B._ Certainly; and you will find that the more obliquely you throw
the ball, the more obliquely it will rebound.
We must now conclude; but I shall have some further observations to make
upon the laws of motion, at our next meeting.
Questions
1. (Pg. 32) On what is the science of mechanics founded?
2. (Pg. 32) In what does motion consist?
3. (Pg. 33) What is the consequence of inertia, on a body at rest?
4. (Pg. 33) What do we call that which produces motion?
5. (Pg. 33) Give some examples.
6. (Pg. 33) What may we say of gravity, of cohesion, and of heat, as
forces?
7. (Pg. 33) How will a body move, if acted on by a single force?
8. (Pg. 33) What is the reason of this?
9. (Pg. 33) What do we intend by the term velocity, and to what is it
proportional?
10. (Pg. 33) Velocity is divided into absolute and relative; what is
meant by absolute velocity?
11. (Pg. 33) How is relative velocity distinguished?
12. (Pg. 34) How do we measure the velocity of a body?
13. (Pg. 34) The time?
14. (Pg. 34) The space?
15. (Pg. 34) What is uniform motion? and give an example.
16. (Pg. 34) How is uniform motion produced?
17. (Pg. 34) A ball struck by a bat gradually loses its motion; what
causes produce this effect?
18. (Pg. 35) If gravity did not draw a projected body towards the earth,
and the resistance of the air were removed, what would be the
consequence?
19. (Pg. 35) In this case would not a great degree of force be required
to produce a continued motion?
20. (Pg. 35) What is retarded motion?
21. (Pg. 35) Give some examples.
22. (Pg. 36) What is accelerated motion?
23. (Pg. 36) Give an example.
24. (Pg. 36) Explain the mode in which gravity operates in producing
this effect.
25. (Pg. 37) What number of feet will a heavy body descend in the first
second of its fall, and at what rate will its velocity increase?
26. (Pg. 37) What is the difference in the time of the ascent and
descent, of a stone, or other body thrown upwards?
27. (Pg. 37) By what reasoning is it proved that there is no difference?
28. (Pg. 38) What is meant by the momentum of a body?
29. (Pg. 38) How do we ascertain the momentum?
30. (Pg. 38) How may a light body have a greater momentum than one which
is heavier?
31. (Pg. 38) Why must we _multiply_ the weight and velocity together in
order to find the momentum?
32. (Pg. 39) When we represent weight and velocity by numbers, what must
we carefully observe?
33. (Pg. 39) Why is it particularly important, to understand the nature
of momentum?
34. (Pg. 39) What is meant by reaction, and what is the rule respecting
it?
35. (Pg. 39) How is this exemplified by the ivory balls represented in
plate 1. fig. 3?
36. (Pg. 40) Explain the manner in which the six balls represented in
fig. 4, illustrate this fact.
37. (Pg. 40) What must be the nature of bodies, in which the whole
motion is communicated from one to the other?
38. (Pg. 40) What is the result if the balls are not elastic, and how is
this explained by fig. 5?
39. (Pg. 40) How will reaction assist us in explaining the flight of a
bird?
40. (Pg. 40) How must their wings operate in enabling them to remain
stationary, to rise, and to descend?
41. (Pg. 41) Why cannot a man fly by the aid of wings?
42. (Pg. 41) How does reaction operate in enabling us to swim, or to row
a boat?
43. (Pg. 41) What constitutes elasticity?
44. (Pg. 41) Give some examples.
45. (Pg. 41) What name is given to air, and for what reason?
46. (Pg. 41) What hard bodies are mentioned as elastic?
47. (Pg. 41) Do elastic bodies exhibit any indentation after a blow? and
why not?
48. (Pg. 42) What do we conclude from elasticity respecting the contact
of the particles of a body?
49. (Pg. 42) Are those bodies always the most elastic, which are the
least dense?
50. (Pg. 42) Give examples to prove that this is not the case.
51. (Pg. 42) All bodies are believed to be porous, what is said on this
subject respecting gold?
52. (Pg. 43) What conjecture was made by sir Isaac Newton, respecting
the porosity of bodies in general?
53. (Pg. 43) If you throw an elastic body against a wall, it will
rebound; what is this occasioned by, and what is this return motion
called?
54. (Pg. 43) What do we mean by a perpendicular line?
55. (Pg. 43) What is an angle?
56. (Pg. 43) What is represented by fig. 1. plate 2?
57. (Pg. 44) Have the length of the lines which meet in a point, any
thing to do with the measurement of an angle?
58. (Pg. 44) What use can we make of compasses in measuring an angle?
59. (Pg. 44) Into what number of parts do we suppose a whole circle
divided, and what are these parts called?
60. (Pg. 44) When are two angles said to be equal?
61. (Pg. 44) Upon what does the dimension of an angle depend?
62. (Pg. 44) What number of degrees, and what portion of a circle is
there in a right angle?
63. (Pg. 44) How must one line be situated on another to form two right
angles? (fig. 1.)
64. (Pg. 44) Figure 2 represents an angle of more than 90 degrees, what
is that called?
65. (Pg. 44) What are those of less than 90 degrees called as in fig. 3?
66. (Pg. 45) If you make an elastic ball strike a body at right angles,
how will it return?
67. (Pg. 45) How if it strikes obliquely?
68. (Pg. 45) Explain by fig. 4 what is meant by the angles of incidence
and of reflection.
CONVERSATION IV.
ON COMPOUND MOTION.
COMPOUND MOTION, THE RESULT OF TWO OPPOSITE FORCES. OF CURVILINEAR
MOTION, THE RESULT OF TWO FORCES. CENTRE OF MOTION, THE POINT AT REST
WHILE THE OTHER PARTS OF THE BODY MOVE ROUND IT. CENTRE OF MAGNITUDE,
THE MIDDLE OF A BODY. CENTRIPETAL FORCE, THAT WHICH IMPELS A BODY
TOWARDS A FIXED CENTRAL POINT. CENTRIFUGAL FORCE, THAT WHICH IMPELS A
BODY TO FLY FROM THE CENTRE. FALL OF BODIES IN A PARABOLA. CENTRE OF
GRAVITY, THE POINT ABOUT WHICH THE PARTS BALANCE EACH OTHER.
MRS. B.
I must now explain to you the nature of compound motion. Let us suppose
a body to be struck by two equal forces in opposite directions, how will
it move?
_Emily._ If the forces are equal, and their directions are in exact
opposition to each other, I suppose the body would not move at all.
_Mrs. B._ You are perfectly right; but suppose the forces instead of
acting upon the body in direct opposition to each other, were to move in
lines forming an angle of ninety degrees, as the lines Y A, X A, (fig.
5. plate 2.) and were to strike the ball A, at the same instant; would
it not move?
_Emily._ The force X alone, would send it towards B, and the force Y
towards C; and since these forces are equal, I do not know how the body
can obey one impulse rather than the other; and yet I think the ball
would move, because as the two forces do not act in direct opposition,
they cannot entirely destroy the effect of each other.
_Mrs. B._ Very true; the ball therefore will not follow the direction of
either of the forces, but will move in a line between them, and will
reach D in the same space of time, that the force X would have sent it
to B, and the force Y would have sent it to C. Now if you draw two
lines, one from B, parallel to A C, and the other from C, parallel to A
B, they will meet in D, and you will form a square; the oblique line
which the body describes, is called the diagonal of the square.
_Caroline._ That is very clear, but supposing the two forces to be
unequal, that the force X, for instance, be twice as great as the force
Y?
_Mrs. B._ Then the force X, would drive the ball twice as far as the
force Y, consequently you must draw the line A B (fig. 6.) twice as long
as the line A C, the body will in this case move to D; and if you draw
lines from the points B and C, exactly as directed in the last example,
they will meet in D, and you will find that the ball has moved in the
diagonal of a rectangle.
_Emily._ Allow me to put another case. Suppose the two forces are
unequal, but do not act on the ball in the direction of a right angle,
but in that of an acute angle, what will result?
_Mrs. B._ Prolong the lines in the directions of the two forces, and you
will soon discover which way the ball will be impelled; it will move
from A to D, in the diagonal of a parallelogram, (fig. 7.) Forces acting
in the direction of lines forming an obtuse angle, will also produce
motion in the diagonal of a parallelogram. For instance, if the body set
out from B, instead of A, and was impelled by the forces X and Y, it
would move in the dotted diagonal B C.
We may now proceed to curvilinear motion: this is the result of two
forces acting on a body; by one of which, it is projected forward in a
right line; whilst by the other, it is drawn or impelled towards a fixed
point. For instance, when I whirl this ball, which is fastened to my
hand with a string, the ball moves in a circular direction, because it
is acted on by two forces; that which I give it, which represents the
force of projection, and that of the string which confines it to my
hand. If, during its motion you were suddenly to cut the string, the
ball would fly off in a straight line; being released from that
confinement which caused it to move round a fixed point, it would be
acted on by one force only; and motion produced by one force, you know,
is always in a right line.
_Caroline._ This circular motion, is a little more difficult to
comprehend than compound motion in straight lines.
_Mrs. B._ You have seen how the water is thrown off from a grindstone,
when turned rapidly round; the particles of the stone itself have the
same tendency, and would also fly off, was not their attraction of
cohesion, greater than that of water. And indeed it sometimes happens,
that large grindstones fly to pieces from the rapidity of their motion.
_Emily._ In the same way, the rim and spokes of a wheel, when in rapid
motion, would be driven straight forwards in a right line, were they not
confined to a fixed point, round which they are compelled to move.
_Mrs. B._ Very well. You must now learn to distinguish between what is
called the _centre_ of motion, and the _axis_ of motion; the former
being considered as a point, the latter as a line.
When a body, like the ball at the end of the string, revolves in a
circle, the centre of the circle is called the centre of its motion, and
the body is said to revolve in a plane; because a line extended from the
revolving body, to the centre of motion, would describe a plane, or flat
surface.
When a body revolves round itself, as a ball suspended by a string, and
made to spin round, or a top spinning on the floor, whilst it remains on
the same spot; this revolution is round an imaginary line passing
through the body, and this line is called its axis of motion.
_Caroline._ The axle of a grindstone, is then the axis of its motion;
but is the centre of motion always in the middle of a body?
_Mrs. B._ No, not always. The middle point of a body, is called its
centre of magnitude, or position, that is, the centre of its mass or
bulk. Bodies have also another centre, called the centre of gravity,
which I shall explain to you; but at present we must confine ourselves
to the axis of motion. This line you must observe remains at rest,
whilst all the other parts of the body move around it; when you spin a
top, the axis is stationary, whilst every other part is in motion round
it.
_Caroline._ But a top generally has a motion forwards besides its
spinning motion; and then no point within it can be at rest?
_Mrs. B._ What I say of the axis of motion, relates only to circular
motion; that is to say, motion round a line, and not to that which a
body may have at the same time in any other direction. There is one
circumstance to which you must carefully attend; namely, that the
further any part of a body is from the axis of motion, the greater is
its velocity: as you approach that line, the velocity of the parts
gradually diminish till you reach the axis of motion, which is perfectly
at rest.
_Caroline._ But, if every part of the same body did not move with the
same velocity, that part which moved quickest, must be separated from
the rest of the body, and leave it behind?
_Mrs. B._ You perplex yourself by confounding the idea of circular
motion, with that of motion in a right line; you must think only of the
motion of a body round a fixed line, and you will find, that if the
parts farthest from the centre had not the greatest velocity, those
parts would not be able to keep up with the rest of the body, and would
be left behind. Do not the extremities of the vanes of a windmill move
over a much greater space, than the parts nearest the axis of motion?
(plate 3. fig. 1.) The three dotted circles represent the paths in which
three different parts of the vanes move, and though the circles are of
different dimensions, each of them is described in the same space of
time.
_Caroline._ Certainly they are; and I now only wonder, that we neither
of us ever made the observation before: and the same effect must take
place in a solid body, like the top in spinning; the most bulging part
of the surface must move with the greatest rapidity.
_Mrs. B._ The force which draws a body towards a centre, round which it
moves, is called the _centripetal_ force; and that force, which impels a
body to fly from the centre, is called the _centrifugal_ force; when a
body revolves round a centre, these two forces constantly balance each
other; otherwise the revolving body would either approach the centre or
recede from it, according as the one or the other prevailed.
_Caroline._ When I see any body moving in a circle, I shall remember,
that it is acted on by two forces.
_Mrs. B._ Motion, either in a circle, an ellipsis, or any other
curve-line, must be the result of the action of two forces; for you
know, that the impulse of one single force, always produces motion in a
right line.
_Emily._ And if any cause should destroy the centripetal force, the
centrifugal force would alone impel the body, and it would, I suppose,
fly off in a straight line from the centre to which it had been
confined.
_Mrs. B._ It would not fly off in a right line from the centre; but in a
right line in the direction in which it was moving, at the instant of
its release; if a stone, whirled round in a sling, gets loose at the
point A, (plate 3. fig. 2.) it flies off in the direction A B; this line
is called a _tangent_, it touches the circumference of the circle, and
forms a right angle with a line drawn from that point of the
circumference to the centre of the circle C.
_Emily._ You say, that motion in a curve-line, is owing to two forces
acting upon a body; but when I throw this ball in a horizontal
direction, it describes a curve-line in falling; and yet it is only
acted upon by the force of projection; there is no centripetal force to
confine it, or produce compound motion.
_Mrs. B._ A ball thus thrown, is acted upon by no less than three
forces; the force of projection, which you communicate to it; the
resistance of the air through which it passes, which diminishes its
velocity, without changing its direction; and the force of gravity,
which finally brings it to the ground. The power of gravity, and the
resistance of the air, being always greater than any force of projection
we can give a body, the latter is gradually overcome, and the body
brought to the ground; but the stronger the projectile force, the longer
will these powers be in subduing it, and the further the body will go
before it falls.
_Caroline._ A shot fired from a cannon, for instance, will go much
further, than a stone projected by the hand.
_Mrs. B._ Bodies thus projected, you observe, describe a curve-line in
their descent; can you account for that?
_Caroline._ No; I do not understand why it should not fall in the
diagonal of a square.
_Mrs. B._ You must consider that the force of projection is strongest
when the ball is first thrown; this force, as it proceeds, being
weakened by the continued resistance of the air, the stone, therefore,
begins by moving in a horizontal direction; but as the stronger powers
prevail, the direction of the ball will gradually change from a
horizontal, to a perpendicular line. _Projection_ alone, would drive the
ball A, to B, (fig. 3.) _gravity_ would bring it to C; therefore, when
acted on in different directions, by these two forces, it moves between,
gradually inclining more and more to the force of gravity, in proportion
as this accumulates; instead therefore of reaching the ground at D, as
you suppose it would, it falls somewhere about E.
_Caroline._ It is precisely so; look Emily, as I throw this ball
directly upwards, how gravity and the resistance of the air conquer
projection. Now I will throw it upwards obliquely: see, the force of
projection enables it, for an instant, to act in opposition to that of
gravity; but it is soon brought down again.
_Mrs. B._ The curve-line which the ball has described, is called in
geometry a _parabola_; but when the ball is thrown perpendicularly
upwards, it will descend perpendicularly; because the force of
projection, and that of gravity, are in the same line of direction.
[Illustration: PLATE III.]
We have noticed the centres of magnitude, and of motion; but I have not
yet explained to you, what is meant by the _centre of gravity_; it is
that point in a body, about which all the parts exactly balance each
other; if therefore that point be supported, the body will not fall. Do
you understand this?
_Emily._ I think so; if the parts round about this point have an equal
tendency to fall, they will be in equilibrium, and as long as this point
is supported, the body cannot fall.
_Mrs. B._ Caroline, what would be the effect, were the body supported in
any other single point?
_Caroline._ The surrounding parts no longer balancing each other, the
body, I suppose, would fall on the side at which the parts are heaviest.
_Mrs. B._ Infallibly; whenever the centre of gravity is unsupported, the
body must fall. This sometimes happens with an overloaded wagon winding
up a steep hill, one side of the road being more elevated than the
other; let us suppose it to slope as is described in this figure, (plate
3. fig. 4.) we will say, that the centre of gravity of this loaded wagon
is at the point A. Now your eye will tell you, that a wagon thus
situated, will overset; and the reason is, that the centre of gravity A,
is not supported; for if you draw a perpendicular line from it to the
ground at C, it does not fall under the wagon within the wheels, and is
therefore not supported by them.
_Caroline._ I understand that perfectly; but what is the meaning of the
other point B?