Part 5

_Mrs. B._ In speaking of the air, I think we defined elasticity to be a property, by means of which bodies that are compressed, return to their former state. If I bend this cane, as soon as I leave it at liberty, it recovers its former position; if I press my finger upon your arm, as soon as I remove it, the flesh, by virtue of its elasticity, rises and destroys the impression I made. Of all bodies, the air is the most eminent for this property, and it has thence obtained the name of an elastic fluid. Hard bodies are in the next degree elastic; if two ivory, or hardened steel balls are struck together, the parts at which they touch, will be flattened; but their elasticity will make them instantaneously resume their former shape.

_Caroline._ But when two ivory balls strike against each other, as they constantly do on a billiard table, no mark or impression is made by the stroke.

_Mrs. B._ I beg your pardon; you cannot, it is true, perceive any mark, because their elasticity instantly destroys all trace of it.

Soft bodies, which easily retain impressions, such as clay, wax, tallow, butter, &c. have very little elasticity; but of all descriptions of bodies, liquids are the least elastic.

_Emily._ If sealing-wax were elastic, instead of retaining the impression of a seal, it would resume a smooth surface, as soon as the weight of the seal was removed. But pray what is it that produces the elasticity of bodies?

_Mrs. B._ There is great diversity of opinion upon that point, and I cannot pretend to decide which approaches nearest to the truth. Elasticity implies susceptibility of compression, and the susceptibility of compression depends upon the porosity of bodies; for were there no pores or spaces between the particles of matter of which a body is composed, it could not be compressed.

_Caroline._ That is to say, that if the particles of bodies were as close together as possible, they could not be squeezed closer.

_Emily._ Bodies then, whose particles are most distant from each other, must be most susceptible of compression, and consequently most elastic; and this you say is the case with air, which is perhaps the least dense of all bodies?

_Mrs. B._ You will not in general find this rule hold good; for liquids have scarcely any elasticity, whilst hard bodies are eminent for this property, though the latter are certainly of much greater density than the former; elasticity implies, therefore, not only a susceptibility of compression, but depends upon the power possessed by the body, of resuming its former state after compression, in consequence of the peculiar arrangement of its particles.

_Caroline._ But surely there can be no pores in ivory and metals, Mrs. B.; how then can they be susceptible of compression?

_Mrs. B._ The pores of such bodies are invisible to the naked eye, but you must not thence conclude that they have none; it is, on the contrary, well ascertained that gold, one of the most dense of all bodies, is extremely porous; and that these pores are sufficiently large to admit water when strongly compressed, to pass through them. This was shown by a celebrated experiment made many years ago at Florence.

_Emily._ If water can pass through gold, there must certainly be pores or interstices which afford it a passage; and if gold is so porous, what must other bodies be, which are so much less dense than gold!

_Mrs. B._ The chief difference in this respect, is I believe, that the pores in some bodies are larger than in others; in cork, sponge and bread, they form considerable cavities; in wood and stone, when not polished, they are generally perceptible to the naked eye; whilst in ivory, metals, and all varnished and polished bodies, they cannot be discerned. To give you an idea of the extreme porosity of bodies, sir Isaac Newton conjectured that if the earth were so compressed as to be absolutely without pores, its dimensions might possibly not be more than a cubic inch.

_Caroline._ What an idea! Were we not indebted to sir Isaac Newton for 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.

_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.