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, thatreflected motionis 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 calledreflected 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.
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 calledthe angle of incidence, the otherthe 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.
Questions1.(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 wemultiplythe 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 inplate 1. fig. 3?36.(Pg.40) Explain the manner in which the six balls represented infig. 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 byfig. 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 byfig. 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 2represents an angle of more than 90 degrees, what is that called?65.(Pg.44) What are those of less than 90 degrees called as infig. 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 byfig. 4what is meant by the angles of incidence and of reflection.
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 wemultiplythe 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 inplate 1. fig. 3?
36.(Pg.40) Explain the manner in which the six balls represented infig. 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 byfig. 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 byfig. 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 2represents an angle of more than 90 degrees, what is that called?
65.(Pg.44) What are those of less than 90 degrees called as infig. 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 byfig. 4what is meant by the angles of incidence and of reflection.
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, andyou 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 indeedit 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 thecentreof motion, and theaxisof 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 movewith 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 thecentripetalforce; and that force, which impels a body to fly from the centre, is called thecentrifugalforce; 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 atangent, it touches the circumferenceof 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.Projectionalone, would drive the ball A, to B, (fig. 3.)gravitywould 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 aparabola; 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.
Plate iii.
We have noticed the centres of magnitude, and of motion; but I have not yet explained to you, what is meant by thecentre 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?
Mrs. B.Let us, in imagination take off the upper part of the load; 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 bemost 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. Thusyou 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 infig. 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.