Part 6

_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.

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?

_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 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.

_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.