Questions1.(Pg.46) If a body be struck by two equal forces in opposite directions, what will be the result?2.(Pg.46) What isfig. 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 infig. 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 byfig. 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 byfig. 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 atangent, as shown infig. 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 thecentre of gravitydefined 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 byfig. 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 isfig. 7, plate 3, intended to illustrate? Whatfig. 8; whatfig. 9?
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 isfig. 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 infig. 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 byfig. 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 byfig. 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 atangent, as shown infig. 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 thecentre of gravitydefined 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 byfig. 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 isfig. 7, plate 3, intended to illustrate? Whatfig. 8; whatfig. 9?
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: Thelever, thepulley, thewheelandaxle, theinclined plane, thewedgeand thescrew; 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, thefulcrum; 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. Thelever 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.
Plate iv.
Caroline.The arms of the lever being different in length, it now exactly resembles the steelyards, with which articles are so frequently weighed.
Mrs. B.It may in fact be considered as a pair of steelyards, by which the same power enables us to ascertain the weight of different articles, by simply increasing the distance of the power from the fulcrum; you know that the farther a body is from the axis of motion, the greater is its velocity.
Caroline.That I remember, and understand perfectly.
Mrs. B.You comprehend then, that the extremity of the longest arm of a lever, must move with greater velocity than that of the shortest arm, and that its momentum is greater in proportion.
Emily.No doubt, because it is farthest from the centre of motion. And pray, Mrs. B., when my brothers play atsee-saw, is not the plank on which they ride, a kind of lever?
Mrs. B.Certainly; the log of wood which supports it from the ground is the fulcrum, and those who ride, represent the power and the resistance at the ends of the lever. And have you not observed that when those who ride are of equal weight, the plank must be supported in the middle, to make the two arms equal; whilst if the persons differ in weight, the plank must be drawn a little farther over the prop, to make the arms unequal, and the lightest person, who may be supposed to represent the power, must be placed at the extremity of the longest arm.
Caroline.That is always the case when I ride on a plank with my youngest brother; I have observed also that the lightest person has the best ride, as he moves both further and quicker; and I now understand that it is because he is more distant from the centre of motion.
Mrs. B.The greater velocity with which your little brother moves, renders his momentum equal to yours.
Caroline.Yes; I have the most weight, he the greatest velocity; so that upon the whole our momentums are equal. But you said, Mrs. B., that the power should be greater than the resistance, to put the machine in motion; how then can the plank move if the momentums of the persons who ride are equal?
Mrs. B.Because each person at his descent touches and pushes against the ground with his feet; the reaction of which gives him an impulse which produces the motion; this spring is requisite to destroy the equilibrium of the power and the resistance, otherwise the plank would not move. Did you ever observe that a lever describes the arc of a circle in its motion?
Emily.No; it appears to me to rise and descend perpendicularly; at least I always thought so.
Mrs. B.I believe I must make a sketch of you and your brother riding on a plank, in order to convince you of your error. (fig. 4. plate 4.) You may now observe that a lever can move only round the fulcrum, since that is the centre of motion; it would be impossible for you to rise perpendicularly, to the point A; or for your brother to descend in a straight line, to the point B; you must in rising, and he in descending, describe arcs of your respective circles. This drawing shows you also how much superior his velocity must be to yours; for if you could swing quite round, you would each complete your respective circles, in the same time.
Caroline.My brother's circle being much the largest, he must undoubtedly move the quickest.
Mrs. B.Now tell me, do you think that your brother could raise you as easily without the aid of a lever?
Caroline.Oh no, he could not lift me off the ground.
Mrs. B.Then I think you require no further proof of the power of a lever, since you see what it enables your brother to perform.
Caroline.I now understand what you meant by saying, that in mechanics, velocity is opposed to weight, for it is my brother's velocity which overcomes my weight.
Mrs. B.You may easily imagine, what enormous weights may be raised by levers of this description, for the longer, when compared with the other, that arm is to which the power is applied, the greater will be the effect produced by it; because the greater is the velocity of the power compared to that of the weight.
Levers are of three kinds; in the first the fulcrum is between the power and the weight.
Caroline.This kind then comprehends the several levers you have described.
Mrs. B.Yes, when in levers of the first kind, the fulcrum is equally distant from the power and the weight, as in the balance, there will be an equilibrium, when the power and the weight are equal to each other; it is not then a mechanical power, for nothing can in this case be gained by velocity; the two arms of the lever being equal, the velocity of their extremities must be so likewise. The balance is therefore of no assistance as a mechanical power, although it is extremely useful in estimating the respective weights of bodies.
But when (fig. 5.) the fulcrum F of a lever is not equally distantfrom the power and the weight, and the power P acts at the extremity of the longest arm, it may be less than the weight W; its deficiency being compensated by its superior velocity, as we observed in thesee-saw.
Emily.Then when we want to lift a great weight, we must fasten it to the shortest arm of a lever, and apply our strength to the longest arm?
Mrs. B.If the case will admit of your putting the end of the lever under the resisting body, no fastening will be required; as you will perceive, when a nail is drawn by means of a hammer, which, though bent, is a lever of the first kind; the handle being the longest arm, the point on which it rests, the fulcrum, and the distance from that to the part which holds the nail, the short arm. But let me hear, Caroline, whether you can explain the action of this instrument, which is composed of two levers united in one common fulcrum.
Caroline.A pair of scissors!
Mrs. B.You are surprised; but if you examine their construction, you will discover that it is the power of the lever, that assists us in cutting with scissors.
Caroline.Yes; I now perceive that the point at which the two levers are screwed together, is the fulcrum; the power of the fingers is applied to the handles, and the article to be cut, is the resistance; therefore, the longer the handles, and the shorter the points of the scissors, the more easily you cut with them.
Emily.That I have often observed, for when I cut paste-board or any hard substance, I always make use of that part of the scissors nearest the screw or rivet, and I now understand why it increases the power of cutting; but I confess that I never should have discovered scissors to have been double levers; and pray are not snuffers levers of a similar description?
Mrs. B.Yes, and most kinds of pincers; the great power of which consists in the great relative length of the handles.
Did you ever notice the swingle-tree of a carriage to which the horses are attached when drawing?
Emily.O yes; this is a lever of the first kind, but the fulcrum being in the middle, the horses should draw with equal power, whatever may be their strength.
Mrs. B.That is generally the case, but it is evident that by making one arm longer than the other, it might be adapted to horses of unequal strength.
Caroline.And of what nature are the other two kinds of levers?
Mrs. B.In levers of the second kind, the weight, instead of being at one end, is situated between the power and the fulcrum, (fig. 6.)
Caroline.The weight and the fulcrum have here changed places; and what advantage is gained by this kind of lever?
Mrs. B.In moving it, the velocity of the power must necessarily be greater than that of the weight, as it is more distant from the centre of the motion. Have you ever seen your brother move a snow-ball by means of a strong stick, when it became too heavy for him to move without assistance?
Caroline.Oh yes; and this was a lever of the second kind, (fig. 7.) the end of the stick, which he thrusts under the ball, and which rests on the ground, becomes the fulcrum; the ball is the weight to be moved, and the power his hands, applied to the other end of the lever. In this instance there is a great difference in the length of the arms of the lever; for the weight is almost close to the fulcrum.
Mrs. B.And the advantage gained is proportional to this difference. The most common example that we have of levers of the second kind, is in the doors of our apartments.
Emily.The hinges represent the fulcrum, our hands the power applied to the other end of the lever; but where is the weight to be moved?
Mrs. B.The door is the weight, which in this example occupies the whole of the space between the power and the fulcrum. Nut crackers are double levers of this kind: the hinge is the fulcrum, the nut the resistance, and the hands the power.
In levers of the third kind (fig. 8.) the fulcrum is again at one extremity, the weight or resistance at the other, and the power is applied between the fulcrum and the resistance.
Emily.The fulcrum, the weight, or the power, then, each in its turn, occupies some part of the lever between its extremities. But in this third kind of lever, the weight being farther than the power from the centre of motion, the difficulty of raising it seems increased rather than diminished.
Mrs. B.That is very true; a lever of this kind is therefore never used, unless absolutely necessary, as is the case in raising a ladder in order to place it against a wall; the man who raises it cannot place his hands on the upper part of the ladder, the power, therefore, is necessarily placed much nearer to the fulcrum than to the weight.
Caroline.Yes, the hands are the power, the ground the fulcrum, and the upper part of the ladder the weight.
Mrs. B.Nature employs this kind of lever in the structure of the human frame. In lifting a weight with the hand, the lower part of the arm becomes a lever of the third kind; the elbow is the fulcrum, the muscles of the fleshy part of the arm, the power; and as these are nearer to the elbow than to the hand, it is necessary that their power should exceed the weight to be raised.
Emily.Is it not surprising that nature should have furnished us with such disadvantageous levers?
Mrs. B.The disadvantage, in respect to power, is more than counterbalanced by the convenience resulting from this structure of the arm; and it is that no doubt which is best adapted to enable it to perform its various functions.
There is one rule which applies to every lever, which is this: In order to produce an equilibrium, the power must bear the same proportion to the weight, as the length of the shorter arm does to that of the longer; as was shown by Emily with the weights of 1lb.and of 3lb.Fig. 3. plate 4.
We have dwelt so long on the lever, that we must reserve the examination of the other mechanical powers, to our next interview.
Questions1.(Pg.54) How many mechanical powers are there, and what are they named?2.(Pg.54) What is a mechanical power defined to be?3.(Pg.54) What four particulars must be observed?4.(Pg.54) Upon what will the velocities depend?5.(Pg.55) What is a lever?6.(Pg.55) Give a familiar example.7.(Pg.55) When and why do the scales balance each other, and where is their centre of gravity? (fig. 1. plate 4.)8.(Pg.55) Why would they not balance with unequal weights?9.(Pg.55) Were the fulcrum removed from the middle of the beam what would result?10.(Pg.55) What do we mean by the arms of a lever?11.(Pg.56) How may a pair of scales be false, and yet appear to be true?12.(Pg.56) If the fulcrum be removed from the centre of gravity, how may the equilibrium be restored?13.(Pg.56) How is this exemplified byfig. 3. plate 4?14.(Pg.56) What proportion must the weights bear to the lengths of the arms?15.(Pg.57) On what principle do we weigh with a pair of steelyards, and what will be the difference in the motion of the extremities of such a lever?16.(Pg.58) How is this exemplified byfig. 4. plate 4?17.(Pg.58) What line is described by the ends of a lever?fig. 4. plate 4.18.(Pg.58) How many kinds are there; and in the first how is the fulcrum situated?19.(Pg.58) When may the fulcrum be so situated that this lever is not a mechanical power, and why?20.(Pg.59) What is represented byfig. 5. plate 4?21.(Pg.59) Give a familiar example of the use of a lever of the first kind.22.(Pg.59) In what instruments are two such levers combined?23.(Pg.59) How may two horses of unequal strength, be advantageously coupled in a carriage?24.(Pg.60) Describe a lever of the second kind. (Fig. 6. plate 4.)25.(Pg.60) What is represented infig. 7. plate 4, and in what proportion does this lever gain power?26.(Pg.60) What is said respecting a door?27.(Pg.60) Describe a lever of the third kind.28.(Pg.60) In what instance do we use this?29.(Pg.61) What remarks are made on its employment in the limbs of animals?30.(Pg.61) What are the conditions of equilibrium in every lever?
Questions
1.(Pg.54) How many mechanical powers are there, and what are they named?
2.(Pg.54) What is a mechanical power defined to be?
3.(Pg.54) What four particulars must be observed?
4.(Pg.54) Upon what will the velocities depend?
5.(Pg.55) What is a lever?
6.(Pg.55) Give a familiar example.
7.(Pg.55) When and why do the scales balance each other, and where is their centre of gravity? (fig. 1. plate 4.)
8.(Pg.55) Why would they not balance with unequal weights?
9.(Pg.55) Were the fulcrum removed from the middle of the beam what would result?
10.(Pg.55) What do we mean by the arms of a lever?
11.(Pg.56) How may a pair of scales be false, and yet appear to be true?
12.(Pg.56) If the fulcrum be removed from the centre of gravity, how may the equilibrium be restored?
13.(Pg.56) How is this exemplified byfig. 3. plate 4?
14.(Pg.56) What proportion must the weights bear to the lengths of the arms?
15.(Pg.57) On what principle do we weigh with a pair of steelyards, and what will be the difference in the motion of the extremities of such a lever?
16.(Pg.58) How is this exemplified byfig. 4. plate 4?
17.(Pg.58) What line is described by the ends of a lever?fig. 4. plate 4.
18.(Pg.58) How many kinds are there; and in the first how is the fulcrum situated?
19.(Pg.58) When may the fulcrum be so situated that this lever is not a mechanical power, and why?
20.(Pg.59) What is represented byfig. 5. plate 4?
21.(Pg.59) Give a familiar example of the use of a lever of the first kind.
22.(Pg.59) In what instruments are two such levers combined?
23.(Pg.59) How may two horses of unequal strength, be advantageously coupled in a carriage?
24.(Pg.60) Describe a lever of the second kind. (Fig. 6. plate 4.)
25.(Pg.60) What is represented infig. 7. plate 4, and in what proportion does this lever gain power?
26.(Pg.60) What is said respecting a door?
27.(Pg.60) Describe a lever of the third kind.
28.(Pg.60) In what instance do we use this?
29.(Pg.61) What remarks are made on its employment in the limbs of animals?
30.(Pg.61) What are the conditions of equilibrium in every lever?
OF THE PULLEY. OF THE WHEEL AND AXLE. OF THE INCLINED PLANE. OF THE WEDGE. OF THE SCREW.
MRS. B.
The pulley is the second mechanical power we are to examine. You both, I suppose, have seen a pulley?
Caroline.Yes, frequently: it is a circular, and flat piece of wood or metal, with a string which runs in a groove round it: by means of which, a weight may be pulled up; thus pulleys are used for drawing up curtains.
Mrs. B.Yes; but in that instance the pulleys are fixed; that is, they retain their places, and merely turn round on their axis; these do not increase the power to raise the weights, as you will perceive by this figure. (plate 5. fig. 1.) Observe that the fixed pulley is on the same principle as the lever of a pair of scales, in which the fulcrum F being in the centre of gravity, the power P and the weight W, are equally distant from it, and no advantage is gained.
Emily.Certainly; if P represents the power employed to raise the weight W, the power must be greater than the weight in order to move it. But of what use then is a fixed pulley in mechanics?
Mrs. B.Although it does not increase the power, it is frequently useful for altering its direction. A single fixed pulley enables us to draw a curtain up, by pulling the string connected with it downwards; and we should be at a loss to accomplish this simple operation without its assistance.
Caroline.There would certainly be some difficulty in ascending to the head of the curtain, in order to draw it up. Indeed I now recollect having seen workmen raise weights to a considerable height by means of a fixed pulley, which saved them the trouble of going up themselves.
Mrs. B.The next figure represents a pulley which is not fixed; (fig. 2.) and thus situated, you will perceive that it affords us mechanical assistance.
A is a moveable pulley; that is, one which is attached to the weight to be raised, and which consequently moves up or down with it. There is also a fixed pulley D, which is only of use to change the direction of the power P. Now it is evident that the velocity of the power, will be double that of the weight W; for if the rope be pulled at P, until the pulley A ascends with the weight to the fixed pulley D, then both parts of the rope, C and B, must pass over the fixed pulley, and consequently the hand at P, will have descended through a space equal to those two parts; but the weight will have ascended only one half of that distance.
Caroline.That I understand: if P drew the string but one inch, the weight would be raised only half an inch, because it would shorten the strings B and C half an inch each, and consequently the pulley with the weight attached to it, can be raised only half an inch.
Emily.But I do not yet understand the advantage of moveable pulleys; they seem to me to increase rather than diminish the difficulty of raising weights, since you must draw the string double the length that you raise the weight; whilst with a single pulley, or without any pulley, the weight is raised as much as the string is shortened.
Mrs. B.The advantage of a moveable pulley consists in dividing the difficulty; we must, it is true, draw twice the length of the string, but then only half the strength is required that would be necessary to raise the weight without the assistance of a moveable pulley.
Emily.So that the difficulty is overcome in the same manner as it would be, by dividing the weight into two equal parts, and raising them successively.
Mrs. B.Exactly. You must observe, that with a moveable pulley the velocity of the power, is double that of the weight; since the power P (fig. 2.) moves two inches whilst the weight W moves one inch; therefore the power need not be more than half the weight, to make their momentums equal.
Caroline.Pulleys act then on the same principle as the lever; the deficiency of weight in the power, being compensated by its superior velocity, so as to make their momentums equal.
Mrs. B.You will find, that all gain of power in mechanics is founded on the same principle.
Emily.But may it not be objected to pulleys, that a longertime is required to raise a weight by their aid, than without it? for what you gain in power, you lose in time.
Mrs. B.That, my dear, is the fundamental law in mechanics: it is the case with the lever, as well as the pulley; and you will find it to be so with all the other mechanical powers.
Caroline.I do not see any advantage in the mechanical powers then, if what we gain by them in one way, is lost in another.
Mrs. B.Since we are not able to increase our natural strength is not any instrument of obvious utility, by means of which we may reduce the resistance or weight of any body, to the level of that strength? This the mechanical powers enable us to accomplish. It is true, as you observe, that it requires a sacrifice of time to attain this end, but you must be sensible how very advantageously it is exchanged for power. If one man by his natural strength could raise one hundred pounds only, it would require five such men to raise five hundred pounds; and if one man performs this by the help of a suitable engine, there is then no actual loss of time; as he does the work of five men, although he is five times as long in its accomplishment.
You can now understand, that the greater the number of moveable pulleys connected by a string, the more easily the weight is raised; as the difficulty is divided amongst the number of strings, or rather of parts into which the string is divided, by the pulleys. Two, or more pulleys thus connected, form what is called a tackle, or system of pulleys. (fig. 3.) You may have seen them suspended from cranes to raise goods into warehouses.
Emily.When there are two moveable pulleys, as in the figure you have shown to us, (fig. 3.) there must also be two fixed pulleys, for the purpose of changing the direction of the string, and then the weight is supported by four strings, and of course, each must bear only one fourth part of the weight.
Mrs. B.You are perfectly correct, and the rule for estimating the power gained by a system of pulleys, is to count the number of strings by which the weight is supported; or, which amounts to the same thing, to multiply the number of moveable pulleys by two.
In shipping, the advantages of both an increase of power, and a change of direction, by means of pulleys, are of essential importance: for the sails are raised up the masts by the sailors on deck, from the change of direction which the pulley effects, and the labour is facilitated by the mechanical power of a combination of pulleys.
Plate v.
Emily.But the pulleys on ship-board do not appear to me to be united in the manner you have shown us.
Mrs. B.They are, I believe, generally connected as described infigure 4, both for nautical, and a variety of other purposes; but in whatever manner pulleys are connected by a single string, the mechanical power is the same.
The third mechanical power, is the wheel and axle. Let us suppose (plate 5. fig. 5) the weight W, to be a bucket of water in a well, which we raise by winding round the axle the rope, to which it is attached; if this be done without a wheel to turn the axle, no mechanical assistance is received. The axle without a wheel is as impotent as a single fixed pulley, or a lever, whose fulcrum is in the centre: but add the wheel to the axle, and you will immediately find the bucket is raised with much less difficulty. The velocity of the circumference of the wheel is as much greater than that of the axle, as it is further from the centre of motion; for the wheel describes a great circle in the same space of time that the axle describes a small one, therefore the power is increased in the same proportion as the circumference of the wheel is greater than that of the axle. If the velocity of the wheel is twelve times greater than that of the axle, a power twelve times less than the weight of the bucket, would balance it; and a small increase would raise it.
Emily.The axle acts the part of the shorter arm of the lever, the wheel that of the longer arm.
Caroline.In raising water, there is commonly, I believe, instead of a wheel attached to the axle, only a crooked handle, which answers the purpose of winding the rope round the axle, and thus raising the bucket.
Mrs. B.In this manner (fig. 6;) now if you observe the dotted circle which the handle describes in winding up the rope, you will perceive that the branch of the handle A, which is united to the axle, represents the spoke of a wheel, and answers the purpose of an entire wheel; the other branch B affords no mechanical aid, merely serving as a handle to turn the wheel.
Wheels are a very essential part of most machines; they are employed in various ways; but, when fixed to the axle, their mechanical power is always the same: that is, as the circumference of the wheel exceeds that of the axle, so much will the energy of the power be increased.
Caroline.Then the larger the wheel, in proportion to the axle, the greater must be its effect?
Mrs. B.Certainly. If you have ever seen any considerable mills or manufactures, you must have admired the immense wheel, the revolution of which puts the whole of the machinery into motion; and though so great an effect is produced by it, a horse or two has sufficient power to turn it; sometimes a stream of water is used for that purpose, but of late years, a steam-engine has been found both the most powerful and the most convenient mode of turning the wheel.
Caroline.Do not the vanes of a windmill represent a wheel, Mrs. B.?
Mrs. B.Yes; and in this instance we have the advantage of a gratuitous force, the wind, to turn the wheel. One of the great benefits resulting from the use of machinery is, that it gives us a sort of empire over the powers of nature, and enables us to make them perform the labour which would otherwise fall to the lot of man. When a current of wind, a stream of water, or the expansive force of steam, performs our task, we have only to superintend and regulate their operations.
The fourth mechanical power is the inclined plane; this is generally nothing more than a plank placed in a sloping direction, which is frequently used to facilitate the raising of weights, to a small height, such as the rolling of hogsheads or barrels into a warehouse. It is not difficult to understand, that a weight may much more easily be rolled up a slope than it can be raised the same height perpendicularly. But in this, as well as the other mechanical powers, the facility is purchased by a loss of time (fig. 7;) for the weight, instead of moving directly from A to C, must move from B to C, and as the length of the plane is to its height, so much is the resistance of the weight diminished.
Emily.Yes; for the resistance, instead of being confined to the short line A C, is spread over the long line B C.
Mrs. B.The wedge, which is the next mechanical power, is usually viewed as composed of two inclined planes (fig. 8:) you may have seen wood-cutters use it to cleave wood. The resistance consists in the cohesive attraction of the wood, or any other body which the wedge is employed to separate; the advantage gained by this power is differently estimated by philosophers; but one thing is certain, its power is increased, in proportion to the decrease of its thickness, compared with its length. The wedge is a very powerful instrument, but it is always driven forward by blows from a hammer, or some other body having considerable momentum.
Emily.The wedge, then, is rather a compound than a distinctmechanical power, since it is not propelled by simple pressure, or weight, like the other powers.
Mrs. B.It is so. All cutting instruments are constructed upon the principle of the inclined plane, or the wedge: those that have but one edge sloped, like the chisel, may be referred to the inclined plane; whilst the axe, the hatchet, and the knife, (when used to split asunder) are used as wedges.
Caroline.But a knife cuts best when it is drawn across the substance it is to divide. We use it thus in cutting meat, we do not chop it to pieces.
Mrs. B.The reason of this is, that the edge of a knife is really a very fine saw, and therefore acts best when used like that instrument.
The screw, which is the last mechanical power, is more complicated than the others. You will see by this figure, (fig. 9.) that it is composed of two parts, the screw and the nut. The screw S is a cylinder, with a spiral protuberance coiled round it, called the thread; the nut N is perforated to receive the screw, and the inside of the nut has a spiral groove, made to fit the spiral thread of the screw.
Caroline.It is just like this little box, the lid of which screws on the box as you have described; but what is this handle L which projects from the nut?
Mrs. B.It is a lever, which is attached to the nut, without which the screw is never used as a mechanical power. The power of the screw, complicated as it appears, is referable to one of the most simple of the mechanical powers; which of them do you think it is?
Caroline.In appearance, it most resembles the wheel and axle.
Mrs. B.The lever, it is true, has the effect of a wheel, as it is the means by which you turn the nut, or sometimes the screw, round; but the lever is not considered as composing a part of the screw, though it is true, that it is necessarily attached to it.
Emily.The spiral thread of the screw resembles, I think, an inclined plane: it is a sort of slope, by means of which the nut ascends more easily than it would do if raised perpendicularly; and it serves to support it when at rest.
Mrs. B.Very well: if you cut a slip of paper in the form of an inclined plane, and wind it round your pencil, which willrepresent the cylinder, you will find that it makes a spiral line, corresponding to the spiral protuberance of the screw. (Fig. 10.)
Emily.Very true; the nut then ascends an inclined plane, but ascends it in a spiral, instead of a straight line: the closer the threads of the screw, the more easy the ascent: it is like having shallow, instead of steep steps to ascend.
Mrs. B.Yes; excepting that the nut takes no steps, as it gradually winds up or down; then observe, that the closer the threads of the screw, the less is its ascent in turning round, and the greater is its power; so that we return to the old principle,—what is saved in power is lost in time.
Emily.Cannot the power of the screw be increased also, by lengthening the lever attached to the nut?
Mrs. B.Certainly. The screw, with the addition of the lever, forms a very powerful machine, employed either for compression or to raise heavy weights. It is used by book-binders, to press the leaves of books together; it is used also in cider and wine presses, in coining, and for a variety of other purposes.
Emily.Pray, Mrs. B., by what rule do you estimate the power of the screw?
Mrs. B.By measuring the circumference of the circle, which the end of the lever would form in one whole revolution, and comparing this with the distance from the centre of one thread of the screw, to that of its next contiguous turn; for whilst the lever travels that whole distance, the screw rises or falls only through the distance from one coil to another.
Caroline.I think that I have sometimes seen the lever attached to the screw, and not to the nut, as it is represented in the figure.
Mrs. B.This is frequently done, but it does not in any degree affect the power of the instrument.
All machines are composed of one or more of these six mechanical powers we have examined; I have but one more remark to make to you relative to them, which is, that friction in a considerable degree diminishes their force: allowance must therefore always be made for it, in the construction of machinery.
Caroline.By friction, do you mean one part of the machine rubbing against another part contiguous to it?
Mrs. B.Yes; friction is the resistance which bodies meet with in rubbing against each other; there is no such thing as perfect smoothness or evenness in nature; polished metals, though they wear that appearance more than most other bodies, are farfrom really possessing it; and their inequalities may frequently be perceived through a good magnifying glass. When, therefore, the surfaces of the two bodies come in contact, the prominent parts of the one, will often fall into the hollow parts of the other, and occasion more or less resistance to motion.
Caroline.But if a machine is made of polished metal, as a watch for instance, the friction must be very trifling?
Mrs. B.In proportion as the surfaces of bodies are well polished, the friction is doubtless diminished; but it is always considerable, and it is usually computed to destroy one-third of the power of a machine. Oil or grease is used to lessen friction: it acts as a polish, by filling up the cavities of the rubbing surfaces, and thus making them slide more easily over each other.
Caroline.Is it for this reason that wheels are greased, and the locks and hinges of doors oiled?
Mrs. B.Yes; in these instances the contact of the rubbing surfaces is so close, and they are so constantly in use, that they require to be frequently oiled, or a considerable degree of friction is produced.
There are two kinds of friction; the first is occasioned by the rubbing of the surfaces of bodies against each other, the second, by the rolling of a circular body; as that of a carriage wheel upon the ground: the friction resulting from the first is much the most considerable, for great force is required to enable the sliding body to overcome the resistance which the asperities of the surfaces in contact oppose to its motion, and it must be either lifted over, or break through them; whilst, in the second kind of friction, the rough parts roll over each other with comparative facility; hence it is, that wheels are often used for the sole purpose of diminishing the resistance from friction.
Emily.This is one of the advantages of carriage wheels, is it not?
Mrs. B.Yes; and the larger the circumference of the wheel the more readily it can overcome any considerable obstacles, such as stones, or inequalities in the road. When, in descending a steep hill, we fasten one of the wheels, we decrease the velocity of the carriage, by increasing the friction.
Caroline.That is to say, by converting the rolling friction into the rubbing friction. And when you had casters put to the legs of the table, in order to move it more easily, you changed the rubbing into the rolling friction.
Mrs. B.There is another circumstance which we have already noticed, as diminishing the motion of bodies, and which greatlyaffects the power of machines. This is the resistance of the medium, in which a machine is worked. All fluids, whether elastic like air, or non-elastic like water and other liquids, are called mediums; and their resistance is proportioned to their density; for the more matter a body contains, the greater the resistance it will oppose to the motion of another body striking against it.
Emily.It would then be much more difficult to work a machine under water than in the air?
Mrs. B.Certainly, if a machine could be worked invacuo, and without friction, it would not be impeded, but this is unattainable; a considerable reduction of power must therefore be allowed for, from friction and the resistance of the medium.
We shall here conclude our observations on the mechanical powers. At our next meeting I shall endeavour to give you an explanation of the motion of the heavenly bodies.