We learn at the same time from this experiment, that the phenomenon in question requires (1.) That the point should be small compared with the distance of the parts of the top from the axis ND; and (2.) That these parts should turn with considerable rapidity; and the success will be more or less complete, as these conditions are more or less perfectly fulfilled.

It will be seen, moreover, that upon an inclined plane the top must have a tendency not to a vertical but to a perpendicular to the plane. But as it must at the same time slide along the plane, and as this motion would cause a great vacillation in passing over the inequalities of the plane, it will not so easily preserve its perpendicular position as if the plane were horizontal.

Fig.140.

Fig.141.

Of the Stiffness of Cords.

253. The stiffness of ropes and cords, or the difficulty with which they are bent into a given curve, is also one of the causes which diminish the effect of forces applied to machines.

In order to understand in what manner this stiffness impairs the effect of forces, let us suppose the wheel or pulley ABC to be moveable about the axle I, without friction. The two weights p and q being equal, if we make a very small addition to one of them, as q, for example, no motion will follow, unless the cord p ABC q be perfectly flexible. Indeed, if we imagine that this cord, instead of being perfectly flexible, is perfectly inflexible, so that the parts Ap, Cq, are stiff rods firmly fixed to the body of the pulley; it is evident that the pulley being moved by an external force in the direction ABC, the two weights p and q will take the situations p' and q'; but they will tend to return to their first position, and can be prevented only by the constant exertion of a particular force. If, then, the cord is neither perfectly inflexible, nor perfectly flexible, the effect of this imperfect flexibility will be, that the point A passing to A', and the point C to C', the parts A' p', C' q', will be a little bent, and in such a manner that the weight p' will be farther from I, and the weight qʻ nearer to it, than they would be if the cord were perfectly flexi

ble; so that a certain force is required in order to bring the parts A'O, CC', into the direction of tangents to the points A and C; in other words, a force must be employed which would be unnecessary but for this want of flexibility.

The pulley being always supposed to move with perfect ease upon its axle I, if instead of a cord a ribbon be employed, a very small increase in the weight q will cause the pulley to turn. But if the cord be replaced, it will evidently be necessary to augment the weight q; (1.) According as the sum of the weights p and 9, or, in general, the whole force by which the cord is stretched, is more considerable; because, other things being the same, the resistance occasioned by the weights p and q, when by the stiffness of the cord they take the positions A'Op', CC' q', will increase as the weights themselves increase.

(2.) The addition to be made to q, must be greater according as the radius of the pulley, (or of the surface over which the cord passes) is less. For the resistance which the power meets with arises from this, that the cord, instead of adapting itself to the revolving surface, remains at a certain distance, forming a curve p' OA' and making with the surface an angle OA'A; and this resistance will evidently be the greater, according as the curvature A'O of the cord departs more from the curvature of the surface; that is, according to the smallness of the radius of this surface.

(3.) The power applied must also be increased in proportion to the diameter of the cord. Indeed it is manifest, that, other things being the same, the cord will bend the less according as the thickness is greater; but we have just seen that the resistance to the power is greater according as the curve A'O differs more from the curve A'A; it is therefore the greater according as A'O differs less from a straight line, or the position of an inflexible rod, that is, according as the cord has a greater diameter or radius.

254. Let us suppose that k is the addition to be made to a power to render it sufficient to overcome the resistance arising from the stiffness of the cords, when the entire force by which the cord is stretched is p, the diameter of the cord which bears

the weight being 6, and the radius of the surface R. We wish to know what this addition must be, when the weight is p', the diameter of the cord d', and the radius of the surface R'. It will be observed, after what has been said, that if there were no difference except with respect to the entire weight by which the cord is stretched, we should arrive at a solution by the proportion

But if, beside the difference in the weights, there is also a difference in the curvature of the surfaces; then, by the second of the above remarks; namely, that the additions arising from this cause are in the inverse ratio of the radii of the surfaces, we should obtain the addition in question, together with that due to a change of weight, by the following proportion,

Regard being had to the third remark, we shall obtain the addition to be made on account of the three causes united, by the proportion

The resistance in the first case, therefore, will be to the resistance in the second

that is, the resistances arising from the stiffness of the cords are as the weights which stretch the cords, multiplied by the diameters of these cords, and divided by the radii of the surfaces over which they pass.

These conclusions, it may be observed, are not perfectly rigorous; but they may be regarded as sufficiently exact for practice, till experiment has thrown new light upon the subject. Indeed, experiment shows that the resistance arising from the stiff

ness of cords, agrees nearly with this law; but all the experiments that have been made upon this subject have not hitherto agreed so perfectly with the theory as might be wished.

255. We shall now illustrate the foregoing principles by an example. For this purpose, let us suppose the common radius of the pulley to be 2 inches, that of the axle of the same quanti- Fig. 92. ty, and the diameter of the cords to be of an inch. Let us take, moreover, the result of experiments on this subject, namely, that a cord of half an inch diameter, loaded with 120lb, and passing over a pulley of of an inch, occasions, by its stiffness, a resistance of 8lb.

This being established, the weight p being 400, as in article 242, the branches 1 and 2 will be loaded, both on account of this weight and the force added to q to overcome the friction, by a force equal to 209, 6. Multiplying, therefore, this weight by of an inch, the diameter of the cord, and dividing by 2 inches, the radius common to all the pulleys; multiplying also the weight 120, used in our experiment, by, the diameter of the cord, and dividing by 2, the radius of the pulley, in the experiment in question; we shall have for the results, 34, 93,; and 40. Now in order to obtain the force required by the stiffness of the cord 1, 2, which passes beneath the moveable pulley, we must use this proportion,

40: 34, 93: gib: 6, 99 or 7b nearly;

this fourth term is the quantity which it would be necessary to add to the tension of the cord 2, if the lower pulley were fixed; but since it is moveable, which diminishes the tension by, as we shall see below, we have only 3, 5 to be added to 109, 6, by which this cord is already stretched; the whole tension will thus be 113, 1.

We have seen that on account of friction, the tension of the 242: eord 3 ought to be equal to 120, 7; therefore the whole force by which the cord 2, 3, which passes over the fixed block, is extended, is 113, 1 + 120, 7 on 233, 8. Multiplying this force as above, by, and dividing by 2, we shall have 38,97. Seeking as before the value of the stiffness of the cord 2, 3, which passes over the fixed block, we shall have the proportion.

242.

40 38, 97 :: glb: 7lb 79.

Adding now these 7b, 79 to 120, 7, the quantity before found for the tension of the cord 3, we shall have 128, 5 for the force of tension, both on account of friction and the stiffness of the cords.

=

The cord 4, on account of friction is stretched by a force 130lb, 3; the whole force of tension upon the cord 3, 4, which embraces the moveable block, is therefore 258, 8; with this quantity, the dimensions of the pulleys and cords remaining the same, we shall find, as above, that the allowance for the stiffness of the cords, would be 8b, 62, if this cord did not embrace the moveable block; but proceeding as we have done above, and for the same reason, we must take but half of this quantity; thus the tension of the cord 4, all things considered, will be 134, 6 nearly.

=

The cord 5, on account of friction, is stretched with a force 143b, 5; the whole force by which the cord 4, 5, embracing the fixed pulley, is stretched, is therefore 278,1. With this value, the dimensions remaining the same, we shall find the force required for the stiffness of the cords to be 9lb, 3; the whole tension of the cord, therefore, is 152b, 8. Thus friction and the stiffness of the cords together require the weight q, which would otherwise be but 100b, to be 153b nearly, a quantity greater by more than one half.

We have taken only half of what the calculation furnished as the quantity to be added to cords 2 and 4, on account of the stiffness. The reason of this is, that the cord by which the moveable pulley is made to revolve, may be considered as turning it on a centre placed at the point where this pulley touches the other cord; and consequently the case is similar to that of a fixed pulley, having a radius equal to the diameter of the moveable pulley; and since the forces required on account of the stiffness of the cords, are in the inverse ratio of the radii of the pulleys, the force in this case must be diminished one half.