in which equation it is to be observed, that the symbol b does not appear; that element of the resistance (which is constant), affecting the tensions t, and t, equally, and there fore eliminating with T, and T. Let by a, then

a

a+1

t=at.-W. Similarly, t,=at,— “W,

a

a

be represented

(143).

Eliminating between these equations precisely as between the similar equations in the preceding case (equation 140), observing only that here is represented by -W, and that the equations (143) are n-1 in number intead of n, we have

Also adding the preceding equations (143) together, we have

t1 + t2+ +t2-1= a(t2+t2+

...

a W

t1)—(n-1)

a

Now the pressure P, is sustained by the tensions t,, t,, &c. of the different strings attached to the bar which carries it. Including P, therefore, the weight of the bar, we have

2

t1+t2+...+t-1+t=P1; .t+t2+.. +t-1=P2-t; and t2+...+t1=P,—t1;

..P,—t=a(P,—t,)—(n−1)a W.

a

Substituting this value of t in equation (144),

1

=

(1+a-1)”—1' 1—un a-n-1 (1+a-1)" — 1'

t=

a-1P2 W

(1 + a−1)” — 1

Now P1=at,+b;

P,

.: P1 = (1 + a−1)" — 1

+

a

n

and

a

1 -a

= a;

=

Whence observing that when a=1, (1+a-1)"-12"-1, we obtain for the modulus of uniform motion (equation

A

A TACKLE OF ANY NUMBER OF SHEAVES.

162. If an number of pulleys (called in this case sheaves) be made to turn on as many different centres in the same block A, and if in another block B there be similarly placed as many others, the diameter of each of the last being one half that of a corresponding pulley or sheave in the first; and if the same cord attached to the first block be made to pass in succession over all the sheaves in the two blocks, as shown in the figure, it is evident that the parts of this cord 1, 2, 3, &c. passing between the two blocks, and as many in number as there are sheaves, will be parallel to each other, and will divide between them the pressure of a weight P, suspended from the lower block: moreover, that they would divide this pressure between them equally were it not for the friction of the sheaves upon their bearings and the rigidity of rope; so that in this case, if there were n sheaves, the tension upon each would be P,; and a. pressure P, of that

amount applied to the extremity of the cord would be suffi cient to maintain the equilibrium of the state bordering upon motion. Let T,, T,, T, &c. represent the actual tensions upon the strings in the state bordering on motion by the preponderance of P,, beginning from that which passes from P, over the largest sheaf; then

P1=a,T,+b,, T, =a,T2+b2, T,=a,T,+b,

&c. &c., Ta2T2+b12;

=

=

where a,,a,, &c., b1, b,, &c. represent certain constant coefficients, dependent upon the dimensions of the sheaves and the rigidity of the rope, and determined by equation (131). Moreover, since the weight P, is supported by the parallel tensions of the different strings, we have

P1=T,+T,+. . . . +T„.

It will be observed that the above equations are one more in number than the quantities T,, T., T., &c. ; the latter may therefore be eliminated among them, and we shall thus obtain a relation between the weight P, to be raised and that P, necessary to raise it, and from thence the modulus of the system."

To simplify the calculation, and to adapt it to that form of the tackle which is commonly in use, let us suppose another arrangement of the sheaves. Instead of their being of different diameters and placed all in the same plane, as shown in the last figure, let them be of equal diameter and placed side by side, as in the accompanying figure, which represents the common tackle. The inconvenience of this last mode of arrangement is, that the cord has to pass from the plane of a sheaf in one block to the plane of the corresponding sheaf in the other obliquely, so that the parts of the cords between the blocks are not truly parallel to one another, and the sum of their tensions is not truly equal to the weight P, to be raised, but somewhat greater than it. So long, however, as the blocks are not very near to one another, this deflection of the cord is inconsiderable, and the error resulting from it in the calculation may be neglected. Supposing the different parts of the cord between the blocks then to be parallel, and the diameters of all the sheaves and

their axes to be equal, also neglecting the influence of the weight of each sheaf in increasing the friction of its axis, since these weights are in this case comparatively small, the co-efficients a, a, a, will manifestly all be equal; as also b1, by, bs ;

.. P1=aT, +b, T=aT,+b, T,=aT,+b, } &c.=&c., T-,=aT„+b

}

+T2:

(147);

Multiplying equations (147) successively (beginning from the second) by a, a, a, and a"-1; then adding them together, striking out the terms common to both sides, and summing the geometric series in the second member (as in equation 140), we have

Adding equations (147), and observing that T,+T2+ +T=P, and that P,+T,+T2+

P1+P-T, we have

P1+P‚¬T=aP,+nb.

....

+T2-1=

To determine the modulus let it be observed, that, neglecting friction and rigidity, a becomes unity; and that for this value of a, becomes a vanishing fraction, whose

a"(a−1)
a"-1

value is determined by a well known method to be 1*. Hence (Art. 152.),

* Dividing numerator and denominator of the fraction by (a-1) it becomes

a

-1+ a

a

n

which evidently equals when a=1. The modulus + +1' may readily be determined from equation (148). Let S, and S. represent the spaces described by P, and P2 in any the same time; then, since when the blocks are made to approach one another by the distance S2, each of the n por tions of the cord intercepted between the two blocks is shortened by this dis

Hitherto no account has been taken of the work expended in raising the rope which ascends with the ascending weight. The correction is, however, readily made. By Art. 60. it appears that the work expended in raising this rope (different parts of which are raised different heights) is precisely the same as though the whole quantity thus raised had been raised at one lift through a height equal to that through which its centre of gravity is actually raised. Now the cord raised is that which may be conceived to lie between two positions of P, distant from one another by the space S,, so that its whole length is represented by nS,; and if represent the weight of each foot of it, its whole weight is represented by uns: also its centre of gravity is evidently raised between the first and second positions of P, by the distance. S,; so that the whole work expended in raising it is represented by fans," or by 1, since SnS,. Adding this work expended in raising the rope to that which would be necessary to raise the weight P, if the rope were without weight, we obtain*

n

2

an-1

which is the MODULUS of the tackle.

THE MODULUS OF A COMPOUND MACHINE.

163. Let the work of a machine be transmitted from one to another of a series of moving elements forming a compound machine, until from the moving it reaches the working point of that machine. Let P be the pressure under which the work is done upon the moving point, or upon the first moving element of the machine; P, that under which it is

tance S2, it is evident that the whole length of cord intercepted between the two blocks is shortened by nS2; but the whole of this cord must have passed over the first sheaf, therefore S1 =nS. Multiplying equation (148) by this equation, and observing that U1 =P1S, and U, PS2, we obtain the modulus as given above.

* A correction for the weight of the rope may be similarly applied to the modulus of each of the other systems of pulleys. The effect of the weight of the rope in increasing the expenditure of work on the friction of the pulleys is neglected as unimportant to the result.