a2

(137),

U1 =2( 42 )U,+ (b− Wa)s, . . . . . (138).

1+

2

It is evident that, since the co-efficient of the second term of the modulus of this system is less than that of the first system (equation 136) (the quantities a and b being essentially positive), a given amount of work U, may be done by a less expense of power U,, or a given weight P, may be raised to a given height with less work, by means of this system than the other; an advantage which is not due entirely to the circumstance that the weight of the moveable pulley in this case acts in favour of the power, whereas in the other it acts against it; and which advantage would exist, in a less degree, were the pulleys without weight.

A SYSTEM OF ONE FIXED AND ANY NUMBER OF MOVEABLE PULLEYS.

160. Let there be a system of n moveable pulleys and one fixed pulley combined as shown in the figure, a separate string passing over each moveable pulley; and let the tensions on the two parts of the string which passes over the first moveable pulley be represented by T, and t,. those upon the two parts of the string which passes over the second by T, and t2, &c. Also, to simplify the calculation, let all the pulleys be supposed of equal dimensions and weights, and the cords of equal rigidity;

... T1=at1+b1, and T2+W=T1+t1;

. ́. eliminating, T, = (1 4a) T2+

Wa+b1
1+ a

(139).

Let the co-efficients of this equation be represented by

a

... T1 =αT2+ß.

4

2

Similarly, TaT ̧+ß, T ̧=aT1+ß, T1 =œT¿+ß, &c.=&c., T2-1=aT2+ß, T„=αP2+ß.

Multiplying these equations successively, beginning from the second, by a, a2, a3, &c., a”-1, adding them together, and striking out terms common to both sides of the resulting equation, we have

or summing the geometrical progression in the second member, T1 = a'P, +8 (=)

απ

(140);

Substituting for a and ẞ their values from equation (139), and reducing

a

a

.. P1 = a ( 1 ) P2 + a(Wa+b; ) { 1-(+)"} +b...(141).

2

Whence observing, that, were there no friction, a would

α

become unity, and and (14a)": (1 —-—-a)" = (-)". We have (equation

121) for the modulus of this system,

U1 = a(12,4a) "U, + {a(Wa+b,) {(1 − (1 —-a)"} +b }S, ... ... (142).

161. If each cord, instead of having one of its extremities attached to a fixed obstacle, had been connected by one extremity to a moveable bar carrying the weight P, to be raised (an arrangement which is shown in the second figure), then, adopting the same notation as before, we have

T1 = at1+b,

2

at,+b=T2, T2=T1+t1+W. Adding these equations together, striking out terms common to both sides, and solving in respect to T1, we have

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 t2 equally, and therefore elimi

a

nating with T, and T. Let be represented by a, then

x

a+1

t=at,-W. Similarly, t1 = at,—~W,

(143).

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

a

a- -1

(144).

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

W

t1 + t2 + ... + tn_1=a(t2 + tz +
+ tm1=a(tg+tz + • • • t») — (n− 1) a \ .

2

...

a

Now the pressure P, is sustained by the tensions t1, tą, &c. of the different strings attached to the bar which carries it. Including P,, therefore, the weight of the bar, we have

t1 + t2+...+t+t=P2 ; .'. t1 + t2 + . . + t1 = P2-tn ;

2

Substituting this value of t, in equation (144),

2

aW (a21-1).

'1 = (1 − a )x"−1 P2 + a "l1 + (n − 1) a "W_ a W (a"

a

Whence observing that when a=1, {(1+a ̄1)”—1 =2′′—1, we obtain for the modulus of uniform motion (equation

A TACKLE OF ANY NUMBER OF SHEAVES.

162. If any 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 the rope; so that in this case, if there were ʼn sheaves, the tension upon each would be 1P,; and a

n

n

pressure P, of that amount applied to the extremity of the cord would be sufficient to maintain the equilibrium of the state bordering upon motion. Let T1, T2, T3, &c. represent the actual tensions upon the strings in the state bordering on motion by the preponderance of P1, beginning from that which passes from P, over the largest sheaf; then

P1=a1T1+b1, T1=a2T2+b2, T2=a3T3+b3,

2

&c.=&c., T1_1=ɑnTn+bn ;

where a,, a,, &c., b1, b2, &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=T1+T2+. ... +T„.

It will be observed that the above equations are one more in number than the quantities T1, T2, T3, &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 P1 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