932
THEORY AND PRACTICE OF ENGINEERING.
Book II.
condition of sucli an effect may be thus expresseddistinct ropes.
More power may he gained by an arrangement in which eachrope is finally attached to the weight. The first rope sustains a partof the weight which is equal to the power : the tension of thesecond rope is twice that of the first, and therefore sustains a partof the weight equal to twice the power ; the third sustains a partequal to four times, and so on, the part sustained by each ropebeing double that sustained by the preceding one. C is the fixedpulley, B and A are the movable, and the three ropes are attachedto the weight W, at D, E, and F.
When a rope passes over a single wheel or fixed pulley, as itstension is uniform throughout the length, it follows that in such acase the power and weight are equal; for the latter stretches thatportion of the rope which is between the weight and the pulley,while the former stretches that between the power and the pulley :and because the tension throughout the whole length is the same,the weight must be equal to the power; no mechanical advantageis therefore gained from the single pulley, hut there are various ap-plications of it where considerable benefit is derived.
In directing the power against the resistance, two fixed pulleysare often found useful; thus, in elevating a weight, the rope maybe carried over the pulley at top, and returning downwards, broughtby means of another on an horizontal plane or the ground.
A single pulley merely serves to change the direction of motion;that placed on the ground, around which the rope passes, maywind round a windlass, or have the power applied in a horizontaldirection, which may he calculated according to the general rule.
The mechanical power of pulleys admits of being almost inde-finitely increased by combination; systems of pulleys may bedivided into two classes, those in which a single rope is used, andthose which consist of several distinct ropes. In both cases theweight is attached to a movable block, in which are fixed two ormore wheels, and whatever the system, every wheel has a separateaxle, and there is a distinct wheel for every turn of the rope at eachblock : each wheel is attended with friction on its axle, and alsobetween the sheave and block, which destroys a great part of thepower employed. When the pulley is heavy, and acts in the samedirection as the resistance, its weight must be considered as part ofthat resistance; and if its direction differs from that of gravity, inwhich the weight of the pulley acts, an allowance must be made forthe loss of power. A parallelogram may be drawnto express this, of which one side will represent theweight of the pulley, the other the resistance ; itsdiagonal will then show the real resistance and itsdirection, which will be greater or less than theother, according as the direction of gravity coincideswith or opposes that of the resistance.
Whatever the arrangement of the pulley one lawholds good, viz. that as force is diminished, velocityincreases, or the power passes over a space greaterthan that gone over by the weight, by as manytimes as the weight exceeds the power.
It matters not how these several wheels are ar-ranged; they may be combined in a variety ofways so as to gain an advantage, or as it is termeda purchase , which is more or less according tothe number of the pulleys, and their mode of com-bination : the purchase gained is computed bycomparing the celerity of the weight raised withthat of the moving power, according to the prin-ciple of virtual velocities: where the several portionsof the cords are parallel to each other, and theweight rises an inch, the blocks w'ill approach eachother by that quantity, and consequently the lengthof cord connecting a single pair of pulleys wouldbe shortened two inches.
W = 2 n P, n expressing the number of
Fig. 1261.
i\
Fig. 1262
Fig. 1263.