In Fig. 52 the pull P is in the same direction that W is to be moved and is only one-half of W. As explained in Art. 94, the mechanical advantage of a machine can be obtained by comparing the distances moved by the force and the weight. If a force must move five times as far as it lifts the weight, then the mechanical advantage is 5, and the force is one-fifth of the weight. In Fig. 52 the mechanical advantage is 2. This can be seen by raising W a certain distance. The rope on each side will be slacked this same distances and, therefore, P must be drawn up twice this distance in order to remove the slack. Since P moves twice as far as W, the force P will be one-half of W, and the mechanical advantage will be 2.

In Fig. 53 we have merely added, to the device of Fig. 52, a fixed block above to change the direction of P. It makes no change in the relation of P and W, except as to direction.

In Fig. 54 we have two pulleys in the fixed block and two in the movable block. Other cases might have even more pulleys, but the principle is the same, and a general rule for calculating their mechanical advantages will be worked out for all cases. Proceeding as before, let us imagine that W and the movable block of Fig. 54 are lifted 1 ft. The four ropes supporting W will each be slacked 1 ft., and it will be necessary to move P 4 ft. to remove this slack. Hence, the mechanical advantage of this system is 4, and P is of W, or W is 4 times P.

In general, we can say that the mechanical advantage is equal to the number of ropes supporting the movable block and the load. The best way to find the mechanical advantage is to draw. a sketch of the blocks and to count the number of ropes that are pulling on the movable block. This number represents the mechanical advantage.

Whenever convenient, it is best to use as the movable block the one from which the free end of the rope runs. This means that P will pull in the same direction that W is to be moved. The

mechanical advantage is greater by 1 if P is pulling in the direction of motion. Notice in Fig. 54 that, if we turned these blocks around and pulled the other way, fastening W to what is in the figure the fixed block, the mechanical advantage would be 5 instead of 4.

In all problems where there is any doubt, draw a rough sketch and count the number of ropes pulling on the movable block (see Fig. 55).

Example: How great a weight can be lifted by a pull of 150 lb. with a pair of pulley blocks, one being a three sheave and the other a two sheave block? Calculate, first, using the three sheave as the movable block and, second, using the two sheave block as the movable one.

Explanation: To avoid confusion, the sheaves are drawn one above the other, instead of parallel. The free end of the rope must run from the three sheave block. Starting from P, we wind the rope in and find that the inner end must be fastened to the two sheave block. We count the ropes pulling on each block and find that, with the three sheave block as the movable one, the mechanical advantage is 6, and the weight lifted would be

150×6=900 lb., First Answer.

With the two sheave block as the movable one, the mechanical advantage is 5 and the weight would be

150X5=750 lb., Second Answer.

In practice, about 60% of these theoretical weights would be raised, the rest being lost in overcoming friction. Likewise, to lift a certain load, the actual pull required will be about 100 or 13 of the theoretical pull.

97. Differential Pulleys.-In lifting heavy weights by hand, a very satisfactory apparatus to use is a Differential Hoist. This is a very simple and cheap apparatus but it is not very efficient and is, therefore, not to be recommended for continuous use.

The pulleys are arranged as shown in Fig. 56. In the fixed block are two pulleys, A and B, A being somewhat larger than B.

These pulleys, A and B, are fastened solidly together and rotate as one about a fixed axis; the pulley C is in the movable block. An endless chain passes over the pulleys as shown, the rims of the pulleys being grooved and fitted with lugs to prevent the chain from slipping. The loop np hangs free and is the pulling loop.

From the figure it is easily seen that, if we pull down on p until pulley A is turned once around, the branch m will be shortened a length equal to the circumference of A. Since B is attached to A, it also will turn once around and the branch o will be lengthened a distance equal to the circumference of B.

Hence, the loop mo will be shortened by an amount equal to the difference of the circumferences of A and B; and the pulley C will rise one-half this amount.

We can express the difference in the distances moved by m

where D and d represent the diameters of large and small pulleys, A and B. Hence C will move up one-half of this or of л× (D–d). To cause this motion of C upward, the chain p was moved a distance of

XD.

The mechanical advantage of the hoist is obtained by dividing the motion of P by the motion of W.

This can be simplified by cancelling out of both numerator and denominator of the fraction, leaving

This formula might be written as a rule in the following words: "The mechanical advantage of a differential hoist is obtained by dividing the diameter of the larger pulley in the upper block by half the difference between the diameters of the larger and smaller pulleys."

A differential hoist can actually lift about 30% of the theoretical load with a given pull; that is, the efficiency is about 30%. Likewise, to lift a given weight will require about 100 or 3

times the theoretical force. In other words, the actual force must be such that the 30% that is really effective will equal the theoretical force.

Example:

Calculate the actual pull required to lift 600 lb. with a differential hoist having 10 and 8 in. pulleys and an efficiency of 30%.

Explanation: In this case the letters D and d of the formula are 10 in. and 8 in., and we find the M. A. to be 10. It should, therefore, only require a force of 60 lb. to raise the 600 lb. weight. But we find that this type of hoist has only an efficiency of 30%, that is, it only does 30% of what we might expect it to do from our theories. Then to lift 600 lb. will require a force such that 30% of it will be 60 lb. This necessary force is 200 lb.