the weight of the next pulley will support three times its own weight, and the weight of the next to that will support just its own weight. The weight of the highest pulley will not give any aid. Thus, finally, we have the following result:-The weight W is equal to the sum of 15 P, together with seven times the weight of the lowest pulley, three times the weight of the next, and the weight of the next to that.

EXAMPLES.

Ex. 1. A weight of 3 tons is lifted by a pair of blocks of two sheaves each, the rope is fastened to the upper block: what strain is on each single of the rope?

60 cwt.
15 cwt.

= 4.

Ex. 2. A boiler weighing 7 tons has to be lifted by a pair of treble blocks: what is the strain on each single of the rope?

7'5 X 2240

16800 lbs. 6 = 2800 lbs.

16800 lbs.

1 ton 5 cwt. Answer.

Ex. 3. A pair of double blocks are used to lift a tons weight: what would be the pull if 20 per cent. be lost by friction?

2 tons X 22404480 lbs.

Pull without friction 4480 lbs. 4 1120 lbs.

With friction: Since 20 per cent. is lost by friction, if we suppose the pull to be 100 lbs., then the weight lifted would only be 80 lbs.; therefore, the pull without friction, viz., 1120 lbs., must be increased in the ratio of 80: 100, or as 8: 10,

Ex. 4. A weight of 4 tons is lifted by a pair of blocks of two sheaves each, the rope is fastened to the upper block: what must be the pull at the free end of the rope, supposing 10 per cent. to be lost at each sheave?

If 10 per cent. is lost at each sheave, the first lifting rope will exert a lifting force equal to 9 of the pull of the free end of the rope.

The others will exert a force equal to 9 of the preceding rope.

The sum of these calculated for all the ropes is the divisor to be used instead of the number of sheaves.

100

That is, the rope, after passing the 1st sheave, will have only %, or f 10 per cent. of the full pull. 9 is the first power of 9. The rope, after passing the 2nd sheave, will have 9 of 981 of the full pull, which is the second power of '9, and so on to as many powers as there are sheaves, and the work may be written thus:

90

Ex. 5. A weight of 2 tons is lifted by a pair of blocks, there are 3 sheaves in each block, the rope is fastened to the upper block: what must be the pull at the free end, supposing 12 per cent. be lost at each sheave?

Calculating as many powers as there are sheaves, thus:

NOTE.-It is quite unnecessary to retain more than four places of decimals in the work.

Weight 2 tons 2 X 22404480 lbs.

3'9277)4480*0000(1140'6 lbs.

39277

55230

39277

159530
157108

242200

I.

2.

3.

4.

5.

EXAMPLES FOR PRACTICE.

What weight on the end of a lever 8 feet in length will exert a force equal to 100 lbs. on the other end, 6 inches being the distance of the fulcrum from the 100 lbs.? Ans., 6.667 lbs.

What weight on the end of a lever 12 feet in length will equipoise 2 tons on the other end, the fulcrum being distant 9 inches from the weight to be equipoised, and the weight of the lever not taken into account? Ans., 1947 lbs.

Four weights, 1, 3, 5, and 7, are at equal distances of 8 inches from each other on a straight lever: how far from each is the fulcrum on which they will balance? Ans., 17 inches, 9 inches, 1 inch, and 7 inches.

The arms of a lever are respectively 15 and 16 inches: find what weight at the end of the short arm will balance 30 lbs. at the end of the long arm, and what weight at the end of the long arm will balance 30 lbs. at the end of the short arm. Ans., 32 lbs.; 28 lbs. 2 ozs.

What force will be required to work the handle of a windlass, the resistance to be overcome being 1156 lbs., the radius of the axle being 6 inches, and of the handle 2 feet 8 inches. Ans., 216.75 lbs.

6. A weight of 17 lbs. just balances a weight of 79 lbs. on a wheel and axle: what will be the radius of the axle if that of the wheel be 17 inches? Ans., 3'65 inches.

7. In a wheel and axle, the radius of the wheel is 20 in., and that of the axle 4 in.; if the power applied is 70 lbs.: what must be the weight? Ans., 350 lbs.

8.

9.

10.

II.

12.

If the radius of the axle be 3 in.: what will then be the weight? Ans., 4663 lbs. What power must be applied, in Ex. 7, in order to support a weight of 400 lbs.? Ans., 80 lbs.

With a wheel and axle a power of 8 lbs. sustains a weight of 12 lbs.: what is the radius of the axle, that of the wheel being 24 in. ? Ans., 16. The circumference of wheel and axle are respectively will sustain a weight of 1 tons. Ans., 12'5 cwt.

1 yd. and 15 in.: what power

12'5 cwt.

Twelve sailors, exerting each a force of 80 lbs., push a capstan with a length of lever equal to 6 ft.; the diameter of capstan is 2 ft. Calculate the resistance which this force is capable of sustaining. Ans., 2 tons 11 cwt. I qr. 20 lbs.

13. In the last example, suppose 10 per cent. of the effort is lost through friction and the rigidity of the rope: what would have been the weight lifted? Ans., 2 tons 6 cwt. I qr. 4 lbs.

14. The diameter of a steering wheel is 5 ft., and the diameter of barrel is 15 in.; if a man applies a power equal to 200 lbs.: what resistance would be overcome? Ans., 800 lbs.

15.

If the diameter of the barrel of the wheel be 8, and the diameter of the rope 1"; weight 200 lbs.: what force must be applied to the handle whose radius is 18? Ans., 15 lbs.

16. In a wheel and axle the diameter of barrel was 6", and that of the rope, the crank handle was 14", and the weight 174 lbs.: what was the force resisted, allowing 16 per cent. for friction? Ans., 49'5589 lbs.

17. In a wheel and axle the barrel is 9" diameter, the rope is 14" diameter, the crank handle is 14′′ radius, and the weight to be lifted 300 lbs.: what is the force applied to the handle, adding 10 per cent. for friction ? Ans., 120.8035 lbs. A machine is formed by the combination of three wheels and axles; the radius of each wheel is four times that of the corresponding axle: if the power is 4 lbs., find the weight. Ans., 256 lbs.

18.

19. In the chinese wheel the length of the power-arm is 30 in., the radius of the larger cylinder is 3 in., and the radius of the smaller 3 in.; if the power is 4 lbs., find the weight. Ans., 480 lbs.

20.

21.

22.

A power P acting by means of a rope going over a wheel whose diameter is 6 feet, supports a weight of 320 lbs., the diameter of the axle is 8 inches, and the rope is 2 inches thick: what is the value of P, supposing the thickness of the rope to be taken into account?

The length of an inclined plane is 80 feet, height 2 feet, the weight of the body 2 tons: what pressure will be required to sustain the body on the plane, friction being neglected? Answer, I cwt.

If the height of the plane be 3 feet: what will be the pressure? Answer, 1} cwt. 23. If the friction in Ex. 21 be of the load: what must be the pressure to move the body up the plane? Answer, 261 lbs.

24. If the friction be of the load: what will then be the pressure? Answer, 3 cwt. 25. The diameter of a screw is 7 inches, and the distance between the threads } inch: what power applied at the circumference of the screw will support a weight of 110 lbs. Ans., 1.25 lbs.

26. When the circumference of the screw is 12 inches, and there are three threads to the inch: find the weight that will be supported by a power of 10 lbs. Ans., 360 lbs. 27. If the circumference described by the end of the lever be 10 feet, the power 10 lbs., and there be three threads in 2 inches: find the resistance supported. Ans., 1800 lbs.

28. If the interval between the threads of a screw be of an inch, and the circumference of the circle described by the powers be 2 feet: what resistance will a power of 13 lbs. sustain. Ans., 1 ton 10 cwt. 2 qrs. 16 lbs.

29. If the interval be of an inch, and the radius of the circle described by the power be 2 feet: find resistance which a power of 32 cwt. will sustain. Ans., 643 tons 8 cwt. 30. In Hunter's screw the circumference of the circle which has the power-arm for radius is 6 feet; the threads of one screw are half-an-inch apart, and those of the other one-third of an inch: show that the advantage of the machine is 432.

31. A weight of 30 lbs. balances a weight of 20 lbs. at the extremities of a straight lever 15 feet long: find the length of the arms.

(30 lbs. + 20 lbs.) : 30 lbs.
:: 15 ft. : x
.*.*= 9 ft., and 15 ft. — 9 ft. = 6 ft. Ans.

32. If one end of a bar rests on a beam, and a weight of 50 lbs. be suspended from it at one-fifth of its length from the beam, what power at the other end will support the weight, and what will be the pressure on the beam ?

Let the length of beam. Then,

1XW=1X P, .' . P = W = 10 lbs.

Pressure on beamW-P=501040 lbs.

33. A weight of 5 tons is lifted by a pair of blocks, there are 4 sheaves to each block, the rope is fastened to the upper block: what must be the pull at the free end, supposing 7 per cent. be lost on each sheave?

Ang.-'93 +8649 + ·8044 + 7481 + ·6957 + ·6470 + ·6017 +*5596 = 5.8514 the divisor; and 5 tons = 5 × 2240 = 11200 lbs.; then 11200 ÷ 5·8514 = 19140 lbs., the pull required.

34. A weight of 7 tons is lifted by a pair of blocks, there are 2 sheaves to each block, the rope is fastened to the upper block: what must be the pull on the free end, supposing 12 per cent. be lost at each sheave?

Ans.-'88+7744 + ·6815 + ·5997 =2'9356; and 7 tons=7 X₤2240 = 15680 lbs. ; then 156802·9356 = 53413 lbs., the pull required.

LLL

MISCELLANEOUS EXAMPLES.

Ex. 1. The square of the diameter of the funnel in feet is equal to the whole width of the fire grate: the fire bars are 5 feet 6 inches long, how many square feet of fire grate surface are there to the square foot of funnel ?

Ex. 2. 4 boilers with 2 furnaces each, the diameter of each fire grate is 2 feet 10 inches: how many tons of coal will be consumed per day, and how many H.P. will be indicated? In this question it is assumed that I ton of coal is burnt for every foot width of fire bars (see Answers to Elementary Questions, No. 181), and that 2} lbs, of coal per hour is the consumption for 1 H.P.

Hence 2' 10" X 4 X 2 = 22′ 8′′ = 22} tons per day.

And 22 tons X 2240 ÷ 24 = 2115*55 lbs. per hour,
.*. 2115′55 ÷ 2°5 = 846·2 I.H.P.

Answer. 22 tons per day, and 846.2 I.H.P.

Ex. 3. The steam pipe is 11 inches diameter and goes inside the boiler; the part inside has slots 5 inches long and 4 inch wide: how many slots must there be to make an area equal to double the sectional area of the pipe, and what are those slots for ?

Double the area = 11a X 7854 X 2 = 190*0668.

The slots are placed there to prevent the water coming into the cylinder when priming.

Ex. 4. The formula for rectangular cross bars for combustion chamber crowns is—

B=

72000 d 2 t
DLS

Where B = boiler pressure.

D= dist. apart of bars.

L= length of bars.

d = depth of bars.

t = thickness or width of bars.

The cross bars are 5′′ × § and 24′′ apart, length of bars 34": what pressure will be allowed in the boiler if there are 3 stays?

Ex. 5. What would be the pressure if there were four stays?

Former pressure 40'549 + 1·68942.238 lbs. pressure. Answer.