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Py=Wx+eWx+Poy,

where e and Po are constant for all values of W and of P. This is said to be the law of efficiency of lifting tackles or pulley systems. Two experiments will be sufficient to determine the values of e and P. One may be with W = o, i.e. no weight at all; then we have PP (from above), so that P, is the effort required to bend and unbend the ropes, and to turn the sheaves on the axles against the friction caused by the weights of the block and ropes. The remaining part eWr of the waste work is the loss due to pressures caused by the actual load lifted.

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There appears then to be a separation into two distinct kinds of waste, but we cannot separate them perfectly because they interact on each other. The equation or law given is on the whole fairly accurate.

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and the counter efficiency = I +e+ Pay/Wx, so that if W be a very large weight compared with P, 1+e is practically the counter efficiency, so we could roughly determine I +e by lifting a very large load and measuring the value of the counter efficiency Py/Wx.

We have dealt pretty fully with the cases we have so far considered, as they exhibit the methods we must use in the great majority of machines. Space prevents our examining any more of the simple mechanisms of chap. v, but the method we have used will apply.

EXAMPLES.

1. A tram car weighing 4 tons, resistance on the level 15 lbs. per ton, is pulled up an incline of 1 in 18. Find the pull required, and also the amount of error which would be made by taking the resistance to be the same on the slope as on a level. Ans. 557 lbs.; .114 lbs.

2. If 8 passengers, each weighing 130 lbs., enter the car, what increase of slope is this equivalent to?

Ans. I in 1200 about.

3. The draught of a waggon is 40 lbs. per ton, and the coefficient of friction between the skid and road 1. In going down an incline, with one of the four wheels skidded, the speed is the same as on a level road with no skid, the horses exerting the same pull. Find the slope. Ans. I in 17.

4. The engine of a goods train can just pull it on a level at 25 miles per hour. Resistance 17 lbs. per ton. Two points A and B on the line are 10 miles apart, and B is 60 ft. below A; the slope from A to C, an intermediate point, is I in 300, and from C to B I in 600. Find at what point of the down grade the speed which was 25 miles per hour passing A will again be the same, and what was the speed at the summit. Also what reduction of power must then be made to keep the speed from rising. Ans. 2.12 miles from the summit, 8

by .22 of original power.

miles per hour. Reduce

5. A locomotive weighs 45 tons, of which .48 rests on the driving wheels. What must be the coefficient of friction between the surfaces of the driving wheels and the rails that the engine may just draw a train, total weight 200 tons, at 50 miles per hour without slipping, up an incline of 1 in 300. Resistance 45 lbs. per ton. (This friction is called the adhesion.)

Ans. .217.

6. A ship weighing 2000 tons is launched. Find what slope of the ways is necessary for uniform motion when once started. Also, what should be the area of bearing surface so that the pressure shall not exceed 2 tons per sq. ft., and so force out the tallow? Coefficient .14. Ans. 8°; 800 sq. ft.

7. The trucks of a double incline weigh 4 tons, and are loaded with 5 tons. Find the slope so that the loaded truck would run steadily down. Resistance 17 lbs. per ton. Ans. 1° 9'.

8. The actual slope in the preceding being 30°, find what frictional moment must be applied to a pulley 8 ft. diameter to keep the motion uniform. Ans. 9 ft.-tons.

9. If the bearings of the pulley be 6 ins. diameter, and the friction coefficient, solve the preceding, taking account of the friction of these bearings. Ans. 9.57 ft.-tons.

10. The wheels of a railway carriage are 3 ft. 6 ins., weight 18 tons. The coefficient of friction between the brake blocks and wheels is .4. Find the total pressure between the blocks and wheels, so that if detached on an incline of 1 in 120 the

carriage may not run down. Ordinary resistance, 14 lbs. per ton. Ans. 210 lbs.

II. In question 2, page 131, the diameter of the axles of the pulleys is in. Coefficient of friction. Find the efficiency, omitting stiffness of the ropes. Ans. .88.

12. In question 6, page 110, the mean diameter of the thrust rings is 15 ins. Coefficient of friction .05. Find the efficiency of the thrust block-shaft pair.

Ans. Work done per revolution, 120 × 27 inch-tons. Work wasted, 10.56 inch-tons.

.. Efficiency=.986.

13. In question 7, page 91, the bearings are 7 ins. diameter. Coefficient. Draw a curve of frictional moment and calculate the work lost in friction per lift.

Ans. Total weight is constant, therefore curve is a straight line.
Work lost 5.9 ft.-tons.

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14. In question 11, page 133, calculate the efficiency, if the diameter of bearings for each shaft be 3 ins., and the driven pulley 2 ft. 3 ins. diameter. Coefficient. Ans. .939.

15. Find the efficiency of the screw jack of question 9, page 38. Coefficient .06. Depth of thread of the pitch. Diameter Ans. .45.

3 in. 16. Find the law of efficiency of a pair of three-sheaved blocks in which a 12 lbs. pull raises 40 lbs., and a 70 lbs. pull 300 lbs. Ans. P=.223W +31 in lbs.

CHAPTER VIII

THE DIRECT ACTING ENGINE-MOTION

Of all machines the above is probably the most important, and hence we will examine it thoroughly in detail, so far as the limits of the present work will allow.

Fig. 112 shows the construction of a vertical engine

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for marine propulsion, and will serve as a type of all direct actors.

A is the cylinder; B, B, B the framing; C the

piston; D the piston rod; E the connecting rod; and F the crank shaft.

The shape or construction of the parts is not, to us, of importance except in so far as it governs the motion, i.e. we care only about the positions and shapes of the bearing surfaces.

We have then as essentials

1st. A fixed piece A and B, B, B.

This we consider

as only one piece, because, although it is actually made in parts, this is only for constructive reasons, and as far as the motion is concerned it might be one solid casting.

[In some small engines it is actually so.]

2d. A piece, C and D, which slides in A (see chap. i. page 19).

3d. A piece E, which is connected to D by a pin joint, so that E turns relatively to D. The centre of

the pin is in the centre line of C and D.

4th. A piece F, which can turn in the end of E, and also in a bearing in B, ie. in A. So we have returned to A again. The centre line of the last bearing, or bearings, must meet the line of stroke, as shown at O in the skeleton figure.

The machine consists then of four pieces, connected as here shown:

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The kind of relative motion of these pieces we have given us by the connections, and we now wish to find the relations which exist between their amounts.

For this purpose we do not need the outlines and cross dimensions as given in Fig. 112 (a), but simply the dimensions of the skeleton figure (b). In (b) the framing

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