mean

Pm+'m is called the mean effective pressure of the

2

card as a whole.

This mean effective pressure is to exerted through the whole double stroke.

Total energy in double stroke=

as before.

Pm+pm

2

be supposed

Whence

A x 2s ft.-lbs.,

=(Pm+p'm)As ft. -lbs.,

Notice that, if P be per sq. ft., A must be in sq. ft.

sq. ins.

Comparing now with chap. iii., page 74, we have

[It is often now the practice to measure the area directly by means of an instrument called a Planimeter, but it is doubtful whether this gives any increase in accuracy, because there is often a difficulty in determining exactly the length of the diagram, which is of course necessary so as by division by it to obtain the mean breadth or mean effective pressure.]

EXAMPLES.

I. In question 3, page 74, if the throw of the pedals be 5 ins., and the driving wheel 54 ins. diameter. Find the mean pressure the rider exerts, assuming that he presses vertically during the downward half revolution of each crank. Ans. 15.27 lbs.

2. Assuming the resistance to a cutting tool to vary as the depth of cut. Compare the work done in planing a flat 2 ins. wide on an iron cylinder 4 ins. diameter, 12 ins. long, with that done in planing it down to the middle, and also with the work done in reducing its diameter to 3 ins. Ans. 1.82:32:15.

3. A spring-loaded safety-valve is 3 ins. diameter, loaded to 135 lbs. per sq. inch. The original length of the spring was 24 ins., and it is compressed to 20 ins. when fully loaded. Find the work done in lifting the valve through I inch.

Ans. 122 ft. -lbs.

4. A cylinder 6 ft. long, 2 sq. ft. sectional area, contains I lb. of air at atmospheric pressure. Find the work done in compressing it to four atmospheres according to the hyperbolic law. Ans. 44,240 ft.-lbs.

5. In question 10, page 75, draw a curve showing the resistance at any point of the lift of the anchor-Ist, neglecting buoyancy; 2d, taking account of the buoyancy of the water. S.G. of the iron 7.5. Calculate in each case the work done during each half of the lift.

Ans. 216,1943; 192.45; 180.12 ft.-tons.

6. If the engine of question 4, page 74, drive a pair of drums so that the rope unwinds from one while winding on the other. Draw a curve of resistance for one lift, assuming equal lengths of rope to wind and unwind in the same time. Weight of rope 8 lbs. per yard. Find the work done in each third of the lift. Ans. 674,420, 166 ft.-tons.

7. The drums in the preceding are conical, varying from 20 ft. diameter to 30 ft., so that the winding and unwinding are not actually equal. Draw a curve of resisting moment due to the rope and coals lifted, and also a curve of the moment exerted by the descending rope; and by combining these obtain a curve showing the moment exerted at any point by the driving engine. Hence calculate the work done during each quarter lift. simplicity take only the complete turns made.

For

Ans. There are 15 complete turns, so take 15 equal distances on the base line, on a scale of angle each representing

2π. The ordinates are then in order

(21 cwt. + × 375 lbs.) 20 ft.,

(21 cwt. +355π lbs.) 20 ft. for 1st turn.

(21 cwt.+ × 355π lbs.) 20 ft.,

(21 cwt. + × 334π lbs.) 20 ft. for 2d turn.

thus giving a stepped curve, assuming the rope wound in parallel circles of diameter 20 ft., 203 ft. etc., the moment changing suddenly at the commencement of each turn. Actually, the rope being wound continuously, the curve would be continuous, but the investigation is outside our present limits. The results are practically unaffected. A similar curve is obtained for the other rope, omitting the 21 cwt.

8. A steam engine is supplied with steam of 45 lbs. absolute, and discharges into the atmosphere, the back pressure being 17 lbs. absolute, and steam admitted during the whole stroke. Estimate the gain of work per cent: Ist, by fitting a condenser so that the back pressure is reduced to 3 ĺbs. ; 2d, by further cutting off the steam at half stroke, so as to use only half the quantity per stroke, assuming expansion according to the hyperbolic law. Ans. 50; 70.

.9, .75, .64, .6, .5, .3 ins. .85, .7, .66, .6, .45, .2

9. The ordinates of an indicator diagram areFront end 1.5, 1.48, 1.45, 1.1, Back 1.6, 1.5, 1.3, 1.1, The indicator spring is compressed 1 inch by 16 lbs. Diameter of cylinder 45 ins., stroke 3 ft. 9 ins., revolution 65. Calculate the I.H.P. If the piston rod be 5 ins. diameter, and there be no tail rod, find what correction should be made in the preceding result. Ans. 341.7; 2.13.

10. The curve of stability of a vessel is a segment of a circle of radius twice the ordinate of maximum statical stability, which is 2500 tons-feet; estimate the total dynamical stability of the vessel, the angle of vanishing stability being 85°.

Ans. When a ship is forcibly heeled over, the water offers a resisting moment, commencing at zero, increasing to a maximum, and then decreasing again to zero, as the ship is heeled through an increasing angle; any heel will capsize her altogether. The curve of stability is a curve whose base shows angles of heel and ordinates corresponding resisting moments, so it is the curve of resistance to rotation; the resisting moment at any angle is called the Statical Stability, and the work done in heeling the ship to that angle the Dynamical Stability; the angle at which the statical stability vanishes is called the angle of vanishing stability. Hence the required dynamical stability is the total area of the curve of stability, viz. 2630 ft.-tons.

11. Calculate the H.P. of a turbine working for 10 hours a day, supplied from a reservoir at an elevation of 50 feet, containing 100,000 cubic feet, which is emptied at the end of 10 hours' work. The reservoir is continuously supplied by a stream which is capable of just filling it during the period of 14 hours Assume .3 of the energy wasted. Ans. 20.5.

rest.

CHAPTER V

SIMPLE MACHINES

IN order to do work we must have some source of energy, and in some instances the source can supply energy of exactly the nature required. For example, to draw a waggon we require say, a pull of 100 lbs. on the traces, this being the resistance offered by the road; then a horse is capable of exerting such a pull, and will move the waggon. In such a case

Energy exerted = work done,

[In the actual case above it is all work wasted.] and also

Effort resistance.

But often the energy which the source can exert is of a different character to the work to be done. For example, to lift 1 ton through 1 foot,

Work done = 1 ft.-ton.

Now a man is capable of exerting I ft.-ton of energy if he have time, but he cannot exert it in the form of an effort of I ton, but of say only I cwt. And then to do the above work an effort of I cwt. must be exerted through 20 ft. For the man therefore to lift 1 ton he must apply his effort to some part of an intermediate agentsay to the end of the rope of a set of pulleys—which agent is capable of changing the form of the energy to

that of the work, then the ton weight must be attached to some other point of the agent-in the case above to the other end of the rope-and the agent must be such as to magnify the effort 20 times, and also to reduce the velocity 20 times. Then will

I cwt. exerted through 20 ft. lift I ton through I ft.

We say then that the man has to use a machine, and in most cases of doing work we have to use a machine, hence the necessity of the task which we now commence, i.e. an examination of the working of various machines.

The steam engine, for example, is a machine, transmitting the energy of the steam to the point at which we require work to be done, e.g. in a colliery winding engine, to the rope which lifts the cage or coals up the shaft, and in the transmission altering the character or nature from

to

Steam pressure x distance moved by piston

Weight of coals x distance through which they are lifted.

This of course implies that there is no waste of energy in transmission, which is never the case in practice; actually some will be wasted and the remainder transmitted.

The effect which the man seeks in using a set of pulleys is the magnifying of his effort, an effect which is expressed by saying the machine gives or has a Mechanical Advantage; the magnitude of the mechanical advantage being measured by the number of times the effort is increased, i.e. by the ratio of resistance to effort.

In old treatises on machinery these latter are called generally weight and power, but we have already given a definite meaning to power, and the resistance is not in all cases a weight, so we will keep to the terms resistance, effort.

Now although the mechanical advantage is what is sought, yet there is another effect which invariably