A favourite question, and good marks were nearly always obtained. A mistake sometimes made was to add ps to pw in plotting the entropy line for steam. Q. 50. The temperatures on two sides of an iron plate 0.5 inch thick differ by 10 centigrade degrees, how much heat (in centigrade pound water units) passes per square foot per second? The conductivity of iron is 0.18 in C.G.S. units (centigrade gramme water units). A fairly favourite question, and the answers were often quite correct. Not infrequently, however, candidates went astray in the units. Q. 51. Give the ordinary statical theory of the Watt or Hartnell form of governor, taking friction into account. If you choose the Watt it must be a modern loaded governor. This was one of the favourite questions. It was sometimes done well but there was too much evidence of mere cram. Proofs evidently learnt by heart from a book would be given without a real knowledge of the symbols involved, and the figure would sometimes be quite inconsistent with the so-called proof. Q. 52. Prove the formula for efficiency in the hypothetical Otto cycle (the diagram being two adiabatics and two constant volume lines), showing how efficiency is greater as clearance is less. In what way does this hypothetical diagram differ from reality? If it differs greatly, why are such calculations of any use. Seldom attempted. The formula for efficiency in terms of the ratio of expansion was generally satisfactorily established, but little knowledge was shown in the answers to the other parts of the question. Q. 53. When steam is throttled or passes through a non-conducting reducing valve show that the value of H remains constant; H is the total heat required to convert 1 lb. of water at 0° C. to dry or wet steam at 0° C. Find an expression for H in terms of the dryness fraction. Steam 95 per cent. dry at 2033 lb. per square inch passes through a non-conducting reducing valve and lowers in pressure to 1019 lb. per square inch; what is its dryness? Use the following information. L is the latent heat of 1 lb. of dry saturated steam. This was a favourite question and the last part was generally correctly worked. Nearly every one tried the first part, but not one gave a valid reason why H remains constant. The following is typical of the line of argument: "The heat H is in the stuff at the higher pressure; none of it is lost in passing through the non-conducting valve; therefore H must be the heat after the fall of pressure." HONOURS. Results 1st Class, -; 2nd Class, 5; Failed, 12; Total, 17. As in the other stages, candidates who sketch well do not generally seem to be strong in theory, and those who do well in theory are weak in sketching. Average men seem to be divided in this way into theoretical and practical men. The best candidates are those who are interested in mechanical details of engines, but who are also interested in the scientific principles involved. 9291, H On the whole, both sketching and theory are of a fairly good standard. No candidate has the thorough acquaintance with theory which is expected from the highest class of student in this subject. Many candidates know what is contained in the best text-books, but they ought also to know of articles recently published in magazines and in the proceedings of Engineering Societies. There is, for example, but little knowledge of the Steam Turbine. The second law of Thermo-dynamics is easy to state, but we fear that a thorough knowledge of what the statement means is often wanting. Q. 68. It has been proposed, instead of much superheating in the ordinary way to have less of this, and to superheat in the receivers between the cylinders. Assuming that the thing can be done in a good practical way, what is your opinion of its merits? Of course you will give your reasons. No candidate suggested the use of the tp diagram in considering Question 68. Q. 69. What information is necessary if we wish to convert an indicator diagram into a pv diagram? How is it done? What assumptions do we make ? Although these assumptions are untrue, show that we can use the pv, or the corresponding to diagram in obtaining practical knowledge. No candidate seemed to know exactly what is meant by a p v diagram in answering Question 69. Q. 70. If t is temperature difference, and R is heat resistance, t/R is heat per second flowing. What is the heat resistance of an iron plate 0'5 inch thick, per square foot of metal, from face to face, the conductivity of iron being 0.18 in C.G.S. units (gramme water centigrade heat units)? If no boiler passes more than 5,500 centigrade pound heat units per hour per square foot of metal, and the average temperature difference (flue gases and water) is 800 centigrade degrees, what is the average total heat resistance (flue gases to water) Compare this with what you previously calculated. In Question 70 the arithmetic of most of the candidates was very incorrect. Q. 71. A Hartnell governor on a turbine shaft has a law F sr-a where Flb. is the outward radial force on each ball which would just keep the governor (if motionless) in the position in which each ball is r feet from the axis. What do s and a depend upon ? When the balls move let there be friction e dr dt Write out the equation of motion radially outwards. Let the method of regulation be such that the torque on the revolving part of the turbine is cbr where b and c are constants. The speed being steady, the resisting torque having been constant, let the torque suddenly alter to another constant value; write out the differential equation connecting speed and time. State algebraically the condition required for stability of motion (that is, no hunting). Of all the candidates who attempted Question 71, an interesting question, carefully arranged so that the proper line of reasoning might be followed, not one gave the correct answer. Q. 72. Steam is flowing from a vessel through an orifice which gets larger outside. What do we know about the changes of pressure and speed of a portion of the steam as it flows? Take the inside pressure as say 100 lb. per square inch, and the pressure outside as 2 lb. per square inch. What is the speed in the threat or narrowest place? How is it that outside the throat the speed gets greater although the cross section is greater? The subject of Question 72 is of very great practical interest, yet there was no correct answer. Q. 73. Explain how you would find the magnitude and line of action of the force corresponding to the acceleration of the connecting rod of a steam engine for any specified position of the crank. You may assume that the necessary dimensions and speed of the engine, and the kinetic elements of the rod are known. Question 73 might easily be answered from mere text-book information, yet the answers were unsatisfactory. In spite of these unsatisfactory details, the Examiners are fairly well satisfied with the results of the examination. Even when a candidate gets only half marks for an answer he often shows evidence of individuality, and proves that he has been thinking things out for himself. The Examiners are confident that in many cases where the reasoning is not altogether good and the real point seems to be evaded, the candidate shows signs that he will in time see things more clearly. Report on the Examination in Practical Mathematics. The number of candidates in this subject is increasing rapidly, as the following table for three successive years will show : This shows a total increase of 133 per cent. in two years, and the rate of increase is seen to be greater in the higher stages. It is satisfactory to be able to record that along with this increase in number there is no lowering of the average quality of the work. In fact, after making due allowance for the somewhat easier questions set this year, there is a distinct improvement noticed in the work of all the stages. There were many classes in the lower stages in which nearly all the candidates passed in the first class. There were some in which the work was almost wholly bad, but it is believed that these unprepared classes are fewer than formerly. Logarithms are now used with facility and with much more certainty than was the case a few years ago. There is still some looseness in writing down expressions, and especially in the use of the sign of equality “=.” Thus it is not uncommon to find such a statement as 66 The candidate as a rule knows quite well what he is doing, but when dealing with more complicated expressions such slackness leads to confusion and mistakes, and should never be permitted. A persistent feature of the examination is that although candidates can use contracted methods, as shown in their answers to Questions 1, 21 and 41, they rarely employ these methods in their arithmetical computations when working any of the other questions of the paper. With regard to plotting, scales are, as a rule, well selected and plainly figured. There is, however, a tendency in such questions as 5, 11, 12, 13 and 27, after having marked the points, to join them by straight lines or chords, instead of by a fair continuous curve, and to measure results from this figure. 9291. H 2 The significance of the slope of a curve is slowly being acquired, but, in Stage 1 especially, candidates have great difficulty in expressing themselves intelligently. The mere plotting of a curve bas little value apart from the lessons the curve teaches, and students should be induced to think more about their work, and not be content with mere mechanical graphing on squared paper. STAGE 1. Results 1st Class, 930; 2nd Class, 595; Failed, 574; Total, 2,099. Q. 1. The four parts (a), (b), (c), and (d) must all be answered to get full marks: (a) Compute by contracted methods, to four significant figures only, and without using logarithms 3214 × 0.74237'912. (b) Write down the logarithms of 32170, 32:17, 03217, 0'003217. (c) Compute, using logarithms 8405 × 01357 ÷ 1·163. (d) Express £0 178. 9d. as the decimal of a pound. (a) Contracted methods well known. Mistakes as a rule only in the fourth figure. Decimal point almost always correctly placed. (6) Few mistakes; these generally in the characteristics of the logarithms. (c) The symbol of the square root instead of the index gave some trouble in applying logarithms. (1) Answer generally correct. The usual method was to divide the number of pence in 178. 9d. by 240. Very few used the florin as the basis of their calculation. Q. 2. The three parts (a), (b), and (c) must all be answered to get full marks :: (a) If A = P(1+)" find 4 when P = 200, r = 4, n = 12. 100 (b) On board a ship there were 1,312 men, 514 women, and 132 children. State these as percentages of the total number of persons. (c) When x and y are small we may take 1 + x as being very nearly equal to 1 + x − - y. What is the error in this when x = 002 and y = 0'03 ? (a) Common mistakes were (1 + '04)1o = 12:48, 1+OO = 125 or 14, 200 (104)12 = 20812. One candidate multiplied the latter out in full, obtaining an answer with thirty figures. () Fairly well done; a few good graphical solutions. () Not well answered as a rule. Q. 3. The four parts (a), (b), (c), and (7) must all be answered to get full marks: (a) Write down algebraically :-The principal P multiplied (b) Divide 17:24 into two parts such that one quarter of the first added to one third of the second make 5'06. (c) What are the factors of 2-10 ? (d) A wheel is 3:45 feet in diameter; it makes 1020 revolutions rolling along a road; what is the distance passed over? 100. (b) Pretty well done, but some, after stating the equations in x and y, could not work them out correctly. (c) The factors of x2. 9 would evidently have been correctly given by almost all, but the 10 proved a stumbling block. Some said, "There are no factors." Others gave such answers x (x — 10), 10 (2-1). 10 Common mistakes were (x−3·3) (x + 3·3), (x − 5), (x — 2). as (d) Fairly well done. Some think 2 is the circumference of a circle others use 27d, and a not uncommon answer was 3'45 × 1020. Q. 4. If y = x2 −3·4x + 2·73, calculate y when x has the values 1, 12, 1'4, 16, 18, 2, and 2'2. Plot these values of x and y and draw a curve. What values of x cause y to be 0 ? Attempted by about 44 per cent. of the candidates, with very satisfactory results. Q. 5. r and t are the distance in miles and the time in hours of a train from a railway station. Plot on squared paper. Describe why it is that the slope of the curve shows the speed; where approximately is the speed greatest and where is it least! 0 15 60 140 190210 215 21.8 230 247 26'8 About half the candidates selected this question. The curve was well plotted, and the places of greatest and least speed successfully indicated. Answers as to why the slope shows speed often amounted, when sifted, simply to the statement it is so. Q. 6. The point B is 4 miles North and 2 miles East of the point 4. What is the distance from A to B and what angle does the line AB make with the due East direction? The favourite optional question. Graphical solutions were common, but in these cases the scale was often too small. Of those who worked by tables, the majority were content with "630" or "64" for the angle, or "between 63° and 64°," or "about 63°." Few obtained the correct angle to within 0.1° by interpolation. Q. 7. The horse-power of the engines of a ship being proportional to the cube of the speed; if the horse-power is 2,000 at a speed of 10 knots, what is the power when the speed is 15 knots? Fairly often attempted. Correct answers were frequent, but many used simple proportion obtaining 3,000 horse-power. Q. 8. There are two maps, one to the scale of 2 inches to the mile, the other to the scale of half an inch to the mile. The area of an estate on the first map is 1'46 square inches, what is the area of this estate on the second map? Pretty frequently attempted but with poor results. Answers like the following occurred |