and use it to prove that in the ordinary lever the moments of the "power" and the " weight" about the fulcrum are equal. AB is a smooth inclined plane, BC its base, AC its height. W rests on the plane, and P, the power, is applied by means of a string that passes from W over a pulley at A. Apply the principle of work to find the ratio P: W in terms of the sides of the triangle. 11. Explain, in the case of the ordinary balance, what is meant by the balance being (i) true, (ii) stable. Prove that, in order that the balance may be true, the arms must be of equal length, and find the condition of stability. If the arms of a balance be unequal, and a body be weighed first in one scale and then in the other, show that the two apparent weights are to one another in the ratio of the squares of the unequal arms. 12. Define pressure at a point inside a fluid, and prove that in a heavy fluid at rest the difference of pressure between two points in the same vertical line is proportional to the difference of level. A rectangular plate is immersed in a vessel of water in such a way that its plane is vertical, and one side lies along the surface of the water. Find the centre of pressure. 13. Find, in direction, magnitude, and line of action, the resultant thrust of water on a body immersed in it. A body weighs 76 lb. when of its bulk is immersed in water, and 64 lb. when only of its bulk is out of the water. Find the weight and specific gravity of the body. 14. Describe the simplest form of air-pump, and make diagrams to show the action of the air and valves (i) during the up-stroke of the piston, (ii) during the down-stroke. If the volume of the receiver be 10 times that of the piston cylinder, find what fraction of the original mass of air remains in the receiver after 3 complete strokes. 15. Describe any apparatus, and the measurements that would be taken, to verify Boyle's law for pressures greater than an atmosphere. A tube is filled about half full of mercury, and inverted with the open end dipping into a cylinder containing mercury. When the column of enclosed air measures 11 inches, the mercury column measures 6 inches. When the air measures 12, the mercury measures 8. Calculate the height of the barometer at the time of the experiments. DYNAMICS. TUESDAY, 1ST APRIL 1902, 3 to 6 P.M. (Candidates may obtain full marks by doing about two-thirds of this paper. Answers, where work is not shown, will not be awarded marks.) 1. A particle moves round a square ABCD with a uniform speed of 200 feet per minute. What uniform speed in feet per second along the diagonal AC would affect the displacement from A to C in the time in which it is actually effected? Express the velocity along AB in centimètres per second, given that a metre is 39 inches. 2. Give a geometrical construction for measuring change of velocity. In question 1, find the direction and magnitude of the change of velocity of the particle as it passes through B. 3. Define force and momentum, and state the relation between them. Describe any method of verifying (approximately) this relation, by observation or experiment. A mass of 10 lb. moving east with a velocity of 10 feet per second comes under the action of two forces, one of 30 poundals west, the other of 40 poundals north. Find the magnitude and direction of its velocity after 10 seconds. 4. In the light of the Third Law of Motion, discuss the motion of a horse pulling a cart, explaining the conditions among the forces which make motion possible. If a mass of 3 lb., free to move, repel a mass of 4 lb., free to move, with a constant force of 1 poundal, and the distance between the masses be 100 feet to begin with, find the distance after 10 seconds. 5. A bullet is projected with velocity u in a direction making an angle a with the horizon: find its range on a horizontal plane drawn through the point of projection, and find for what value of a the range is a maximum. A bullet is projected with a velocity of 1000 feet per second in a direction making an angle of 45° with the horizon. Calculate, neglecting the friction of the air, after how many seconds it will touch the earth again. 6. Show in the case of a falling body that the force of gravity is a measure of the rate of change of kinetic energy per unit of space traversed. A body of mass 10 lb. is thrown to the height of 100 feet. Determine its energies, kinetic and potential, (i) leaving the earth; (ii) at highest point; (iii) after striking the earth. 7. Prove in the case of two intersecting forces that the algebraic sum of their moments about any point is equal to the moment of their resultant about that point. Assuming this to be true for parallel forces, determine the magnitude, direction, and line of action of the resultant of two dissimilar parallel forces. 8. Define a couple, and establish the law of composition of couples. Find the resultant force or resultant couple in each of these two cases: (i) a system of forces represented in magnitude and line of action by AB, BC, CD, DA; (ii) a system of forces represented by AB, AC, DB, DC (ABCD being an irregular quadrilateral). 9. A, B, C are three equal particles at the angles of a triangle: find their centre of mass. If O be any point, and G the centre of mass, prove that the resultant of three forces represented by OA, OB, OC is represented by three times OG. 10. Find the mechanical advantage of the inclined plane by the principle of work, the force being applied parallel to the plane. A wheel 1 foot in diameter has an axle 1 inch in diameter. Find by the principle of work the mechanical advantage of the machine. 11. From the definition of a liquid derive the law of the equal transmission of pressure. Describe the Bramah press, and express the mechanical advantage in terms of the diameters of the ram and plunger. 12. Describe Nicholson's Hydrometer, and explain how it would be used to determine the specific gravity of a piece of lead. On what does the sensibility of Nicholson's Hydrometer depend? Give reasons for your answer. 13. Given a U-tube, a metre measure, mercury, and water, describe an experiment to determine the specific gravity of mercury. A straight barometer tube, 33 inches long, is filled with mercury and inverted in a wide deep cistern of mercury. The column stands at 28.98 inches. Water is poured into the cistern until the level of the water outside is exactly the same as the level of the mercury inside the barometer tube. Calculate the depth of water. (The specific gravity of mercury is 136.) 14. Establish the law of variation of pressure with depth in heavy liquids. Find the centre of pressure on a triangular plate, placed with its plane vertical, in a vessel of liquid, its base being in the surface of the liquid. 15. Investigate the total work done by gravity (i) when unequal masses M and m hung over a pulley move through a space h; (ii) when a piece of lead of specific gravity s and mass m is allowed to sink slowly through water through a distance h, In each case neglect friction. Use (ii) to deduce the principle of Archimedes. PRELIMINARY EXAMINATION PAPERS ENGLISH. FRIDAY, 27TH SEPTEMBER 1901, 9 A.M. TO 12 NOON. (Fight, and not more than eight, questions are to be answered. 1 and 2 must be answered, with either 3 or 4 and either 5 or 6. The remaining four may be any questions not already answered.) 1. Write an Essay, of from two to three pages, on one of the following subjects: (1) The arguments for and against Free University Education. (2) Legal restrictions on Practitioners of Medicine. (3) The reasons of the weakness of the Modern Drama. 2. Paraphrase: At break of day the College Portress came; She brought us Academic silks, in hue The lilac, with a silken hood to each, And zoned with gold; and now when these were on, And we as rich as moths from dusk cocoons, She, curtseying her obeisance, let us know I first, and following thro' the porch that sang Betwixt the pillars, and with great urns of flowers. 3. Give some account of any four of the following: Dunstan ; The Empress Maud; Sir John Chandos; Humphrey, Duke of Gloucester; Sir Thomas More; Lord Clarendon; Pulteney; Lord St Vincent; Mr Cobden. 4. Give some account of any four of the following: King Alfred's chief acts; William of Normandy's claim to the English crown; Edward the Third's to that of France; the Statute of Præmunire; the loss of Henry V.'s conquests; the Pilgrimage of Grace; Monopolies; the Test Act; the Royal Marriage Act; the Anti-Corn Law movement. 5. Give the situation of six of the following, and one or two particulars connected with each: Galveston, Athlone, Constance, Bruges, Muscat, Roraima, Goa, Kashgar, Montpellier, Assisi, the Galapagos. 6. Draw roughly the coast-line from the north of Denmark to Brest, marking the chief rivers, headlands, bays, and coast towns. 7. Rewrite, so as to correct or improve them, the following sentences: (a) We will be extremely glad to see you, and are only sorry we have kept you waiting on an answer so dreadfully long. (b) This is a question which we ought to have expected to have found answered in the Seventh Report of the Commissioners of Her Majesty's Inland Revenue,' but they ignore it. (c) The buzz of insect life, so varied and pleasant, appeals to the sense of hearing, although unpleasantly to the sense of feeling, as they leave tangible evidences on the face and hands of their vigorous attacks. (d) Returning and walking westwards, a fine reach of river comes into view-one of the best sights that is to be seen in this valley. 8. Give a general analysis of the following sentence, and parse the words in italics : Ye see, O friends, How many evils have enclos'd me round; Yet that which was the worst now least afflicts me, 9. (a) Give the derivations of eight of the following words: |