WEDNESDAY, 6Th November. [2 P.M. TO 5 P.M.] DYNAMICS. 1. Define velocity, acceleration, momentum and force. If the units of mass, length and time be increased in certain ratios, shew that the units of these four quantities are increased in the ratios a, b, c, be a and determine in terms of a, b, c the ratios in which the fundamental units are increased. 2. What is meant by the angular velocity of P about a given point 0? 10 Shew how to connect angular velocity with linear velocity. A point P starts from O with constant velocity v in a straight line inclined at an angle a to the straight line OAB through two fixed points A and B. Prove that the angular velocity with which A and B appear to b separate, as seen from P, is v sin a ( where a and b, r and r are the distances of A and B from O and P respectively. Shew that the 3. State Newton's Laws of Motion. Explain how force and mass are 10 measured and point out the distinction between "weight" and "mass." If an engine of horse-power H draws a train of W tons up a plane of inclinationa with a uniform velocity v, against a resistance of P lbs. weight per ton, prove that Wo 4. Explain the principle of the conservation of energy, and verify it in 12 the case of a particle falling freely under gravity. A heavy ring mass m slides on a smooth vertical rod, and is attached to a light string, which passes over a small pulley distant a from the rod, and which has a mass M (>m) fastened to its other end. Shew that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, 2 Mma M2-m2 it will descend a distance before coming to rest, 5. A light string hangs over a smooth pulley. A mass Mis hung on one 12 side, and masses m and m' at two different points on the other side. Find the motion and the tensions of the different parts of the string. A pulley is fixed at the top of a smooth inclined plane of inclination a to the horizon. A string passing over the pulley in a principal plane is attached to a mass M on the inclined plane, and the other end supports & mass m hanging freely. Shew that M will just reach the pulley if is (M + m) sin a detached after it has fallen a distance equal to m (1 + sin a) original distance of M from the top. of the 6. Prove that there are two possible directions of projection in which a 12 particle may be thrown with given velocity from one given point P to pass through a second given point Q, provided the point Qlies within a certain parabola, and that both the paths touch this parabola. Shew that the geometric mean of the two times of flight from P to Qis 2 PQ \ -1 7. A particle is projected from a point O with a velocity of 400 feet 10 per second in a direction making tan with the horizon; find when, where, at what angle, and with what velocity it will strike an inclined plane passing through O, the tangent of its inclination being 5. 8. A sphere of mass m impinges with velocity v on a sphere of mass m'at rest, the line of impact being inclined at an angle a to the direction in which m is moving. Find the magnitude and direction of each sphere after impact, e being the coefficient of restitution. If two unequal spheres moving with equal velocity v impinge directly, prove that the resulting loss of kinetic energy is (1 e) v2, where p is the harmonic mean between the masses of the spheres. 9. Prove the fundamental property of the hodograph and shew that if 10 the moving point travels with constant velocity, the hodograph is a point. A point moves in a circle, and its speed is proportional to its distance from a fixed diameter of the circle; find the hodograph. 10. A bead slides down a smooth circle; shew how to find its pressure on the circle at any moment. A particle slides down the outside of a fixed smooth sphere starting from rest at a point P and leaving the sphere at a point Q. Prove that the height of Q above the horizontal diametral plane is two-thiras that of P. 8 TUESDAY, 5TH NOVEMBER. [2 P.M. TO 5 P.M.] HYDROSTATICS. 1. Prove that the free surface of a liquid at rest is a horizontal plane. 10 An isosceles triangle is immersed with its base horizontal in two liquids of densities p and σ that do not mix, so that th of its median is in the upper fluid (o). Find the depth of the vertex below the free surface of the upper liquid when the pressures on the two parts into which the triangle is divided by the surface of separation are equal, and shew that the problem is lies between n2 - 1 and n3 possible only when σ ρ 1. 2. Find the conditions of equilibrium of a body partly immersed in a 10 liquid and capable of turning about a fixed horizontal axis. Two smooth spheres of equal weights and radii float in water connected by a string of length 27. A ring of mass M and specific gravity σ is hung on the string; shew that the sy heres will be in contact and that there will be a pressure between them of magnitude 3. Define the centre of pressure of a plane area, and shew that the 7 centre of pressure of a circle of radius a wholly immersed in a liquid with its centre at a depth d is at a distance a2 sin a from the centre, a being the 4d inclination of the plane of the circle to the horizon. 4. State Boyle's Law and shew how it may be experimentally verified 11 in the case of pressures less than the atmospheric pressure. A glass cylinder contains water to a depth h; another glass cylinder of depth k whose cross-section is n times that of the first, is pushed open end foremost into the water, till it rests on the bottom of the first cylinder, its base not being immersed. Prove that the height of the water in the second cylinder is given by where H is the height of the water barometer at the time. 5. Explain how the Hydrostatic Balance is used to find the specific gra- 8 vity of a solid and a liquid. A body floats in a liquid of specific gravity s with as much of its volume out of the liquid as would be immersed in a second liquid of specific gravity is if it floated in that liquid. Prove that the specific gravity of the body s',ss' | (s + s′). 6. Describe the common pump. If 1, l'are the heights at which the water stands in the lower cylinder before and after a stroke, shew that the ratio of the sectional areas of the upper and lower cylinders is where a and b are the lengths of the upper and lower cylinders and H is the height of the water barometer. the earth's surface on the supposition that the temperature is constant. Deduce the practical formula for z in feet, viz., z = 60, 300 (log10 ho - log10 h), where h, h are the barometric heights and the constant corresponds to the temperature 32° Fahr., having given the following data: 1 cubic foot of air at barometric height 30 inches weighs '081 lb. at temperature 32°; 1 cubic foot of water weighs 62.5 lbs.; loge 10 = 2.3. 8. Describe Hauksbee's Air-pump. 10 If there be a clearance C at the bottom of the barrel, and A, B are the volumes of the receiver and barrel respectively, shew that the density after n strokes is given by 9. Shew that the resultant thrust on an element of a perfect fluid in 10 motion is perpendicular to the surface of equal pressure passing through the element. A cylindrical vessel is full of water; find how much overflows when the cylinder rates a out its axis which is vertical with uniform angular velocity, distinguishing the cases of w2 >< 2gl] a3, where l is the length and a the radius of the cylinder. 10. Find the tension of a spherical elastic membrane containing gas at a 10 pressure p and surrounded by air at a pressure II. A cylindrical boiler with spherical ends has to withstand a steam pressure of p lbs. per square inch; if t be the tensile strength per square inch and r the radius of the boiler, find the least thickness of the body of the boiler and of its ends. MONDAY, 4TH NOVEMBER. [2 P.M. TO 5 P.M.] CHEMISTRY-PAPER I. N.B.-The two Sections should be kept separate. Definite chemical changes should be represented by equations in all cases. 1. Represent the following equations graphically so as to point out the 14 valency of the different elements: State the rule with respect to the change of valency in the same element. 2. 0139 gramme of pyrolusite, when treated wtih hydrochloric acid, 12 yielded enough chlorine to liberate 0336 gra nme iodine from potassium iodide. Find the percentage of manganese dioxide in pyrolu site. 3. Sulphuretted hydrogen is added to each of the following sub- 12 stances(1) Chlorine, (2) Sulphuric acid, (3) Ammonia, (4) Potassium chromate, (5) Sulphur dioxide, and (6) Ferric chloride. State fully what happens in each case. 4. How is phosphine ordinarily obtained? To what is its spontaneous 12 inflammability due? Compare its properties with those of ammonia. SECTION II. 5. How would you ascertain the quality of boiler water? What treat- 17 ment do you consider to be most economical when the water contains (1) not more than 15 gr. scale-forming material in 10 litres, (2) considerably more than this amount, and is (a) temporarily hard, (b) permanently hard? 6. Compare the Deacon process of manufacturing chlorine with the 17 Weldon process of chlorine-making and manganese recovery, with a view to show, why the latter is practically preferred, though the former is theoretically a more perfect process. 7. What relative quantities of salt and acid of specific gravity 170 (78 16 per cent. pure acid) should a charge for the manufacture of nitric acid theoretically contain, and why is the practical charge different? Describe the condensing system for nitric acid which, in your opinion, would be most adapted to the requirements of this country. TUESDAY, 5TH NOVEMBER. [10 A.M. TO 1 P.M.] CHEMISTRY-PAPER II. N.B.-The two Sections should be kept separate. Definite chemical changes should be represented by equations in all cases. SECTION I. 1. Give two methods for causticising sodium carbonate. To what uses 12 is caustic soda applied? How can sodium be obtained from it? 2. What processes have been proposed for obtaining copper from the 14 burnt pyrites of sulphuric acid works? 3. If you are given in the laboratory silver chloride, bismuth nitrate 12 and red lead, how would you obtain the metal from each ? 4. How would you prepare nickel sulphate and how much of the 12 metal would be required to prepare 798 grammes of the crystallised salt ? SECTION II. 5. Compare the compounds of metals with non-metals, with those of 12 metals with metals. How are the different properties of metals generally affected by alloying them with each other? Why does an alloy which is homogeneous in the liquid state very rarely yield a homogeneous casting? 6. What difficulties presents the lixiviation of "black-ash" and how are 13 they overcome to a great extent ? How are the caustic soda and the sulphide of the "tank-liquor" changed into carbonate ? |