7. A certain square pleasure-ground, containing 5 acres, has in its centre a circular sheet of water occupying 1 acre, 1 rood, 20 perches : find the lengths of the paths reaching from each of the angles to the water's edge. 8. Prove the rule for finding the area of the surface of a right circular cone. Also, given the expression for the volume of a right circular cone in terms of the radius of its base and its height, deduce that for the volume of the frustum of such a cone in terms of the radii of its two ends, and of its height. 99 9. The volume of a spherical orange is cubic inches: the thick. 7 ness of the rind is į inch : after the rind is removed the orange is divided into equal parts by planes through a diameter making angles of 30° with one another: find the volume of each of these parts 22 to be taken (=to 10. A rectangular tank is excavated whose sides are vertical, but whose ends slope at an angle of 45°, of length 382 feet, and of depth 5 feet: the earth from the excavation is employed to fill up a pit 21 feet deep, of circular section, whose radius at the top is 8 feet and at the bottom 7 feet. How broad must the tank be so that the earth may just fill the pit ? STATICS AND DYNAMICS. Examiner-DR. H. W. M'Cann. 1. Define a “force" and show how it is possible to represent a force completely by a straight line. Define and explain the principle of the transmissibility of a force to any point in its line of action. 2. If any number of forces acting on a particle be represented in magnitude and way of action by the sides of a polygon taken in order, they will keep the particle in equilibrium. If the forces, instead of all acting at a point, actually act along the sides of the polygon, what is their resultant ? Forces represented in magnitude and way of action by the sides AB, BC, CD, AD of a quadrilateral act on a particle: find their resultant. If these same forces do not act on a particle, but act along the sides of the quadrilateral, find their resultant. 3. Define the centre of gravity of a body or a system of bodies, and find that of a plane triangular lamina. From a plane triangular lamina the triangle obtained by joining the middle points of its sides is cut away; find the centre of gravity of the remainder. 4. Explain what is meant by the “tension” of a string at any Find the ratio of Power to Weight in the 3rd system of pallies. If the strings, instead of being fastened to a weight, are fastened to a scale-pan in which a man, weight W, stands, find with what force he must pull down the free end of the string passing over the lowest pully in order to support himself, the strings being all vertical. 5. State the three laws of friction. A uniform rod rests with one end against a rough vertical wall, the other end being supported by a string of equal length fastened to a point in the wall; prove that the least angle 0 which the string can make with the wall is given by the formula y tan 8 3, where M is the coefficient of friction between the rod and the wall, 6. State and discuss Newton's Three Laws of Motion. 7. A body starts with velocity u and is acted on by a uniform force in the direction of the velocity during time t; iff be the acceleration due to the uniform force, and s the space described in time t, then proves ut + fta. If a body be projected vertically upwards with velocity u, find for what time it will rise, what height it will reach, and after what time it will return to the point of projection, neglecting the resistance of the air. 8. Prove that a body projected in any direction not vertical and acted on by gravity will describe a parabola. From two points A, B, not in the same vertical line, two particles are projected at the same instant towards one another: prove that the line joining the particles is always parallel to AB, and that the particles will meet; and find the time of meeting. 9. Two bodies, masses m, m', moving with velocities v, u', in the same direction impinge ; find expressions for their velocities after impact. Show that the momentum of the system is unaltered, but the vis viva diminished, unless the bodies be perfectly elastic. An engine weighing 40 tons, and 3 coal-trucks, each weighing 15 tons are at rest on a horizontal line; there is an interval of one foot between the engine and the first truck, and between each truck and the next. The engine starts off and strikes the first truck which then strikes the second, and so on. Supposing the engine to be constantly impelled by a force equal to the weight of one ton, and the bodies to be inelastic, find the velocity with which the last truck starts, and the whole time occupied in starting the train, neglecting friction, and taking g 32. 10. Explain what is meant by a "simple pendulum", write down the time of a small oscillation, and show how it may be used to determine the force of gravity at any place. A pendulum which oscillates in a second at one place gains 5 beats an hour if carried to another place. Compare the weights of the same substance at the two places. HYDROSTATICS AND OPTICS. Examiner-DR. W. H. M CANN. l. Distinguish carefully between pressure at a point, and pressure on a point in a fluid. Prove that the pressure at any point within a heavy inelastic homogeneous fluid at rest, not subject to external pressure, is equal to the weight of a column of Auid whose base is a unit of area and whose height is equal to the depth of the point below a horizontal plane through the highest point of the fluid. How is this result modified if the atmospheric pressure be taken into account ? 2. State the rules for determining the vertical and horizontal components of the pressure of a flaid at rest under the action of gravity on any surface in contact with it, and hence deduce the resultant pressure of a fluid upon the surface of a solid either wholly or partially immersed in it. A body floating on an inelastic fluid is observed to have volumes V1, V2, respectively above the surface at times when the density of the surrounding air is Pi, Pz; find the density of the inelastic fluid in terms of V, V., P1, P2. 3. Find the centre of pressure of a triangular lamina immersed in liquid with its base in the surface (i) neglecting the atmospheric pressure (ii) taking it into account. If a quadrilateral lamina ABCD in which AB is parallel to CD be immersed in liquid with the side AB in the surface, the centre of pressure will be at the point of intersection of AC and BD if AB2 = 3 CD. 4. Explain the construction and use of the common hydrometer. The hydrometer being graduated upwards, its readings for two different fluids are x1, xy, and for a mixture of equal parts of these x; show that the volume of a unit of length of the stem is to the volume of the whole instrument below the zero point as *, + x2 – 2x : **, + 3X2 — 2X, X,. 5. Describe the Forcing Pump, and explain why the air-vessel is necessary for its efficiency. 6. Explain the terms geometrical focus, principal focus, applied to a pencil of rays reflected or refracted at a spherical surface. In direct reflection at a spherical concave or convex surface, 2 radius r, prove that FQ. Fq where F is the principal focus, Q, q, a pair of conjugate foci. 7. Rays of light diverging from a point are refracted directly at a spherical surface; write down the formulæ (i) connecting the distances of the conjugate foci from the centre of the surface. (ii.) connecting the distances of the conjugate foci from the centre of the sphere. An eye is placed close to the surface of a sphere of glass (n = }) which is silvered at the back; shew that the image which the eye sees of itself is of its natural size. 8. Prove that when a ray of light is refracted from one medium into another, the deviation increases as the angle of incidence increases, whether refraction be from a denser into a rarer medium or vice versa. Hence show that the axis of a pencil of rays which passes through a prism in a principal plane is turned from, or towards, the edge of the prism, according as the prism is denser or rarer than the surrounding medium. 9. Find the geometrical focus of a pencil of rays after direct refraction through a lens the thickness of which is neglected. Hence show that in any lens whatever the position of the principal focus is the same whichever side be turned towards the incident light. If Q, q, be conjugate foci and the lens be concave, trace the change in position of q as Q moves from a great distance on one side to a great distance on the other. 10. Describe Galileo's telescope, giving a diagram of the path of rays from a distant point to the eye. What is the best place for the eye in using this telescope ? Define the "magnifying power" of a telescope, and prove that in this telescope the magnifying power is the ratio of the focal length of the object-glass to the focal length of the eye-glass. GEODESY. Examiner-CAPT. W. H. JOHNSTONE, R. E. Assoc. Inst. C. E. 1. Describe the method of observing horizontal angles with a theodolite. If not pressed for time how could you improve the accuracy of your observed angles ? 2. You are required to carry a triangulation over a piece of country comprising about 30 square miles. Only horizontal distances are required, not vertical heights. You will be supplied with a 6-inch Everest theodolite and whatever chains, staves, &c. you may require. Describe how you would proceed, indicating the successive steps of the process. Give a diagram of your triangulation and mention the length of your base. 3. Describe fully how you would test the correct adjustment of an Everest theodolite (whose antecedents were unknown to you) before commencing the above triangulation. 4. Describe how you would set out a right angle in the ground :(1.) With a 100-feet chain. (2.) With a piece of rope, the length of which you have no means of measuring. 5. What is meant by the “line of Collimation" of the telescope of a theodolite or level ? Describe how you would “collimate" a Dumpy level. How is it that defective collimation produces no errors as long as the distances observed are equal ? 6. Describe the process of “traversing' with a theodolite. Under what circumstances would you employ this method of survey. ing, and what are its advantages ? 7. "I set up a levelling staff upon a point A and turning my level upon the staff, I read 2.23 feet, I now send the staffman forward, in the direction in which I wish to level, and halt him at a convenient point B about 200 feet off. Directing the level upon the staff at B I read 7.89. I now take up my level and carry it forward to a convenient distance beyond the staff. Having set up the level I read 3:11 upon the staff at B. But I now see, away to the right, a Bench mark, the reduced level of which I know to be 153:07 feet. I therefore send the staff to it and read 5:36. Again sending the staff forward to C, I read 4:73. Shifting the level as before I read 6•04 back upon C and 3:12 forward upon D.” Enter the above flying levels in a suitable form of level book and deduce the reduced level of the points A, B, C and D. Mention the difference in level between the points A and D. 8. You are standing on the sea-shore which is unobstructed and practically level. Show how you would find your distance to a buoy which is moored a short distanco out to sea : (i.) When provided with only a 100-feet tape and a ball of string. (2.) When provided with a 100-feet tape and a pocket sextant. (3.) When provided with a 100.feet tape and an optical square or cross staff. N. B. You are supposed to have pencil and note book but no logarithmic or trigonometrical tables. 9. Explain clearly the principle of the Vernier. You have a scale of inches divided into tenths,—what will be the length of a Vernier scale which will enable you read to the eightieth part of an inch? 10. What is the best and most accurate method of finding the variation of the compass with a theodolite ? Give a detailed explanation of the successive steps of the process. CARPENTRY AND MECHANISM. Examiner- MAJOR ECKFORD, R. E. 1. Give one or two methods of fixing the end of the rafter to the tie beam, also of attaching an iron king rod and two wooden struts to a wooden tie beam. 2. What is the relative thickness of the tongue and the cheeks of a mortise. 3. Give two examples of scarfs for lengthening timber. (2), When the beam is not required to be of the same dimensions throughout, describe an easy way to join two pieces of timber together. 4. Give a sketch of a trussed beam. And of a truss for a roof of 50 feet span, using queen posts and struts to support the rafter. 5. How is the centering for a bridge over a stream (which can be diverted or which is only full during the rains) usually supported in India ? Describe the method of lowering the centres by sand cylinders. |