and if we xp where c is a constant. Find p in terms of h. If p a wt, express t in terms of h introducing constant. (33) 48. The total cost C of a ship per hour (including interest, depreciation, wages, coal, &c.) is in pounds (33) Express the total cost of a passage of 3,000 miles in terms of s. What value of s will make this total cost a minimum? At speeds 10 per cent. less and greater than this, compare the total cost with its minimum value. 49. The curve y = a + bc passes through the three points x = 0, y = 26·62; x = 1, y =35 70; x = 2, y=49.81, What is the area of the curve from the 0 to the ordinate at a = 2? find a, b, and c. ordinate at a (42) 50. Describe a method of finding whether a given curve follows, approximately, the law y = a + bxn or y = b (x + a)n or y = a + benx. Logarithmic paper must not be used; the work can be done on ordinary drawing paper using Tee and set squares. (42) 51. If y = a sin qt and x = b sin (qtc) where t is time and a, q, b, c are constants; if q = 2π/T where T is the periodic time. Find the average value of xy during the time T. (42) 52. Q being the rate of flow of water per second over a sharpedged notch of length 7, the height of the surface of nearly still water (some distance back) above the sill being h; it has been proved that the empirical formula obtained by Dr. Francis is also a rational formula; it is Show that for a given 1, although a constant c may be found which will give a correct answer for one value of h, it must give incorrect answers for all other values of h. (33) 53. If i is -1, write down the values of 2, 3, 4, 5. Find √17+ 30i, √i, 1÷i each in the shape a + bi. If a + bi operating upon sin qt (where t is the variable and q is a constant) gives a sin qt+b cos qt, find three answers, the effects of operating with √17 + 30i, ✅i and upon sin qt. 1 Vi (33) 54. Get instructions from Q. 53. The voltage applied at the sending end of a long telephone line being vo sin qt, the current entering the line is where, per unit length of cable, r is resistance, is inductance, s is leakance, and k is permittance, or capacity. If r = 6 ohms, = 0.003 Henries, k = 5 × 10-9 farads, s = 3 × 10-6 Mho, and if q = 6,000, find the (42) current. NOTE.-There is a quicker method of working than what is indicated in Q. 53, using Demoivre. You may use it if you please. SUBJECT VI. THEORETICAL MECHANICS. INSTRUCTIONS. Read the General Instructions on page 3. Division I.-SOLIDS. Stage 1. You are not permitted to answer more than seven questions. 1. What are the British standards of length and weight? Where are they kept and by whom? If V be the volume of a body in cubic inches and D its specific gravity, what further information is necessary before you can ascertain its weight in pounds avoirdupois ? 2. Define "force." (12) Give three examples of forces met with in Nature. How can it be shown experimentally that a given mass of matter has not the same weight at all points of the surface of the earth? (12) 3. What is meant by transmission of force? How is force transmitted through a solid body? A flat body rests on a smooth horizontal table, and can turn freely on the table round a fixed point A; one end of a string is tied to a point B of the body, and the other end carries a weight, which hangs over the edge of the table; show the position in which the body comes to rest. Explain briefly how this example illustrates the transmission of force. (12) 4. Define the centre of gravity of a body. State where the centre of gravity is situated in the case of ::-(a) a square, (b) a circle, (c) a sphere, (d) a triangle-all of uniform density. Two circular plates of equal small thickness, and of the same material, lie side by side in contact on a table; their radii are 6 in. and 5 in. Find where the centre of gravity of the whole is situated. If their thicknesses were uniform but unequal, and if their centre of gravity were at the point of contact, what would be the ratio of their thicknesses? 5. Explain the principle of the lever. (14) A uniform horizontal straight lever AB can turn freely round a point C between A and B; AC and CB are 2 ft. and 4 in. respectively; the end A rests on a fixed support. If the weight of the lever is 12 lbs., find the pressure on A. Find also the weight hung from B, which will reduce the pressure on A to 2 lbs. (14) 6. Explain the reason why a pair of nut-crackers is effective. A nut is placed one inch from the joint and a pressure of 5 lbs. applied at a distance of 3 inches from the joint; if the pressure just suffices to crack the nut, what resistance is offered by the latter? 7. State what is meant by the tension of a thread or rope. (12) In a "tug-of-war," each party is pulling with a force of 500 lbs.; what is the tension of the rope? A thread passes over a fixed point and carries equal weights, one at each end. If the tension is supposed to be the same at each point of the thread, what suppositions are implied as to the thread and the fixed point? (12) 8. A body of given weight rests on a smooth inclined plane, and is supported by a certain force; find the pressure on the plane when the force acts (a) horizontally, (b) along the plane. Show that the weight is a mean proportional to the two pressures. 9. What is a foot-poundal of work? (12) A horse, pulling with a force equal to 1 cwt., walks at the rate of 3 miles per hour; how many foot-poundals of work is it doing every second, and at what horse-power is it working? (14) 10. Give a formula for determining the height to which a body will rise if projected vertically upwards; also one for the time of ascent. = A stone was cast vertically upwards so as to reach a height of 300 metres; disregarding the resistance of the air, what was the initial velocity of the body? (g = 9.785.) 11. What condition is necessary for a particle to move in a circular path on a horizontal plane? (14) If the path be a yard in diameter and be traversed uniformly in a second and a half, describe completely the velocity and acceleration of the particle when it is in any position in the path that you choose to select. u 56905. F (16) 12. A bullet is tied to one end of a fine thread, the other end of which is fastened to a fixed point. The bullet moves backwards and forwards through a small arc; write down a formula for the time of one oscillation, explaining the notation. If it were wished to increase the time of an oscillation by one half, by how much must the length of the thread be increased? (16) Stage 2 and Stage 3. You may take either Stage 2 or Stage 3, but you must confine yourself to one of them. The examination in this subject lasts for three hours. Stage 2. You are not permitted to answer more than eight questions. 21. Explain what is meant by relative motion. A is moving toward the east with a given velocity V, and a point B seems to move toward the north from A, with an equal velocity; what is the actual direction of B's motion, and what is its velocity? A long row of posts stands at right angles to a railway. How do the posts seem to move to a passenger in a train going along the railway, and why? (25) 22. Given the masses of two particles and the co-ordinates of their positions; find the co-ordinates of their centre of gravity. A uniform rod BC is divided at 4 into two parts, which are joined by a hinge. If AB is fixed, and AC moves in a plane round A, show that the centre of gravity of the divided rod describes a circle. Also show how to draw the circle, when AC is one-third of the length of the rod. (20) 23. If any portion (volume = v) of a body or system of bodies (whole volume = V) be displaced to another position, the displacement GG' of the centre of gravity of the whole is parallel to gg' the displacement of the centre of gravity of the portion, and its amount is given by Prove the above theorem, and apply it to show that a heavy uniform chain resting on a smooth curve in a vertical plane has its extremities in a horizontal line. (20) 24. Prove the "principle of moments" in the case of parallel forces. A straight stick has a spherical metallic knob at one end; show how to determine the weight of the knob without detaching it. (20) 25. One end of a cord is fastened to a fixed point A, while the other end is carried over a fixed smooth pulley at B, and fastened to a weight W; what weight must be attached to the point (C) of the cord midway between A and B so that there may be equilibrium with the part AC horizontal? Show also that the tension of AC will be when AB is inclined at angle a to the horizontal. (30) 26. Explain briefly what is meant by the coefficient of friction, and by the angle of friction. A ladder rests with one end against a wall, and the other on a horizontal pavement; its centre of gravity is two-sevenths of its length from its foot; the coefficient of friction of the ladder with either wall or pavement is 0.4. Find the inclination of the ladder when on the point of slipping. (N.B., tan 21° 45′ = 0.4.) If the upper end of the ladder had been a little above the wall, how would this have affected the reactions which support the ladder? (25) 27. A string passes once round a rough post (coefficient of friction ), and equilibrium is maintained by tensions P, Qat its extremities respectively. If Q be fixed in amount, show that it is the geometric mean of the limiting values of P. If the greater of these values is e times the lesser prove that Two balls of equal weight are connected by a string of length ; one is placed on the ground and the other projected upwards with the velocity due to the height h where h>; show that the first bail will rise to a height } (h − 1). . (25) 29. Define kinetic energy. If a body moves without rotation, show that its kinetic energy ismv2, where m denotes its mass and v its velocity. A perfectly flexible rope lies at length on a smooth table, with one end just hanging over, so that the end begins to go down. Find, from considerations of work and energy, the velocity of the rope at the instant that one half of its length is hanging. (20) |