v. Find the differential co-efficient of r with respect to x, where r and x are both functions of y. If x=epCoseand q=eeSine, show that dm dr w. If u=0 be the equation of a plane curve in a rational form, prove that at a point of inflection d2u/du\2 d'u du du dudu -2 2 is generally=0. dx dy dxdy dx dy+dy dx, Find the nature of the point at the origin in a curve in which y2m+1= a2(m—n) μ2n+1 m and n being integers such that x. Find the values of the following integrals : y. A portion of a parabola intercepted between the chord joining the vertex and the extremity of the latus rectum revolves about this Σπα chord; show that the volume of the solid so generated = 15/5 where 4a is the latus rectum of the parabola. (15.) 2. If and are dependent on one another by the equation— where F (0)=√1-cSine and a is constant, show that (1) CoseCosp-Sin@Sino ✓✓/1— c2Sin2a=Cosa SUBJECT VI. THEORETICAL MECHANICS. EXAMINER, REV. J. F. TWISDEN, M.A. First Stage or Elementary Examination. INSTRUCTIONS. N.B.-You are at liberty to answer any eight, but not more than eight, of the following questions. Three hours are allowed for working this paper. Full and exact answers are required. All the work should be shown. The value attached to satisfactory answers is indicated by numbers in ( ). In all cases the number of the question must be placed before the answer on the worked paper. 1. Draw two lines AB and AC at right angles to each other. Suppose that a force of 10 lbs. acts on the point A from A to B, and a force of 12 lbs. on the point A from A to C; find by a construction made to scale the magnitude and direction of the force which acting at A would balance these forces. (10.) 2. A rod AB weighs 10 lbs. and is found to balance about a point 8 ft. distant from A. A weight of 6 lbs. is fastened to A; about what point will the rod now balance? (10.) 3. A bricklayer's labourer with his hod weighs 170 lbs.; he puts into the hod 20 bricks weighing 7 lbs. each, he then walks up a ladder to a height of 30 feet. How many units of work does he do; and if he can do 1,500,000 units of work a day, how many bricks will he take up the ladder in a day? (10.) 4. A cube whose side is 18 in. long is placed in water, the top face being exactly on the surface. Find the pressure on its lowest face, and show by a diagram how it acts. (A cubic foot of water may be reckoned to weigh 1000 oz.) (10.) 5. What is meant by the specific gravity of a body? A body whose specific gravity is 7 weighs 18 oz. in water, what is its weight in air? ̄ (10.) 6. State exactly what is meant when it is said that the accelerating force of gravity near the earth's surface is very nearly 322 in feet and seconds. A body falls freely under the action of gravity from rest for 6 seconds, what is the space described in the last 2 seconds of its motion? (20.) 7. Show that when two couples act on a body in the same plane, they will be in equilibrium, if their moments are equal and of contrary signs. Draw a square ABCD, suppose forces of 10 lbs., a piece to act from A to B, B to C, C to D respectively; show that they can be balanced by a force of 10 lbs., and show by a diagram exactly how it acts. (30.) 8. A body placed on a rough horizontal plane is found to be capable of just being dragged along the plane by a force of 27 lbs. applied horizontally. It is known that the coefficient of friction between the body and the plane is 0.18. What is the weight of the body? (15.) 9. State exactly what is meant by a body being (1) in stable (2) in unstable equilibrium? Illustrate your answer by the case of a loaded sphere placed on a horizontal plane. (15.) 10. Describe the suction pump, illustrating your answer by a sketch showing the action of the valves. How and why is the action of the suction pump limited? (15.) 11. State exactly what is meant by the work accumulated in, or vis viva of, a moving point. A railway truck weighs 12 tons, it is drawn from rest by a horse through a distance of 50 feet, and is then moving at the rate of 3 miles per hour; if the resistances are 8 lbs. per ton, how many units of work must the horse have done on the truck? (30.) 12. A and B are two inelastic balls, whose weights are 4 lbs. and 12 lbs. respectively, A with a velocity of 8 feet per second meets B moving with a velocity of 4 feet per second. With what velocity, and in what direction will they be moving directly after impact? (15.) Second Stage or Advanced Examination. N.B.-You are at liberty to answer all the questions. The answers should be full and exact. All the work should be shown. In all cases the letter distinguishing the question must be placed before the answer on the worked paper. a. How does it appear that two equal parallel forces acting towards contrary parts on a body cannot be balanced by any one force? What is such a combination of two forces called? b. What is meant by the centre of gravity of a body? Draw a triangle A, B, C, and suppose it to represent a triangular board weighing 10 lbs. Suppose weights of 5 lbs., 5 lbs., and 10 lbs. are placed at A, B, and C respectively. Where will the centre of gravity of the whole now be? c. A body weighing 54 lbs. is just set in motion on a rough horizontal plane by a horizontal force of 9 lbs. If the force is withdrawn and the plane tilted up, at what inclination of the plane to the horizon will the body begin to slide? d. Explain the method of determining the relation between two forces which are in equilibrium in the state bordering on motion on a body capable of turning round a rough axle. e. State and prove the conditions of equilibrium of a floating body. The specific gravity of cork being 0.26, if 1 oz. of cork is tied by a thread to the bottom of a vessel of water so as to be kept wholly under water, what is the tension of the thread? f. A given quantity of elastic fluid occupies different volumes under different pressures, what is the law connecting the volumes and the corresponding pressures, assuming the temperature to remain unchanged? A bent tube closed at one end, of uniform bore, has the air in the closed end isolated by mercury, which fills the bend. When the height of the barometer is 28 inches the mercury stands at the same level in the two legs, and the inclosed air occupies 8 inches. On the barometer rising the difference of level of the mercury in the two legs is half an inch. What is now the height of the barometer? g. Two bodies weighing 17 lbs. and 15 lbs. respectively are connected by a fine thread passing over a smooth fixed point; explain how their motion is determined. Supposing them to start from the same point, how far apart will they be after 5 seconds from the beginning of the motion (g=32)? h. What is meant by the moment of inertia of a body with reference to a given axis? Find that of a square about an axis at right angles to its plane and passing through the intersection of its diagonals. [N.B.-It may be assumed that the moment of a rod about an axis at right angles to its length and passing through one end equals (mass) × (length).2] i. What is meant by the centre of oscillation of an oscillating body? Show that the centres of oscillation and suspension are reciprocal. j. A and B are two balls whose coefficient of elasticity is 0.5, their weights are 4 lbs. and 12 lbs. respectively. A with a velocity of 8 feet per second meets B moving with a velocity of 4 feet per second. With what velocity and in what direction will they be moving directly after impact? Honours Examination. You are at liberty to answer all the questions. The answers should be full, but not diffuse. All the leading steps, if not absolutely all the rough work, should be shown. In all cases the letter distinguishing the question must be placed before the answer on the worked paper. p. Show that, in the general case, any system of forces can be reduced to a single force and a couple. Perform the reduction in the case of three equal forces acting severally along three edges of a cube which are neither parallel nor intersect. q. An ellipse with its plane vertical is placed on a rough inclined plane; there is sufficient friction to prevent its sliding; determine under what circumstances and in what position it can come to rest on the plane. r. A rod, whose weight can be neglected, is supported on two points A and B; a weight P is placed on it at a given point C; what is the measure of the tendency of P to break the rod at C? Explain how the tendency is resisted. s. The stresses on two planes at right angles to each other are in directions parallel to a third plane at right angles to the given planes. Explain how the stresses can be determined on any fourth plane which is also at right angles to the third plane. t. Explain what is meant by the metacentre of a floating body, and state the conditions of stable equilibrium in the case of such a body. If r is the radius of the base, and h the height of a cylinder, ascertain the relation between r and h, which will determine whether it will float with its axis vertical, or with its axis horizontal. u. Investigate a formula for comparing the density of a mixture of air and vapour with the density of dry air at the same temperature and under the same pressure. [It may be assumed that the ratio of the density of vapour to that of dry air at the same pressure and temperature is 0-625.] v. What is meant by the absolute unit of force? A certain force contains F absolute units of force, the unit of mass being a pound avoirdupois, the unit of space a foot, and the unit of time a second. It would contain F, units, if the unit of mass were a hundredweight, the unit of space a yard, and the unit of time 10 seconds. What is the ratio of F to F1? w. A heavy point slides down the inside of the circumference of a rough circle whose plane is vertical; it starts from rest at an extremity of the horizontal diameter. Find the velocity at the lowest point. x. What is meant by the principal axes of a rigid body? Find the moment of inertia of a square lamina about an axis passing through the intersection of the diagonals and making an acute angle with one diagonal and a right angle with the other. y. A body capable of turning freely round an axis supported at two points is struck; investigate the equations for determining the impulsive strains on the points of support. Take AB, any straight line on the surface of a cylinder whose geometrical axis is CD. The cylinder can turn freely round the points of support A and B; it is struck along a line at right angles to the plane ABCD, meeting CD in a point P, such that 4 CP = CD. The initial angular velocity communicated by the blow being given, determine the impulsive strains on the points A and B. SUBJECT VII. APPLIED MECHANICS. First Stage or Elementary Examination. The student will be expected to answer eight questions, and of these eight, four must be the first four in the paper. The first four questions are valued at 30 each, the next four are valued at 20 each, and the remaining four at 30 each. An imperfect answer to a question of high value may not receive so much credit as a correct reply given to a question of lower value. N.B.-A full and correct answer will in all cases gain more credit marks than an inexact or incomplete one. In all cases the number of the question must be placed before the answer on the worked paper. Three hours allowed for this paper. 1.-30. What is the nature of the property termed malleability which is found in some of the metals, such as lead, copper, and wrought iron? Give a few examples of the advantage taken of this property in applied mechanics. 2.-30. When a founder pours liquid iron into a mould to form a casting, is the said casting, when cold, of the same dimensions as the pattern with which the mould was formed; if not, how do you account for the difference? 3.-30. How many units of mechanical work will be expended in raising 10 cubic feet of water a height of 100 feet? 4.-30. Can a given amount of power be increased by any combination of mechanical arrangements, such as wheels and pinions, fly-wheels, &c.? Give distinct reasons for your opinion. 5.-20. What is the object in having bronze or other peculiar metal as the bearings or bushes of machinery for iron axles to run or work upon? 25011. C |