8. ABC is a triangle, with C vertically below B, and let the lengths of AB, BC, CA be 7, 10, 9 ft. respectively; suppose that AB and AC represent two weightless rods, connected by a smooth joint at A, and fastened to a wall by smooth hinges at B and C. If a weight of 5,000 lbs. is hung at A, find the reactions at B and C; and state the mechanical principle which enables you to determine them. The tendency of the weight is to stretch one of the rods; state which it is, and justify your answer. (16.) 9. Define a "Horse-Power" and express it in foot-poundals per second. What is the Horse-Power of an engine which raises 10,000 gallons of water (each gallon weighs 10 lbs.) to a height of 100 ft. in 40 minutes? (12.) 10. What is meant by the phrase "composition of velocities"? A steamer is travelling with a uniform velocity 22, and a ball is rolled across the deck with uniform velocity 10; determine graphically and by calculation the velocity of the ball in space. (10.) 11. A point moves in a straight line; it has an initial velocity (V), and its velocity undergoes a constant acceleration. Write down formulæ for finding the velocity at the end of a given time, and the distance described in the time, and deduce the formula which gives the velocity acquired during the description of an assigned distance. Two particles, whose masses are 15 lbs. and 9 lbs., are connected by a fine thread which passes over a smooth point (as in Atwood's Machine); if they are allowed to move find the acceleration of the velocity, and the tension of the thread. (14.) 12. A particle moves with a constant velocity in a given circle; state what is known as to the force acting on the particle. If the radius of the circle is 10 ft., the mass of the particle 4 lbs., and the velocity 30 ft. a second, find the magnitude of the force which constrains the particle to move in the circle. Along what line, and in what direction, does the force at each instant act? Advanced Stage. INSTRUCTIONS. Read the General Instructions on page 1. You are not permitted to answer more than eight questions. (16.) 21. Define "Relative Velocity" and "Relative Acceleration." Two particles start at the same moment from the same point in opposite directions, with initial velocities u, u1, and are subject to accelerations f, fi respectively; find the sum of their velocities when the distance between them is s. (20.) 22. State the distinctive property of the centre of a system of parallel forces, and find the position of the centre of two unlike parallel forces acting at given points. ABC is a given triangle having a right angle at C; parallel forces of 12, 10, and 20 units act at A, B, and C respectively, and the force of 20 units is unlike the other two forces; find the position of their centre, and show it in a carefully drawn diagram, (20.) 23. Find the centre of gravity of the curved surface of a cone, supposed to be of uniform density. Also, find by geometrical construction the centre of gravity of a quadrilateral lamina of uniform density. (25.) 24. Define a couple, and show that two couples of equal opposite moments, acting on a rigid body in parallel planes, balance each other. Forces act from A to B, B to C, C to A, along the sides of a triangle ABC; if the forces are proportional to the lengths of the sides along which they act, show that they are equivalent to a couple, and find its moment. (20.) 25. State the laws of Friction. A block of weight w1 lies on a table, and has another block of weight w2 on the top of it; find the relation that the coefficient of friction between the blocks bears to that between the lower block and the table, if it be just possible to move the upper block by horizontally applied force without disturbing the lower one. (20.) 26. Describe the simple machine termed "The Screw," and apply the principle of work to find the relation between the power and the weight. If the screw-cylinder be 6 inches in diameter, the distance between two threads 2√3 inches, and the angle of friction 30°, find the thrust produced by a force of 50 lbs. applied, in a plane perpendicular to the axis, at the end of an arm, whose length, measured from the centre of the axis, is 3 feet. (25.) 27. ABCD is a parallelogram, of which BD is a diagonal; suppose it to represent a frame of five weightless rods connected by smooth joints; let the frame be suspended from A, and carry a weight Whung from C. Given the lengths of the rods, find the stresses in AB and BD, and whether those rods are in compression or extension. (25.) 28. Distinguish between stable and unstable equilibrium. A hemisphere and a cylinder are joined by their bases, which are equal. The body thus formed is placed in equilibrium with the hemisphere in contact with a horizontal table. Find under what circumstances the equilibrium will be stable. N.B.-The centre of gravity of a hemisphere is at a distance of 3-8th of the radius from the centre. (20.) 29. Show how to find the work done by a variable force, and show the merit of graphical construction by taking as an illustration the work done on the piston in a steam engine. A heavy hemisphere rests upon a horizontal plane with its base horizontal and uppermost. How much work must be performed in turning it over on to its base? (25.) 30. A body is projected in a vacuum, at an angle a to the horizontal, and with the velocity due to the height; show that at every instant the vertical height of the body and its horizontal distance from the point of projection are connected by an invariable relation. If a be variable, show that the greatest possible range, on the line of greatest slope of a plane inclined at an angle B to the horizon, when the projection takes place in the vertical plane containing that line, ist 2h 1 + sin B (30.) 31. Draw a circle and suppose its plane to be vertical; let C be the lowest point of the circumference, and the arc CA a quadrant; also let B be a point of the circumference halfway between A and C. A particle is held at A, and is connected with the centre by a thread whose length equals the radius; if the particle is let fall, find its velocity as it passes through B and through C If the length of the radius is 3 ft., and the mass of the particle 10 lbs., find the tension of the thread when the particle is at B, and also when it is at C. (30.) 32. Give an account of the direct impact of imperfectly elastic smooth spheres. Demonstrate that if two smooth spherical balls impinge directly, the velocity of the centre of gravity immediately after impact is the same as immediately before impact. 1899. SUBJECT VI. THEORETICAL MECHANICS. (B.-Division II.) FLUIDS. EXAMINERS: REV. J. F. TWISDEN, M.A., AND MAJOR P. A. MACMAHON, R.A., Sc.D., F.R.S. GENERAL INSTRUCTIONS. If the rules are not attended to, the paper will be cancelled. You may take the Elementary Stage, or the Advanced Stage, or Part I. of Honours, or (if eligible) Part II, of Honours, but you must confine yourself to one of them. Put the number of the question before your answer. The value attached to each question is shown in brackets after the question. But a full and correct answer to an easy question will in all cases secure a larger number of marks than incomplete or inexact answer to a more difficult one. an You are to confine your answers strictly to the questions proposed. Your name is not given to the Examiners, and you are forbidden to write to them about your answers. The examination in this subject lasts for three hours. Elementary Stage. INSTRUCTIONS. You are not permitted to answer more than seven questions. 1. Give the names of the fundamental units of time, distance, and mass commonly used in England, and the names of the corresponding units on the metrical system. Define density. The volume of a substance (A) is onetenth of a cubic metre, and its mass is 500 kilogrammes ; the volume of another substance (B) is 3 cubic centimetres, and its mass is 40 grammes; compare the densities of A (10.) and B. 2. Define momentum. A body, whose momentum expressed in British absolute units is K, is moving in a straight line and is acted on, for one second, by a force of K poundals in the direction opposite to that of the body's movement; determine the final momentum. (10.) 3. A particle, whose mass is 3 lbs., is moving at the rate of 64 feet a second; how many foot-poundals of work can it do against a resistance? How far would it go before being brought to rest, if its motion were directly opposed by a force equal to the weight of 2 lbs. (g = 32)? (12.) 4. Distinguish between specific gravity and specific (or intrinsic) weight as applied to a substance. In what proportion by weight must gold of specific gravity 19 3, and copper of specific gravity 8 6, be taken to produce an alloy of specific gravity 18? (12.) 5. A rectangular vessel, e.g. a cube, is filled to a depth of 6 in. with a liquid whose specific gravity is 1.2; a second liquid, which does not mix with the former and whose specific gravity is 0.8, is poured in till the resultant pressure on the part of a side of the vessel in contact with the upper liquid equals the resultant pressure on the part in contact with the lower liquid; find the depth of the upper liquid. (16.) 6. A body of uniform density is at rest in water and one-tenth of its volume is above the surface :-(a.) If the body is floating, what is its specific gravity? (b.) If it is resting on the bottom, and its specific gravity is 1.5, what part of its weight is supported by the bottom ? (14.) 7. A vertical cylinder, diameter of base 1 foot, and height 3 feet, is half-full of water; find the intensity of the pressure on the base. A cube, edge 3 inches, floats half-immersed in the liquid; find the alteration in the whole pressure on the base. น 8274. F (14.) |