Find the force which when applied to the end of an arm 18 inches long will cause the carriages to be drawn together with a force of I ton, taking the ratio of the circumference to the diameter of a circle as 34. 5. Explain what is meant by the angle of friction. A body is placed on a rough plane whose inclination to the horizon is 45° the angle of friction being 30°. Prove that the force which must be applied to the body in a horizontal direction in order to just prevent it from slipping down is the same as if the plane were made smooth and its inclination to the horizon were decreased to 15°. (This question may be solved either geometrically or analytically.) 6. A slip-carriage is detached from a train and brought to rest with uniform retardation in two minutes during which time it travels two-thirds of a mile. With what velocity was the train travelling when the carriage was detached? The weight of the whole train was 130 tons and that of the slip-carriage IO tons. Before the carriage was detached the train was just kept going with uniform velocity, by the pull of the engine, the resistance due to friction amounting to 35 lbs. weight per ton. Supposing the engine to pull the train with the same force after the carriage is detached, find the acceleration. 7. In Atwood's machine, two weights of 5 lbs. and 7 lbs. are attached to opposite ends of a string which passes over a light smooth pulley, and a rider weighing 4 lbs. is placed over the smaller weight. Find the acceleration. When the system has moved through one foot from rest, the rider is detached by coming in contact with a fixed ring. How far will the 5 lb. weight descend below the ring before coming to rest? 8. A mass of m lbs. is making n complete revolutions per second on a smooth horizontal table in a circle of radius to whose centre it is connected by a string. Find the tension of the string expressed in pounds' weight. In the Watt's governor of a steam-engine, two equal light rods OA, OB, I foot long, are freely hinged at one end O to a vertical axis, and to their other ends A, B are attached equal weights of 7 lbs. The whole system is revolving about the vertical axis so that the rods OA, OB make equal angles with the vertical on opposite sides of it. Taking π = 22, find the number of revolutions per minute when the rods include an angle of 120°. 9. A particle is projected in vacuo from a point O with a velocity whose horizontal and vertical components are 32 feet per second and 80 per second (upwards) respectively. Draw a careful figure exhibiting the positions of the particle after 1, 2, 3, 4, and 5 seconds respectively, and state the coordinates of the particle at these instants, referred to horizontal and vertical axes through O. Find also the coordinates of the focus of the parabola described. IO. A sphere of mass M moving with velocity U overtakes and impinges directly on a sphere of mass m, moving with velocity u in the same straight line. If the elasticity of the spheres be perfect, find by how much the kinetic energy of the first sphere is decreased and that of the second increased. Three equal spheres whose coefficient of restitution (¿.e. elasticity) is are ranged at rest in a straight line. The first sphere is then projected in the same straight line with velocity U, so as to impinge directly on the second. Find the final velocities of the three spheres after the last collision has taken place between them. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1897. OBLIGATORY EXAMINATION. I. EUCLID. [Ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. In the absence of special directions to Candidates, any of the propositions within the limits prescribed for examination may be used in the solution of problems and riders. Great importance will be attached to accuracy.] I. Define carefully the following geometrical terms:-point, plane, plane angle, circle, parallelogram. 2. Prove that if two angles of a triangle are equal to one another the sides which are opposite to the equal angles are also equal to one another. 3. Prove that if the square described on one of the sides of a triangle is equal to the sum of the squares described on the other two sides of it, the angle contained by these two sides is a right angle. 4. Prove that if a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are together equal to twice the rectangle contained by the whole and that part, together with the square on the other part. 5. Prove that in every triangle the square on the side subtending an acute angle is less than the sum of the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the projection on it of the other side. 6. Prove that if two circles touch each other externally, the straight line which joins their centres passes through their point of contact. 7. Prove that if a straight line touch a circle the straight line drawn from the centre to the point of contact is perpendicular to the line touching the circle. 8. Prove that if from any point without a circle two straight lines be drawn, of which one cuts the circle and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, is equal to the square on the line which touches it. 9. Inscribe a circle in a given triangle. IO. Prove that in a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another. II. Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. 12. Prove that in equal circles angles at the centre have the same ratio which the arcs on which they stand have to one another. 13. If two sides of a triangle are produced and the exterior angles between the produced sides and the third side are equal to one another; prove, without using any proposition subsequent to Euclid I. 6 (question 2), that the triangle is isosceles. 14. Within a parallelogram ABCD any point K is taken; a straight line drawn through K parallel to AB meets AD and BC at H and G respectively, and a straight line drawn through K parallel to AD meets AB and CD at E and F respectively. Prove that the squares on EH and FG together exceed or fall short of the sum of the squares on HF and EG by a rectangle independent of the position of K. 15. Construct a square of which one side shall lie along a given tangent to a given circle and of which the opposite side shall be a chord of the circle. 16. The circle inscribed in the triangle ABC touches the side BC at D. Show that the circles inscribed in the triangles BAD and CAD touch each other. 17. Through the extremities of two fixed perpendicular radii of a circle a pair of parallel chords are drawn. Prove that the circle whose diameter is the line joining the middle points of the two parallel chords passes through a fixed point, and also prove that the centre of this circle lies upon a fixed circle. |