13. The exterior angle between any two intersecting tangents to a parabola is equal to the angle which either of them subtends at the focus. Prove that if S be the focus and PQ the points in which two fixed tangents are cut by a third, the triangle SPQ will have its angles constant. 14. Prove that the straight line joining the foci of an ellipse subtends at the pole of any chord half the sum or difference of the angles which it subtends at the extremities of the chord. Distinguish between the cases. VII. MECHANICS. [Full marks will be given for about two-thirds of this paper. Great importance is attached to accuracy. Gravitational acceleration may be taken = 32 feet per second per second, and π= T = 2.] I. Show how to find the magnitude and line of action of the resultant of a system of coplanar forces. Draw a straight line AB, 3 inches long, and upon it as base construct a triangle ABC whose sides CA, CB are 2 and 1 inches respectively. If forces of 5 lbs. act from C to A and from C to B, and a force of 6 lbs. from B to A, find graphically the magnitude and line of action of the resultant of the three forces. 2. If M1, M2 be the algebraical sums of the moments of a system of coplanar forces about two points A, B, prove that the algebraical sum about a point C lying between A and B will be M1. BC+M2. AC; AB and hence show that, if the algebraical sums of the moments of such a system with respect to three points not in a straight line, be separately equal to zero, the sum will be zero for any other point in the same plane. 3. The lower end B of a uniform rod AB, of length 2a, rests on a smooth horizontal plane BD, and the upper end A is supported by a string AC of length b, the extremity C being fixed at a point whose height above the plane is 2a. Find (i.) the horizontal force which, applied at the lower end of the rod, will cause it to rest at a distance b from the point in the plane immediately below C, (ii.) the reaction of the plane at B, and (iii.) the tension of the string AC;—the weight of the rod being W. 4. A uniform wire is bent into the form of a triangle; find the position of its centre of gravity. If the lengths of the sides be 3, 5, and 7 inches, and the triangle be suspended from the obtuse angle, prove that in the position of equilibrium the shortest side will be horizontal. 5. State the laws of Limiting Friction, and explain what is meant by the "coefficient of friction" and the "angle of friction." A lamina in the form of an equilateral triangle rests within a fixed rough hoop, the plane of each being vertical, and the radius of the hoop equal to the side of the triangle. Show that in the limiting position of equilibrium, the base of the triangle will be inclined to the horizon at an angle equal to twice the limiting angle of friction. 6. Prove that the work done in drawing a body up a rough inclined plane is equal to the work done in drawing the body along an equally rough horizontal plane through a distance equal to the length of the base of the inclined plane, together with the work done in lifting the body through the height of the plane. A train whose mass is 96 tons commences the ascent of an incline of I in 80 at the rate of 45 miles an hour, the resistance being 7 lbs. per ton. If after travelling a mile the velocity be reduced to 30 miles per hour, find in foot tons the work done by the engine. 7. Find the tension of a light string which passes round a smooth fixed peg, and has unequal masses my, my attached to its extremities. Two bodies of equal mass hang from the ends of a light string which passes round a fixed peg, and when at rest are 9 feet and 16 feet above a fixed horizontal plane. A piece of the upper body breaks off and strikes the plane at the same moment that the lower body does. Find the ratio of the parts into which the upper body is divided. 8. Show that there are generally two directions in which a particle may be projected with given velocity from a given point O so as to pass through a point P. If R be the difference between the two ranges on a horizontal plane through O, and H the difference between the two greatest heights attained by the particle in the two paths, prove that R = 4H tan 0, where is the angle OP makes with the horizon. 9. A smooth sphere impinges obliquely on a fixed plane; find the motion after impact, the elasticity being imperfect. Two billiard balls P, Q of radius v stand upon a smooth table ABDC, their centres being at distances h1, h2 from the cushion AC and k1, ką from the cushion AB. Find the direction in which P must be struck so that after impinging in succession upon AB, AC it may strike Q directly, taking e to be the coefficient of restitution of the indiarubber cushion. IO. The edges of a groove cut in a smooth horizontal table are two concentric circles whose radii are II and 13 inches. A sphere of 1 lb. mass whose diameter is 3 inches moves in this groove with uniform velocity. Find the greatest number of revolutions it may make per second without leaving the inner edge, and if it make half this number, find the reactions at the points where it rests upon the edges. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1898. 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. Enumerate (without proofs) the cases in which with certain data, two triangles are equal in every respect. Show that the proof of Prop. 4, Book I., may require one of the triangles to be moved out of its plane. 2. Show that if a parallelogram and a triangle have the same base and altitude, the area of the first is double that of the second. Hence prove that the area of a trapezium is half the sum of the parallel sides multiplied by the perpendicular distance between them. |