I. Forces. V. STATICS AND DYNAMICS. [You may assume that g = 32 feet per second per second.] State accurately the principle known as that of the Triangle of A particle is acted upon by three forces of given magnitudes; show how these forces must be arranged so as if possible to produce equilibrium, when (i.) the forces have magnitudes represented by the numbers, 6, 11, 18. (ii.) their magnitudes are represented by 8, 15, 17. 2. Let O be the position of a particle, and OA a right line drawn through O. Find the magnitude and direction of the resultant of forces proportional to 10, 18, 20, 16, acting on the particle, when their lines of action make with OA angles of 0°, 30°, 90°, 135°, respectively, all measured in the same sense (i.e., all outwards from 0, or all inwards towards O). 3. A body whose mass is 520 pounds is placed on a smooth inclined plane, the tangent of whose inclination to the horizon is; find the force necessary to sustain the body 4. (i.) when this force is horizontal; (ii.) when it acts along the inclined plane. Define a couple, and show that its moment is the same about all points in its plane. Prove that the sum of the moments of any two forces in the same plane about any point in the plane is equal to the moment of their resultant about the point. 5. Two parallel forces, P, Q, act on a rigid body; find the magnitude and line of action of their resultant (1) when they act in the same direction; (2) when they act in opposite directions. Two parallel forces of 20 and 25 pounds' weight, of opposite senses, act on a rigid body, the perpendicular distance between their lines of action being 4 inches; find the resultant. 6. Show how to find the position of the centre of gravity of a given system of particles whose masses are m, m2, m, ... occupying given positions. At each vertex of a triangle is placed a particle whose mass is proportional to the length of the opposite side; show that the centre of gravity is the centre of the inscribed circle. From a thin uniform circular plate of radius 13 inches is cut out a circular plate of radius 5 inches, the centre of the latter being 4 inches distant from that of the former; find the position of the centre of gravity of the remainder. 7. Describe the screw press, and find the condition for the equilibrium of the effort and resistance ("power" and "weight") applied to it when there is no friction. 8. Define acceleration. State clearly what is meant by saying that g is about 32 feet per second per second, and describe any method by which this value of g has been obtained. Which is the greater acceleration, 15 miles per hour per minute, or feet per second per second? 9. If a particle moves in a right line with constant acceleration, a, having had an initial velocity, u, show that the distance travelled in time is given by the equation s = ut+at2. A bullet is fired vertically upwards with a velocity of 496 feet per second; 3 seconds afterwards another is fired vertically upwards from the same point with a velocity of 568 feet per second; when and where will they meet? IO. Define an absolute unit of force, and, in particular, the dyne and the poundal. What force uniformly applied to a mass of 12 pounds will give it an acceleration of 8 feet per second per second? (Express the force both in pounds' weight and in poundals.) II. From a given point on a horizontal plane is projected a particle with a velocity u at an elevation a; find the range on the plane. If the particle is projected with a velocity of 500 feet per second at an angle of elevation whose tangent is, find the range. Find also the magnitude and direction of the velocity 10 seconds after the time of projection. FURTHER EXAMINATION. I. VI. PURE MATHEMATICS. [Full marks may be obtained for about two-thirds of this paper.] Sum up the conditions under which Euclid says that two triangles are equal. ABC is a triangle and D a point in BC such that AD bisects the angle A. If O be the centre of a circle which touches AB at A and also passes through D, prove that OD and AC are at right angles and find the magnitude of the angle AOD. 2. Apply the theory of geometrical progression to the evaluation of a mixed recurring decimal. Show that the sum of and the series is unity. + + 38 x 18, 38 x 182,38 × 183 ad inf. Employ logarithms to evaluate the tenth and twentieth terms of the series. 3. A clock strikes at intervals of one second. Determine the intervals as they appear to men travelling in express trains at 60 miles an hour directly towards and directly away from the clock respectively. The velocity of sound may be taken as 1100 feet per second. 4. Find the greatest coefficient in the expansion of (x+y)n. Prove that the greatest coefficients in the expansion of the trinomial N. B.-n! represents the product n(n − 1)(n − 2) ... 3. 2. I. 5. In any triangle show that, in the usual notation, 6. Find the volume of a pyramid on a triangular base and deduce that the volume of a sphere is the product of the area of its surface and one-third of its radius. A sphere is cut by a horizontal plane. If P be a point in the perimeter of the section and A the highest point of the sphere, show that the surface of the sphere above the cutting plane has an area equal to that of a circle of radius AP. 7. In the triangle ABC, sides 4, 3, 5 AB produced is a tangent, BC the axis and AC the tangent at the vertex of a parabola. Draw the triangle in your book and construct geometrically for (i.) the focus, (ii.) the directrix, (iii.) the point P on the parabola at which AB is tangent, (iv.) the other extremity of the focal chord through P, (v.) the extremities of the latus rectum. 8. In an ellipse QQ' is the chord of contact of tangents from an external point P; CK is the perpendicular on QQ' from the centre C; and PG is the perpendicular to QQ', through P meeting the major axis in G. Prove that the semi-minor axis is a mean proportional to KC and PG. 9. Write down the usual forms of equation to a straight line. Through a given point P whose coordinates are (p, 9) a straight line is drawn intersecting the axes in A and B so that P is the middle point of AB. Determine the equation of AB. 10. Find the general equation of the circle when the coordinate axes are inclined to one another at an angle w. The axes being rectangular investigate the condition that must be satisfied by the parameters of the circle in order that it may be possible to find a point or points on the circle at equal perpendicular distances from the axes. find the equation to the circle passing through the vertex and the extremities of the latus rectum. Find also the coordinates of the points where the circle is intersected by those normals to the parabola which make an angle of 45° with the axis. 12. C is the centre of an ellipse and BB' the minor axis. If S be the focus which (to origin at the centre and coordinate axes coincident with the axes of the ellipse) has a positive abscissa; and B'S be produced to meet the curve in P; show that CP makes an angle with the major axis such that 13. Find the equation to the tangent at any point of the hyperbola 4xy = a2+b2. If two tangents at right angles intersect at P find the locus of P. VII. MECHANICS. [Full marks may be obtained for about two-thirds of this paper. Great importance is attached to accuracy. N.B.-g may be taken = 32.] I. If three forces acting on a particle are in equilibrium, each force is proportional to the sine of the angle between the other two. ABCD is a quadrilateral, having the angles at A and D right angles, and CB = CD. Forces P, Q, and R, acting along AB, CA, and AD respectively, are in equilibrium. show that AB = BC or BC. If 2. Prove that the algebraical sum of the moments of a number of coplanar forces acting on a particle about any point in their plane is equal to the moment of their resultant about that point. Pis the orthocentre of a triangle ABC. Forces act along AP, BP, CP, and are proportional to sin (A +0), sin (B+0), sin (C+0). Prove that their resultant passes through the centre of the circumscribed circle. the value of when the forces are in equilibrium? What is 3. Two equal uniform rods, AB, BC, each of weight W, are hinged together by a smooth hinge at B, and rest on a smooth cylinder of radius 6, |