8. Enunciate and prove the parallelogram of velocities. A ship is sailing due north at the rate of three miles an hour, and a passenger on board walks transversely across the deck at the rate of 4'4 feet per second; find his actual directions of motion as he walks one way or the other. 9. A stone is projected vertically upwards with a velocity of 64 feet per second; neglecting the resistance of the air, find the greatest height to which it rises, and the time of its ascent and descent. Also find the times at which its height will be 28 feet. IO. State Newton's Second Law of Motion, and show how it gives a method of measuring force. A body whose mass is 4 lbs. is moving with a velocity of 64 feet per second. If it is brought to rest in 128 feet by applying a constant resistance, find the magnitude of the resistance. II. If a particle move with uniform velocity v in a circle of radius r, prove that its acceleration is in the direction of the centre of the circle, 22 r and is equal to A heavy particle on a smooth horizontal plane is attached by a string to a fixed point on the plane. If the string be straightened, and the particle be projected horizontally in a direction perpendicular to the string, compare the tension of the string with the weight of the particle. FURTHER EXAMINATION. VI. PURE MATHEMATICS. [Full marks may be obtained for about two-thirds of this paper.] I. Find the length of a man's stride, if dividing the number of strides he makes per minute by 30 gives his speed in miles per hour. Contrast the rate of striding of a runner who covers 100 yards in 9.8 seconds, with a stride of 7 feet, with the rate of pedalling of a bicyclist who, on a machine geared so as to be equivalent to one with a driving wheel of 80 inches diameter, rides 30 miles in one hour. 2. Indicate the method of solution of simultaneous linear equations. Three trains, of lengths a, b, c (feet), are travelling with uniform velocities u, v, w (feet per second), in the same direction on equidistant parallel rails with their rear-most carriages in a straight line. Show that the trains may all be cut by some straight line or other for a time 3. Prove that the number of permutations of ʼn different things taken at a time when each of the n things may be repeated is n". In the decimal system of notation, how many numbers are there which consist of four digits? Prove that the sum of all such numbers is 49495500. 4. O is the centre of a circle, and AOB a diameter. Circles are described upon AO and OB as diameters. Show that the circle described, so as to touch the large circle internally and the two smaller circles externally, has a diameter one-third that of the larger circle. 5. ABCD is a square of which BD is a diagonal. Through A a straight line is drawn, meeting BC and BD, produced, if necessary, in H and K. If be the perpendicular distance of K from BC, show that the reciprocal of p is equal to the sum or the difference of the reciprocals of BH and BA, and distinguish the cases. If the line through A be drawn at random, show that it is an even chance that is half the harmonic mean of BH and BA. 6. Construct a quadratic equation, with rational coefficients, so that one root may be 2 sin 18°. 7. Indicate the operations necessary in order to determine the distance between two inaccessible points in the same plane as the positions where the necessary observations are made. 8. Find the polar equation of a straight line in the form r = p sec(0 - a). Find the condition that this straight line may touch the circle 2-2lr cos (0 -ẞ) + l2 — a2 = 0. 9. Find the general equation of a circle whose centre lies on the axis of x. If the abscissae of the centres of two such non-intersecting circles be +a and -a', and their radii and r', find the coordinates of points on the axis of x at which the circles subtend equal angles. Find the equation of the radical axis of the two circles. 10. Taking the principal axes of an ellipse as coordinate axes find its equation. If the ellipse be rotated through an angle in the positive direction show that its equation becomes If the direction of rotation be reversed, how is this equation affected? II. Find the equation of the normal to an ellipse at a given point. x2 The equation of an ellipse is +12 = 1. Find the coordinates of the intersection of normals at the points whose eccentric angles are 75° and 15°. 12. Find the equation of a hyperbola in rectangular coordinates. Show that if a variable line form, with two fixed lines, a triangle of constant area, the locus of a point which divides the intercept made on the variable line in a given ratio, is a hyperbola. 13. In a parabola, prove that an isosceles triangle is formed by the focal distance of a point, the normal at the point and the axis. Find the locus of the foot of the perpendicular from the focus on the normal. (This question is to be solved geometrically.) 14. Prove that the feet of the perpendiculars from the foci on any tangent to an ellipse lie on a circle whose radius is equal to the semi-major axis. In an ellipse, if a line be drawn through a focus making a constant angle with the tangent, prove that the locus of the point of intersection with the tangent is a circle. Find also its centre and radius. (This question is to be solved geometrically.) 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 feet per second per second.] I. Explain the derivation of the triangle of forces from the parallelogram of forces. Two forces are represented by the lines joining the middle points of opposite sides of a quadrilateral. Show that their resultant is represented in magnitude and direction by one of its diagonals. 2. Show that if any number of coplanar forces acting at a point are in equilibrium, the sum of their components resolved in any two directions at right angles to each other must each be zero. A string 31 inches long passes through a small ring of 4 ounces' weight, and has its extremities fixed at two points 25 inches apart, and in the same horizontal line. Find the tension of the string in the position of equilibrium in ounces' weight correct to two places of decimals. Find also the magnitude of the horizontal force which, applied to the ring, will cause it to rest at a point 7 inches from the nearer end of the string. (This question may be solved graphically or analytically.) 3. Prove that a system of coplanar forces will be in equilibrium if the algebraical sum of the moments of the forces about any three points not in the same straight line vanishes in each case. A triangular lamina ABC, whose sides BC, CA, AB are respectively 18, 24, and 30 inches in length, is placed in a vertical plane with BC, CA resting upon two fixed smooth pegs D, E, 20 inches apart and in the same horizontal line. If equal weights W, W be suspended from A and B, and the triangle be kept with AB horizontal by means of a string connecting C with the peg D; find the tension of the string and the pressures on the pegs, neglecting the weight of the lamina. 4. (This question may be solved graphically or analytically.) Define the centre of mass (centre of gravity) of a heavy body, and show that if a heavy body be suspended from a fixed point its centre of mass must be vertically beneath the point. In a lamina of any form a line AB of length c is taken, and it is observed that when the lamina is suspended from A, the line AB dips 30° below the horizon, and 45° when suspended from B. Find the distance of the centre of mass of the lamina from AB. (This question may be solved graphically or analytically.) |