VI. 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 OA and OB represent two forces in direction and magnitude, find the line OR representing their resultant. If OC and OD be two equal lines cut off from OA and OB respectively, and if OR meet CD in G, find the ratio of CG to GD. 2. A rod AB, whose centre of gravity is at G, is supported at an angle of 42° to the vertical, with one end A in contact with the smooth vertical wall AD, by a string attached to the point C of the rod and also to a point in the wall. AB = 3", AC = 1", AG = I§". Find graphically the length of the string and its tension in terms of the weight of the rod, which is 55 ozs. 3. The horizontal rod BEC and the rod DE being without weight, and the latter perfectly rigid, find graphically the actions at B and E when the weight 150 lbs. is suspended from C. BD is vertical and BC = 18, BE = '9, DE = 1'5, BD = I'2. 4. Find the relation between the power and the weight in the smooth wedge whose transverse section is an isosceles triangle. If the pressure between the wedge and the separated points in contact with it be proportional to the distance between these points, prove that the work done is proportional to the square of the distance of penetration. 5. The uniform square ABCD rests vertically with the side BC upon a horizontal plane, coefficient of friction, and has a rope attached at D and passing over a small smooth pulley at the point E in BA produced till EA is equal to AB. If the rope be pulled, find whether the initial motion of the square will be tilting or sliding. 6. Apply the laws of motion to determine the path of a projectile. From a given point in a railway carriage, moving with uniform velocity in a straight line, bullets are fired continuously with a constant velocity at right angles to the rails, and with a constant inclination to the horizon. Find the locus at any instant of all the bullets which have not reached the ground. 7. Find the acceleration down a smooth inclined straight line. Two particles slide down two straight lines, in the same vertical plane, at right angles to one another, starting simultaneously from their point of intersection; prove that their distance apart, at any time, will be equal to the distance either would have descended vertically in that time. 8. What is the experimental principle assumed in determining the velocities of two given elastic balls after direct impact? If the elasticity is perfect, prove that the total kinetic energy is unaltered by impact. Prove also the converse of this proposition. 9. Two equal and perfectly elastic spherical beads, each of radius r, strung upon an inextensible string, are placed on a smooth table and are drawn apart to the greatest possible distance a between their centres. One of them is then projected directly towards the other with a given velocity и. Determine their distance apart, and their velocities, at any time t from projection. Write down the equation to the path of their centre of gravity, supposing the table removed at the instant of projection. 10. Two given weights are connected by a string passing over a smooth pulley. Determine their acceleration and prove that the resultant pressure between the string and pulley is less than it would have been if half the sum of the weights had been suspended at each end of the string. II. Define the terms Work, and Energy (potential and kinetic), stating the units commonly used in each case. A shot of 6 lbs. weight leaves the muzzle of a gun of 6 cwt. in a horizontal direction with the velocity of 1000 feet per second. Find the potential energy of the charge in the ordinary units, assuming that the gun is perfectly rigid, its bore smooth, and the carriage free to move on a horizontal plane. How would the conclusion be affected in the absence of any one of these assumptions. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 1892. OBLIGATORY EXAMINATION. I. EUCLID (Books I. IV. AND VI.). [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. Great importance will be attached to accuracy.] I. Make a triangle of which the sides shall be equal to three given straight lines; and show that if ABC be a triangle with the vertical angle at A greater than either of the base angles, it is not possible to form a second triangle with sides equal to those of ABC and base equal to twice BC. 2. Give Euclid's Axiom on Parallels, and those on the addition and subtraction of equals to and from equals and unequals. Prove that if a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. 3. ABC being an isosceles triangle, and D any point in the base BC; show that the perpendiculars to BC through the middle points of BD and DC divide AB and AC at H and K respectively, so that_BH=AK and AH = CK. 4. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. Prove this; and give its equivalent in algebraical symbols. 5. Prove that in every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. If in a quadrilateral ABCD, the square on AB is greater than the squares on the three other sides BC, CD, DA by twice the rectangle CD, DA, show that the angle ACB is obtuse. 6. Prove that the straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and that no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle. Give the corollary to this proposition. 7. Prove that the angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc. If the diagonals of a quadrilateral inscribed in a circle cut at right angles, show that the angles which a pair of opposite sides of the quadrilateral subtend at the centre of the circle are supplementary. 8. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; prove that 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; and indicate the construction when the circle is required to touch one side externally. 10. Inscribe a regular pentagon in a given circle. If a regular pentagon and a regular hexagon be on the same base and on the same side of it, prove that the pentagon is wholly within the hexagon. II. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, prove that the triangles on each side of it are similar to the whole triangle, and to one another. If AB, BC be two sides of a regular figure, L and M their respective middle points, and O the centre of the inscribed circle, show that the triangle BLM has to the triangle OLM the duplicate ratio of that which a side of the figure has to the diameter of the circle. 12. Prove that in equal circles, angles, whether at the centres or circumferences, have the same ratio which the circumferences on which they stand have to one another. |