ST. JOHN'S COLLEGE. 1819. MECHANICS AND TRIGONOMETRY. 1. State the laws of motion, and the experiments by which the third is proved. 2. Two lines, SP and HP, revolving about the points S and H, represent two forces whose compound force is constant. Find the curve which is the locus of P. 3. Two chords AB, AC, of a circle, represent two forces; one of them, AB, being given, find the position of the other when the compound force is a maximum. 4. When there is an equilibrium on the single moveable pulley whose strings are not parallel; if the whole be put in motion—the velocity of the weight : velocity of the power :: the power : the weight. 5. Find the greatest inclination of a plane upon which a given elliptic cylinder, whose axis is horizontal, can be supported. 6. Two planes have a common base, and are inclined to the horizon at 30°. An inelastic body is projected up one of them with a given velocity, it then descends and oscillates between them. Find the whole space described, and time of motion. 7. Compare the relative velocities before and after impact, in the case of imperfectly elastic bodies. 8. A parabola is placed with its axis vertical. Draw the line of quickest descent from the curve to the focus. 9. Two equal and elastic balls are let fall at the same instant in the same vertical line from two altitudes 9a and 4a above a hori. zontal plane. Find the successive points of impact, and the spaces described by each before the return to their original positions. 10. The time of an oscillation in the arc, LŽP, of a cycloid, is equal to the time of describing the semicircumference Izp, with the velocity at V continued uniform. 11. Two particles of matter are attached at different points to an inflexible line without gravity, which is suspended by its extremity. Find the time of a small oscillation, supposing one of them to lose all its weight, and the other all its inertia. 12. A given weight, P, is attached to a given cylinder, Q, by means of a string wrapped round its circumference, and passing over the common vertex of two inclined planes. P is drawn up one, while Q descends down the other. Compare the lengths of the planes. 13. Having given the velocity and direction of projection, find the range on a given inclined plane passing through the point of projection, and the time of fight. 14. Compare the effect of an uniform force acting down and up alternately, in parallel directions, on a crank, with the effect of the same force acting always at right angles in a complete revolution. 15. An uniform chain is coiled on a smooth horizontal plane, a given length being drawn out, is projected along the plane with a given velocity. Find the velocity after the description of any given space. 16. Shew that an angle varies directly as the arc which subtends it, and inversely as the radius. 17. Having given the sines, cosines, and tangents of two ares; to find the sines, cosines, and tangents of their sum and difference. 18. If an angle be taken whose tangent is to the radius as the greater side of a triangle to the less, the radius will be to the tangent of the excess of this angle above half a right angle, as the tangent of the semi-sum of the angles opposite those sides to the tangent of their semi-difference. Prove this geometrically; and shew the superiority of it as a practical solution, when two sides and the included angle of a triangle are given. 19. The hypothenuse of a right-angled triangle being constant, find the corresponding variations of the sides. 20. Expand ATM, and from thence find the value of x in the equation A' = N. 21. A person at the foot of a hill running east and west observes a tower due north of him, and takes the elevation of it above the hill. He then walks in a direction N.E. till the tower bears due west of him, when he again takes its elevation. Determine from hence the inclination of the hill; and the distance between the points of observation being given, find the height of the tower. TRINITY COLLEGE. MECHANICS. 1. A, B, and C, are three bodies whose perpendicular distances from a given plane are d, d', d" ; C is on the opposite side to A and B; prove A xd + Bx d' -C x d" ={A + B + C} x d" where d" is distance of centre of gravity. 2. BA is perpendicular to the horizon; BDA is a semi-circle, BCG a quadrant; take any plane AC. If a ball is thrown up AC with velocity acquired down BA, it will describe a space equal to AC + CD in the time of falling through BA. 3. In a straight lever the sum of the products of each body, and its distance from the fulcrum, is equal on both sides. 4. In a bent lever of uniform density and thickness, whose arms are (a) and (a'), (a) being parallel to the horizon, and weighing (6) lbs., compare P and W, when the inclination of the arms is (e), and P (acting at the end of arm a) is inclined at 2(0), and W at 2(0) 5. Prove the general proposition of the wedge; apply the result to the case of an equilateral wedge, where the power on the back acts perpendicularly, and the resistances on the sides are equal, and act perpendicularly to the back. 6. A straight lever is parallel to the horizon; given its length, given a weight Phung at one end ; required the variation of the position of the fulcrum, supposing W to vary in arithmetic progression. 7. A body, G, is kept at rest by three forces proportional to AG, BG, CG; G is centre of gravity of the triangle formed by joining A, B, C. 8. If with centre of gravity of any number of bodies as centre, and with any radius, a circle be described, the sum of the products of each body, and the square of its distance from any assumed point in circumference, is constant. 9. Prove that in perfect elasticity Aa? + Bb% = Ap+ Bqe, where a and b are the velocities of A and B before impact, and p and q after. Compare also elasticity and compression when Aa" + Bb" = Ap" + Bg": 10. In a single moveable pulley, where the strings passing under the inoveable pulley are not parallel, compare P and W; first, when the strings are equally, secondly, when they are unequally inclined to the horizon. 11. An imperfectly elastic ball falls perpendicularly from a height (a) :-Required whole space described by ball after 5 rebounds, and the greatest height after last rebound. 12. Assuming the time of oscillation to equal the time of describing semi-circle, &c., investigate the actual value of the time of oscillation, and thence, compare it with time down axis. 13. In inclined planes, P : W:: W's velocity : P's. 14. Determine the expressions for range and greatest height, upon a plane passing through point of projection; and compare greatest height of all parabolas with a given velocity to farthest range. 15. AH is a vertical diameter ; HBA, CEA, two contiguous circles, touching in A. Prove the time down BC, DE, &c., to be constant. 16. If a body is kept at rest by three forces, and lines be drawn at any equal angles to the directions in which they act, forming a triangle, the sides of the triangle represent the quantities of the forces. 17. If 3 forces are represented by 3 sides forming the solid angle of a parallelopiped, the resulting force is the diagonal of the parallelopiped. If the 3 forces are equal, and act in planes perpendicular to each other, compare the compound force with them. 18. A is vertex of triangular pyramid, G is centre of gravity. If upon body at G forces act in directions AG, BG, CG, DG, and proportional to them, it remains at rest. 19. A uniform beam AB, is moveable about fixed point A, and supported by given weight Pover fixed pulley C; AC is equal to AB, and parallel to horizon. Required position in which AB rests? 20. Make a body oscillate in a given cycloid. 21. VP= 4 radius; MN perpendicular the diameter. Cycloid area MVN=hexagon inscribed in the circle. 22. Compare times of describing vertical diameter and any other :-Required also that diameter, the time through which = 2 time down vertical diameter. 23. If the number of oscillations performed in same time by two pendulums (whose lengths are L and I) be as m: m + n, compare force of gravity at the two heights. 24. If one pendulum is at distance of (n) radii from the earth's centre, at what point below the surface must another of equal length be placed to vibrate in same time? ST. JOHN'S COLLEGE. 1820. MECHANICS AND TRIGONOMETRY. 1. The moving forces acting on two bodies are reciprocally proportional to the quantities of matter. Compare the velocities generated in any time. 2. If SA, the least distance in an ellipse, be one third of SM the greatest, and SA be taken to represent one force, then SB, the mean distance, will compound with SA, the least possible force; and the compound force will be a mean proportional between SM and SA. 3. When there is an equilibrium on the inclined plane, if motion be communicated, the velocity of the power is to the velocity of the weight as the weight to the power, 4. Find the inclination of a plane, on which a regular figure of n sides will just be supported. 5. A and B are two elastic balls placed on a billiard table, FC is a reflecting cushion. Produce AB to C. Then if A impinge on B and drive it against the cushion, they will meet, after the reflec. tion of B, if the angle of impact equal 45 degrees minus the of impact. 6.If the number of mean proportionals between two balls A and X be increased without limit; A's velocity : that communicated to X:: ✓X :NA. 7. A weight is fixed to the lowest point in a circle moveable in a vertical plane about its centre; another equal weight is attached to a string wrapped round the circumference. Find the velocity acquired by the descending weight in any space. 8. If the axis of a parabola be horizontal, and the weight W be supported on the curve, by means of a string passing over a pulley in the focus to which a given weight P is attached; then W will vary as the corresponding abscissa. 9. If a body be projected successively in all possible directions, from the same point, with the same velocity, prove geometrically, that the locus of the ultimate intersections of the successive parabolas is a parabola. 10. BLV is a cycloid; DOV the generating circle on the axis. Draw ROL parallel to the base which is horizontal. Then the time down BL : time down DR :: arc DO : chord DO. 11. The velocity of a body in a cycloid varies as the right sine of a circular arc, whose radius equals the length of the arc at the beginning of motion, and versed sine the arc fallen through. 12. Having given the velocity and direction of projection; to find where the body will strike the horizontal plane passing through the point of projection ; and the time of flight. 13. Find the least velocity with which a body projected from the top of a tower of given height, after reflection from the horizontal plane, shall strike the top of another tower, whose distance and height are given. Find also the direction of projection and the time of fight. 14. Two equal weights are attached to a string passing over a cycloid whose base is horizontal. Find the whole pressure; and prove that the pressure estimated in a vertical direction is uniform. 15. A sphere and hollow cylinder of equal weight are suspended by a string passing over a solid cylindrical pulley equal in weight to the former. To determine the circumstances of the motion and tension of the string. 16. Given the three sides of a triangle, explain the different methods of obtaining one of its angles, and shew which is to be preferable when that angle is nearly a right angle 1-cos. A 17. If A be any angle = tau." j A. 1+cos. A |