9. Find the ratio of the power to the resistance in the wheel and axle, without supposing them to act in the same plane. 10. Find, generally, the ratio of the power to the resistance in the toothed wheel; and also, when the teeth are small and prove the principle of virtual velocities on the latter supposition in this mecha nical power. 11. In an arch which is in equilibrium, the weights of the voussoirs are as the differences of the tangents of the angles which their joints make with the vertical. 12. A ball of given elasticity falls from a given height upon a hard plane: determine the whole time before the cessation of the motion. 13. The straight line of quickest descent from a point within a circle to its circumference passes through its highest point, and that from a point without the circle to its circumference passes through its lowest point. 14. A ball having descended to the lowest point of a circle through an arc whose chord is C drives an equal ball up an arc whose chord is c: shew that the common elasticity (e) of the two balls may be found from this proportion 1 e: C: 2c- C. 15. A body is projected in a given direction, and with a given velocity, and is acted on by the constant force of gravity in parallel lines: find the equation to its path; shew that it is a parabola, and construct it. 16. Assuming that the brachystochronic curve between two points is an arc of an inverted semicycloid, with its base horizontal and the extremity of its base at the higher point: shew how this cycloid may be constructed. 17. Find at what point in the direction of its axis a straight rod of small thickness must be suspended that its oscillation may occupy a given time, and find the lowest limit of that time. D 18. If the force to a centre is where D is the distance, A is a A given line, and the force of gravity is the unit of force; prove that a cycloidal pendulum, whose length is A, will oscillate in any time, through the same space as a body drawn by that force from the distance D would move over in that time. 19. A heavy ring R hanging on a thread fastened at A and B, oscillates through a very small are in the vertical plane of A and B : find the time of an oscillation, neglecting the magnitude of the ring, and the inertia of the string. 20. The uniform triangular plate ABC whose weight is known, is supported by three known weights a, b, c, connected with the angular points, A, B, C, by strings passing through a fixed ring at D: find the lengths AD, BD, CD; and the angles which they make with the vertical. 21. If a tennis ball in rapid motion strikes a vertical wall at a very acute angle, it will describe a curve in the air, so as to return to the wall after having rebounded from it: explain this pheno menon. 22. A given moving force will communicate the same velocity to the centre of gravity, to whatever body in a system it is applied. TRINITY COLLEGE, 1823. 1. Two Momenta, which when communicated separately, would cause a body to describe the adjacent sides of a parallelogram, will, when communicated together, cause it to describe the diagonal, with an uniform motion. 2. If the angle at which two given forces act is increased, their joint effect is diminished. 3. Two weights will balance each other on an horizontal lever, when they are inversely as their distances from the fulcrum. 4. There are two wheels, whose diameters are 5ft. and 4ft. on the same axle, the diameter of which is 20 inches. What weight on the axle would be sustained by forces equal to 48 lbs. and 50 lbs. on the larger and smaller wheels respectively? 5. Find the relation between P and W, when they sustain each other by a system of pullies, with the same string round all the pullies on the principle, that if the state of rest be disturbed, P and W will be inversely as their incipient velocities. 6. If two weights sustain each other on two inclined planes, by a string which passes over a pulley at their common vertex, so as to be parallel to the planes respectively; then shall the weights be inversely as the lengths of the planes. 7. Explain the contrivance called a Lewis, for raising heavy stones. 8. Three forces cannot sustain each other on a wedge, unless their directions pass through the same point. 9. The mechanical advantage of the screw, is independent of the radius of the screw. 10. The distance of the common centre of gravity of any number of particles from a given plane, remains the same, however the particles are moved about in planes parallel to, the given plane. 11. With what velocity must a ball impinge on another equal ball moving with a given velocity v, that its velocity may be destroyed by the impact; the common elasticity of the balls being th of perfect elasticity? n 12. Having given the diameters of two balls moving in the same plane, and the velocities and directions of their motion, find the places of their centres when they come into contact. 13. If a space be described with a velocity uniformly accelerated from rest, it will be the half of the space which would have been described in the same time, had the velocity been uniform and equal to that at the end of the time. 14. The time of an oscillation in a cycloidal arc is constant, whatever be the portion of the whole cycloid: prove this, and find the actual time of an oscillation. 15. The path of a projectile near the surface of the earth is a parabola nearly. TRINITY COLLEGE, 1823. 1. EXPRESS the resultant of two given forces in terms of the components, and of the angle at which they act. 2. Enumerate the different principles on which the preceding question has been resolved. 3. Forces, represented by straight lines drawn from the angular points of a triangle to its centre of gravity, will be in equilibrium. 4. The effects of forces, when estimated in given directions, are not altered by composition or resolution. 5. There is no tension sufficient to keep a heavy cord stretched in any line that is not vertical. 6. A ladder rests against a wall; find its pressure against the wall and on the ground respectively. 7. Define the centre of gravity, and prove that every body has a centre of gravity. 8. If two weights support each other upon any machine, and it be put in motion, the centre of gravity of the weights will, at first, neither ascend nor descend. 9. What is the least slope down which a regular hexagonal prism could roll? 10. A weight slides on a thread fastened to the extremities of equal arms of a lever of uniform density: shew that the lever will not rest except in a vertical or an horizontal position; and that, if it be put in motion, it will ultimately rest in a vertical position. 11. Deduce the differential expression for the centre of gravity of a solid of revolution; and apply it to find the centre of gravity of a cone. 12. An endless cord, passing through rings at given points A, B, on the same level, has equal weights attached to rings at C and D, and a ring at E, equally distant from A and B, supported by a force equal to half of either of those weights: find the angles of the figure which the thread will assume; and the length of the thread, that its tension may be to the supporting force, in the subduplicate ratio of 5 to 4. 13. Apply a given force perpendicularly to a straight lever, so as to keep it at rest, when it is acted on perpendicularly at given points, by two given forces not in the same plane. 14. If a piece of timber 17 feet long, be rested on a prop placed 4 feet from one end, it is found that an hundred weight at that end would be balanced by 12 pounds at the other; and that, if the places of the weights are exchanged, the prop must be 8 feet from the other end. Find the weight of the timber, and the place of the which the timber would balance without the weights. prop on 15. Find the equation of equilibrium on the inclined plane, when friction, proportional to the pressure, is taken into the account. 16. The relative velocity after the direct impact of perfectly elastic bodies is the same as before the impact. 17. Find the direction in which a perfectly elastic billiard-ball must be struck, that, moving from a given point on a five-sided billiard-table, it may, after impinging on the first, third, fifth, second, and fourth sides in order, be reflected to a given point. 18. If two bodies begin to fall at the same time from the common vertex of two inclined planes, the line joining them will move parallel to itself. 19. Required the plane of speediest descent from one given sphere to another. 20. The weight A after descending freely through a feet, begins to draw up another greater weight B by a string passing over a pulley: find the extreme height to which B could rise, and the time of rising. 21. A body is projected from the summit of a mountain of 30° elevation, so as just to strike at the bottom, and with double the velocity of projection, which is what would have been acquired in falling down 400 yards of vertical height. Find the height of the mountain, and the greatest height attained by the projectile. 22. Required the length of an inclined plane, the height of which is one half of its base, that a body projected directly up the plane with a given velocity may be as long after leaving the plane, before it again meets the horizon, as it was in ascending the plane: also find the range and the time of flight. 23. Explain the method of applying a pendulum, so as to regulate the motion of a clock, and the method of producing the requisite angular velocity in the hands. |