cycloid, through which a body will pass from A to P in the shortest time possible. 20. PQ is a vertical line terminating in a hard horizontal plane at Q; a perfectly elastic ball being dropped from P meets another perfectly elastic ball rebounding with a known velocity from Q, and both are reflected back; to find where they must meet, in order that they may thus rebound from one another continually. 21. A body is projected from a given point with a given velocity; to find the direction that it may just touch a given plane. 22. From what height must a perfectly elastic ball be dropped on the convex surface of a given hemisphere, so that after reflection it may describe the greatest possible horizontal range? 23. Prove that the oscillations caused by the addition of a small weight to either scale of a balance are isochronous; and show that the isochronous pendulum is equal to the diameter of the circle which passes through the fulcrum, and the points of suspension. TRINITY COLLEGE, 1826. 1. WHAT are the objects of the science of Mechanics? 2. What are the properties attributed to matter in this science? Show the fallacy of the reasoning by which it has been attempted to prove that infinite divisibility is essential to matter. 3. What is weight? Under what limitation may it be taken as a measure of mass? 4. What kind of machine is an oar? 5. How is a wedge shown to be a species of inclined plane? 6. What is the relation of the power and weight in the second system of pullies, where the same string passes round them all? 7. What is the most effective direction for the power on an inclined plane? 8. In what case will a body of a given form be supported, or fall, on a horizontal surface? 9. Find the centre of gravity of a plane triangle. 10. Show that the times of falling down all chords of a circle to the lowest point are the same. 11. Investigate the curve described by a projectile in vacuo. 12. A hollow parabolic conoid being placed with its vertex downwards, find a point at a given height from whence a perfectly elastic ball being let fall, shall after one rebound hit the vertex. TRINITY COLLEGE, 1827. 1. IF three forces acting on a point keep one another at rest, any two are to one another inversely as the sines of the angles which their directions make with that of the third force. 2. Find the proportion between the power and the weight in the screw. 3. Prove the principle of virtual velocities in the case of the lever. 4. A body rests with its base on a horizontal plane; to find when it will be supported. 5. To find the centre of gravity of any curvilineal area. 6. If a heavy body be suspended from any point, it can only be at rest when its centre of gravity is the highest or lowest possible. Point out the different nature of the equilibrium in these two cases. 7. State the three laws of motion and the evidence on which they rest. 9. Determine the velocities after impact of two elastic bodies impinging directly on each other with given velocities. 10. Prove that the curve described by a projectile is a parabola, and that the velocity at any point is that acquired by falling from the directrix. 11. Find the direction in which a body must be projected from a given point with a given velocity, so as to hit a given mark: and shew that there are two different directions which will solve the problem. Also find the time of flight, and the greatest altitude of the projectile above the horizontal plane. 12. The velocity acquired in falling down a system of inclined planes on the supposition that no velocity is lost in passing from one to another is equal to that acquired in falling down the perpendicular height of the system. How is this proposition to be extended to the case of a curve surface? TRINITY COLLEGE, 1827. 1. IF the resultant both of the forces p and m, and also of p and n be in the direction of the diagonals of the parallelograms, the sides of which are proportional to these forces; then will this also be true for the resultant of the forces p and m + n. 2. A point in the vertex of a right-angled triangle is solicited by a number of forces represented in magnitude and direction by lines drawn to equidistant points in the base; required the magnitude and direction of the resultant. 3. Given the magnitudes of three forces acting on a point, and not in the same plane, prove that the resultant is the diagonal of a parallelopiped of which these are the edges; also find its magnitude and the angles which it makes with the components. 4. If two parallel forces act in opposite directions, determine the magnitude and point of application of the resultant. What will be the result in the case when the two forces are equal? 5. A beam of wood of given weight rests with one end on the ground, and with the other on an inclined plane; what is the force necessary to prevent the plane from moving? 6. AC, CB are the equal arms of a straight lever whose fulcrum is C: to C a heavy arm CD is fixed perpendicular to AB. Prove that when different weights are suspended from the extremity A, the tangents of the inclinations of CD to the vertical will be proportional - to the weights. 7. Find the centres of gravity (1). Of a triangle, the density of which in every part is as its distance from the base. (2). Of the area of the logarithmic curve intercepted between two ordinates. 8. Two heavy spheres, of given dimensions, are placed in a hollow hemispherical basin. Determine their position of equilibrium. 9. If two parallel forces act in the same direction on the opposite angles A and C of the parallelogram ABCD, and a third force act on the point B in the direction of the diagonal BD; find the magnitude and point of application of a fourth force which will keep the parallelogram at rest. 10. In what direction must a ball be projected along the interior of a hollow spherical superficies, so that it may pass through a given point? the ball being supposed to be without weight. 11. A number of balls of given weight are projected at the same instant in given directions with given velocities; find the height of their common centre of gravity after a given time, and the highest point to which it will rise. 12. Water is to be raised in a bucket from a well of given depth by means of a given weight hanging over a fixed pulley. What weight of water must be raised each time, so that the greatest possible quantity may be raised in twelve hours? 13. P draws Q up a groove cut out in an inclined plane; find the velocity of Q at any point, the angle which the string makes with the plane varying at every point. 14. OA and OB are the vertical and horizontal radii of a circle; it is required to find a point C in the quadrant AB to which if a tangent be drawn meeting the radius OB produced in D, and a line touching the circle at A in the point E, the time down DE + time of moving along EA with the acquired velocity may be a minimum. 15. From the highest point A of the vertical diameter AB of a circle, draw the line of quickest descent to the cissoid of which B is the origin and AB the axis. 16. A and B are two balls of given elasticity; what must be the magnitude of a third ball, that the velocity communicated from A to B by the intervention of this ball, may equal that communicated immediately from A to B? Determine also the limits within which the problem is possible. 17. Two balls are projected at the same instant from two given points in a horizontal plane, and in opposite directions, so as to de scribe the same parabola. What must be their relative magnitude, and their elasticity, so that after impact one of them may return through the same path as before, and the other descend in a right line? 18. A body moves in a cycloid the plane of which is inclined to the horizon at a given angle; find the time of an oscillation, and the point where the velocity perpendicular to the horizon is a maximum. TRINITY COLLEGE, MAY 1828. 1. If two weights acting perpendicularly upon a lever on opposite sides of the fulcrum, have their distances from the centre inversely as their weights, they will balance each other. 2. Enumerate the definitions and axioms which are requisite in the proof of the preceding proposition. 3. Four weights, 1, 3, 7, 5, are at equal distances on a straight lever. How far from each is the fulcrum on which they will balance? 4. In the system of pullies in which the same string passes round all the wheels, what is the proportion of the power and weight? 5. Explain its construction that all the wheels may revolve in the same time. [White's Pully.] 6. Explain any contrivance by which a reciprocating motion upwards and downwards may be converted into a continued circular motion. 7. What is the use of a fly-wheel? How would the defect be perceived of its being too heavy or too light? 8. Explain and prove the second law of motion. Show from it that a projectile will describe a parabola. 9. On what assumptions can the third law of motion be deduced from the second? Are these assumptions allowable? 10. Explain the principal peculiarities in the different mechanical agents or first movers which we can employ: gravity, water, air, heat, animal strength. |