24. Determine the motion of a given heavy particle, projected along a rod which is supported on a fulcrum at its middle point, and is of a given uniform density; the motion beginning from the middle point, and the first position of the rod being horizontal, TRINITY COLLEGE, 1824. 1. Ir the quantities and directions of two forces acting upon a point be represented by two adjacent sides of a parallelogram; its diagonal will represent the quantity and direction of their resultant. 2. Three weights A, B and C are suspended from given points of a straight lever: where must the fulcrum be placed, in order that the lever may be at rest? 3. What is meant by the principle of virtual velocities? Shew its application in the case of equilibrium upon an inclined plane. 4. Find the distance of the centre of gravity of any number of given bodies from a given plane, the distance of the centre of gravity of each body from the plane being given. 5. What are the laws of motion? Can they be considered as entirely founded upon observation and experiment. 6. P(3) draws up Q (5) by means of a string passing over a fixed pulley: find the force accelerating P's descent, and the space described in t"(10): the weight of the string, and the inertia of the pulley being neglected. 7. Find the time of oscillation in a cycloid. 8. If the length of the seconds pendulum be 39.1386 inches, what must be the length of a pendulum which loses 10" in 24 hours, 1 the force of gravity being diminished by th part of the whole? 1000 9. Find the range and time of flight of a projectile upon an inclined plane passing through the point of projection. 10. In the impact of bodies, whether elastic or not, the velocity of the centre of gravity is the same before and after impact. TRINITY COLLEGE, 1826. 1. How are forces compared and measured? 2. Find a single force equivalent to two given forces acting at the same point in given directions. 3. Exhibit the algebraical value of the compound force, and show in what cases it is equal to the sum or difference of the given forces. 4. In what case can forces acting at different points be compounded, and how? 5. How may two blocks be formed so as to answer the purpose of several pullies in the system where the same string passes round all? 6. What is the principle of virtual velocities? Prove that it obtains in the case of the double inclined plane. 7. Demonstrate that, when weights balancing each other in all positions on a machine are set in motion, the common centre of gravity remains in the same horizontal plane. 8. Explain the graduation of the Danish steelyard, in which the places of the weights are invariable and the fulcrum is moveable. 9. What is rackwork? Explain the construction and mechanical advantage of the jack, used by masons to lift up large stones. 10. How is the equilibrium of forces acting in different directions on a rigid body stated algebraically? 11. Distinguish between stable and unstable equilibrium. What must be the height of a parabolic conoid resting on its vertex in an equilibrium of indifference? 12. Find the centre of gravity of a wedge, of which the sides are cut into the form of a parabola, the flat surfaces being exactly similar and equal. 13. Find the equation to the catenary measuring the co-ordinates from the lowest point. Show that it nearly coincides with a parabola, about the vertex. 14. Prove the equations of motion: 15. Under what conditions may one perfectly elastic ball be made to strike another, so that each shall move, after the impact, in a given direction? 16. How is the velocity, communicated from one mass to another, increased by interposing others between them? What is the utmost extent of the advantage to be obtained in this manner ? 17. When any number of bodies not urged by any forces are set in motion, their common centre of gravity moves uniformly in a straight line. 18. Describe the motion of a ball projected up an inclined plane with such a velocity as to fly over the top of it. Show that the parabola which it describes has the same directrix as that in which it would have moved had the inclined plane not existed. 19. Find the time of oscillation of a pendulum, and show how it may be made to oscillate isochronously. 20. How is the intensity of the force of gravity estimated by observations on a pendulum? 21. Find the correction to be applied to a pendulum which vibrates seconds nearly, but not exactly. How is this correction applied by means of a screw? 22. Find the equations to the motion of a point on a curved surface, and apply them to the case of a hollow parabolic conoid, with it axis vertical. Show that when the path becomes a plane circle, the angular velocity is the same in all cases for the same surface. 23. What is D'Alembert's principle? Apply it to determine the motions of two weights connected by a string hanging over a pulley. 24. Explain the construction of Attwood's Machine, and describe the experiments made with it to illustrate the principles of Mechanics. TRINITY COLLEGE. MAY 1831. 1. IF two weights, acting perpendicularly upon a lever, on opposite sides of the fulcrum, have their distances from the fulcrum inversely as the weights, they will balance each other. 2. Shew that, cæteris paribus, the larger a carriage-wheel is, the less is the force requisite to draw the carriage over a given obstacle. 3. When three forces acting on a point are in equilibrium, each varies as the sine of the angle contained by the directions of the other two. 4. Required the proportion of P to Win the single moveable pulley, the strings not being parallel. 5. Find the proportion of P to W, when there is equilibrium on the inclined plane, P acting in a direction making any angle with the plane; and shew that this proportion is the same as that of W's velocity in the direction of its action, to P's velocity in the direction of its action, supposing a small motion to be given to the weights. 6. Find the centre of gravity of a pyramid whose base is a triangle, and thence derive that of a cone. 7. State the three laws of motion. What are the quantities to which in mathematical language the terms velocity, momentum, accelerative force, and moving force are applied? 8. Explain the nature of impact, and the manner in which time must be considered in estimating its effect. Illustrate your explanation by examples. * 9. The space described by a body uniformly accelerated from rest, is half the space described in the same time with the last acquired velocity. 10. Obtain the equation to the path of a projectile, and the velocity at any point. 11. Find the time of a small oscillation in a circular arc. 12. Give satisfactory reasons for concluding that gravity at the Earth's surface is a constant force, and that it acts on all bodies alike. TRINITY COLLEGE, MAY 1831. 1. FIND the magnitude and the direction of the resultant of two given forces acting in given directions on a point. 2. A given weight W is to be supported by a horizontal rod of given length, on two vertical props, one of which can sustain no more than P, and the other no more than Q; required the point of the rod from which W must be suspended, that the props may support less than the greatest they can support, by the same quantity. 3. Find the proportion of the power to the weight when there is equilibrium on the screw. 4. If a heavy homogeneous triangle be held in any position by three vertical strings attached to its angles, the strings sustain equal portions of it. 5. In the common balance, the weights being unequal, find the position in which it will rest. Hence determine the stability and sensibility of the balance. 6. A flexible chain of given weight is wrapped exactly round a given circle, the plane of which is vertical, and is supported on the circle: required the tensions at the highest and lowest points. 7. A ladder of given weight and dimensions rests in a given position against a vertical wall, and is prevented sliding by friction; having given the ratios of friction to pressure on the horizontal plane and on the wall, find how high a man of given weight may ascend the ladder before it begins to slide. 8. Any number of beams arranged as sides of a polygon, in a vertical plane, support each other, and support also given weights at the angles: it is required to find the horizontal pressure at the points of support. 9. If a couple of equal and opposite forces act in a rigid plane, shew that equilibrium may be produced in an unlimited number of ways by introducing another couple: and assuming arbitrarily a line in which one of the forces of the additional couple shall act, deter mine the line in which the other must act. 10. Describe the construction of a dome: shew that no dome can remain at rest by its own weight, when supported on a horizontal plane; and that the weights of the consecutive rings of voussoirs may increase in any proportion greater than that of the difference of the tangents of the angles which the joints make with the vertical. 11. State the principle of virtual velocities; and prove by means of it that if a uniform rod rests on two straight lines the equations of |