17. If on an inclined plane, of which the length = a, and height =b, a part be taken equal to b, and described in an equal time by a body whose descent began from the top; what is the distance of either extremity of this part from the vertex? 18. If a body begin to oscillate from the highest point of an inverted cycloid, shew that the time of describing any given arc of the cycloid, is measured by the arc of the generating circle intercepted between the ordinates to the axis, drawn from the extremities of the arc. Compare also the time of its describing any arc of the cycloid measured from the summit, with the time of descent down the chord. 19. If a clock gain or lose t seconds in a given time (=T"), then if the error be small, prove that the quantity by which the 2lt length (1) of the pendulum is to be corrected, will be T 20. How far below the earth's surface, or how far above it, must a pendulum, shorter than 39.2 inches by a very small fraction (x), be taken to vibrate seconds? Earth's radius = r. 21. If a body P be connected by a pulley with a weight W on a plane inclined to the horizon at a given angle I;-required the space described by the force of gravity in t", and also the velocity it will acquire. 22. Prove that the elevation at which a projectile must be thrown from a plane of given inclination (I), in order that the range may be n times the space due to the velocity of projection,-is determined by the two values of E in the expression Compare the terms of flight, and the greatest heights, for the two values of E thus found, where the plane is horizontal, and n=1; 23. A body is projected from a given height with a given velocity. Required the locus of all the greatest heights, for every direction in which it may be thrown. 24. A hollow paraboloid being placed with its vertex downwards, and terminated above by an horizontal circular plane perpendicular to the axis, it is required to find by construction a point in any given diameter of the circle, such that a body let fall from it on the surface of the paraboloid, will, after one rebound, hit the vertex. TRINITY COLLEGE, 1819. 1. DETERMINE the quantity and direction of the compound force resulting from any number of given forces acting on a point. 2. 1st, When the forces are in the same plane. 2nd, When they are in different planes. AB is an uniform lever, of given length and weight, moving freely about A. A string is fixed at B distended by a weight passing over a fixed pulley E, placed at the horizontal distance AE = AB. Determine the position in which the lever will rest. 3. In a system of pullies, where each hangs by a separate string, and each string is fixed to the weight; supposing the power and weight in motion, 1st, Prove that velocity of P: velocity of W:: W ; P. 2nd, Compare the velocities with which the different pullies revolve round their axes. 4. If a plane pass through the centre of gravity of a system of bodies, prove that the sum of all the products formed by multiplying each of the bodies into its perpendicular distance from the plane on one side, is equal to the sum similarly taken on the other side of the plane. 5. Determine the proportion of the sides of a right-angled triangle, so that the time down the length may equal the time down the height time of describing the base with the last acquired velocity continued uniform. 6. Prove that if the same triangle be suspended by the centre of the inscribed circle, it will only rest when the shortest side is parallel to the horizon. 7. If two bodies balance each other on two inclined planes by means of a string passing over their intersection, prove that when the bodies are put in motion their centre of gravity will neither ascend nor descend. 8. Determine the sides of the strongest rectangular beam which can be cut out of a given cylindrical piece of wood. 9. If a body be moved from rest through the space S, by the action of an uniform force during T", and acquire the velocity V; TX V prove that S= 10. Having established the preceding equation, deduce the two following: Where M is the moving force, and Q the quantity of matter. L A body projected up an inclined plane where H = describes 32 feet in 2". How far will it ascend before all its velocity is lost? 12. A body, whose elasticity, descends from the height of 10 feet, and rebounds till all its velocity is lost; required the whole time of its motion, and the whole space passed over. 13. If in a common cycloid an ordinate be drawn from the point which bisects the axis, and from the extremity of that ordinate another line be drawn to the vertex; prove that the cycloidal segment thus cut off = of the square upon the axis. 14. Given the force of gravity; determine the time in which a body will vibrate in a cycloid whose axis = A. 15. A pendulum of unknown length is observed to vibrate 59 times in a minute; it is then shortened three inches, and is observed to vibrate 61 times in a minute. What is the length of a pendulum vibrating seconds in the same latitude? 16. When a body vibrates in the complete arc of a cycloid: (1). Compare the tension of the string in every point of the curve arising from gravity, with the tension arising from centrifugal force. (2). Determine the point where the accelerating force down the curve = the tension of the string. 17. If a body oscillate in the complete arc of a cycloid; prove that the time of its descent to any point, is measured by the arc of the circle upon the axis cut off by the ordinate to that point. 18. Given the point of projection and the velocity, (1). Determine the direction, so that the ball may strike a given point in the under surface of a given horizontal plane above the point of projection. (2). Under the same circumstances, determine the direction so that the range may be a maximum. 19. If grape shot be discharged from a cannon; prove that at the end of any given time they will be found in the surface of a given sphere. 20. A heavy beam AB has one end (4) fixed in the ground, and is supported by a prop CD of given length. If the angle A be given, determine the position of CD so that it may sustain the least possible pressure. TRINITY COLLEGE, 1820. 1. WHAT is the proportion between the power and the weight, in the system of pullies where each pulley hangs by a separate string? 2. A weight P upon an inclined plane is supported by a weight Q hanging freely, the string being parallel to the plane. Shew that if they be moved into any other position, the centre of gravity moves in an horizontal line. 3. If P and Q, in the last question, be not in equilibrium, what will be the path of the centre of gravity? 4. Explain and prove the second law of motion. 5. In the collision of elastic bodies, the relative velocity after impact, is to that before, as the imperfect to perfect elasticity. 6. A given weight P, by means of a string passing over a fixed pulley, draws up a chain, which was previously coiled upon an horizontal plane directly below the pulley; find the velocity acquired by P in descending through a given space. TRINITY COLLEGE, 1820. Statics. 1. WHEN forces keep each other in equilibrium round a fixed point, the sum of all their moments is = = 0; those being reckoned negative which tend to turn the system in the opposite direction. 2. Find the resultant of any number of forces in the same plane acting on a point. Apply the formula to the following example : AB, AC, AD are three lines making angles of 120° with each other; the point A is acted on by pulling forces in AB and AC which are as 3 and 4, and by a pushing force in DA which is as 5. Find the force which will keep it at rest. 3. A string fastened at A and passing over a fixed pulley B, has a known weight W hung by a knot at C; find what weight must be appended at B, that CB may be horizontal. 4. A weight Q hanging freely, supports an equal weight P upon an inclined plane, by means of a string passing over a pulley below the plane find the position of equilibrium. 5. When a body is sustained upon a curve whose co-ordinates are x and y, by any forces whose components in those directions are X and Y, shew that Xdx+ Ydy 0. Apply the formula to find the position of equilibrium when a weight Q hanging freely, supports a weight P upon a parabola whose axis is horizontal, by means of a string passing over the focus. 6. Find the centre of gravity of any number of points in the same plane. 7. The sum of the squares of the distances of three equal bodies from each other, is three times the sum of the squares of their distances from their common centre of gravity. 8. Prove the differential expression for the centre of gravity of any solid of revolution; and find the centre of gravity of a hemisphere. |