9. ABCD is a quadrilateral figure of which the two shorter sides AB, BC are equal, as also the two longer AD, DC; and the angle ABC is a right angle: what is the greatest length of the side AD that the figure may stand on the base AB on an horizontal plane without oversetting? 10. Given a bent lever with arms of uniform thickness, moveable in a vertical plane about the angular point: find the position in which it will rest. 11. A given beam considered as a line is supported on two given inclined planes: find the position of equilibrium. 12. Given the pressure upon one of the four legs of a rectangular table of known weight; find the pressures of the other three. Shew that without this datum the problem is indeterminate. 13. ABC is a right-angled isosceles triangle, and three equal forces act in the lines AB, BC, CA. At what point of the plane ABC, produced if necessary, must a force be applied to keep it at rest, and what must be its magnitude and direction ? 14. A beam BC hangs by a string AB from a fixed point A, with its lower extremity C upon an horizontal plane: find the position in which it will rest. Also find the horizontal force which must be applied at C to retain it in a given position. 1 15. A false balance has one of its arms exceeding the other by of the shorter. It is used, the weight being put as often in one scale as the other. What is the shopkeeper's gain or loss per cent.? m 16. 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. Dynamics. 1. A Bow is drawn by a force of 50 lbs. ; the weight of the arrow being lb., compare the force of gravity with the initial accelerating force which the string exerts upon the arrow, when it is let go; neglecting the inertia of the bow. 2. If a, b, be the velocities of two bodies A, B before their direct impact; u, v the velocities after, and ẞ the velocities gained and lost respectively, and e the fraction which measures the elasticity; Aa2 + Bb2 = Au2 + Bv2 + ·(Aa2 + BB2) -e 1+e 3. A and B are two given points in the diameter of a circle: find in what direction a perfectly elastic body must be projected from A, so that after reflection at the circle it may strike B. 4. Prove that if a body be accelerated by a constant force v=ft and s= }ƒto. 5. Find the velocity and direction with which a body must be projected from a given point that it may hit two other given points in the same vertical plane. 6. AB is the vertical diameter of a circle: a perfectly elastic body descends down the chord AC; and being reflected by the plane BC, describes its path as a projectile. Shew that this path strikes the circle at the opposite extremity of the diameter CD. 7. Find the equation to the cycloid; and shew that in the same cycloid the oscillations are isochronous. TRINITY COLLEGE, 1821. 1. IF forces proportional to the sides of a quadrilateral figure, be applied perpendicularly at their middle points, they will keep one another in equilibrium. 2. The moment of the resultant of any forces is the sum of the moments of the components. 3. At what point of a vertical pillar must a rope of given length be fixed, so that a man pulling at the other end may exert the greatest force in upsetting it? 4. If AP, BQ represent two forces acting on the equal arms AC, CB of a lever whose fulcrum is C, also if AP, BQ make given angles with the horizon, and Pp, Qq be perpendicular to ACB, shew that there will be an equilibrium when Ap+ Bq is a maximum. A given weight W is supported by a strings passing over pullies, placed at the angles of a regular polygon, whose plane is horizontal, each string being fastened to an equal weight P: find the position in which W will rest. 6. One sphere is supported by three others which touch, find the pressures on each; also the horizontal pressures necessary to prevent them from sliding. 7. A given weight, suspended by a string of given length, is drawn horizontally by a given force, find the position into which it will be drawn. 8. Investigate the expression for the centre of gravity of an area, and apply it to the quadrant of an ellipse. 9. The sum of the squares of the distances of the centre of gravity of any number of equal bodies from the centre of gravity of each, is equal to the sum of the squares of the distances of the centres of gravity of these bodies, taken two and two, divided by the number of bodies. 10. If any system of bodies, acted on by gravity alone, be in equilibrium, its centre of gravity is either the highest or lowest possible. 11. AC and BD are two uniform beams, moveable in a vertical plane about the fixed points A and B, also AC is equal to AB; required the position in which the beams will support each other. 12. Find the equation between P and W, when the weights of the pulleys are equal, and each hangs by a separate string. 13. If two hemispheres rest, with the convex surface of one placed on that of the other, shew that the equilibrium will be stable, or unstable, according as the radius of the upper one is less, or greater than three-fifths of the radius of the under. 14. 4 and B are two equal balls at rest; required their position, so that if a perfectly elastic ball C, impinge upon them in a direction perpendicular to, and bisecting the line joining their centres, the relative velocities of A and B after impact may be the greatest possible, 15. Draw geometrically the line of quickest descent from the focus of a parabola to the curve, the axis being vertical, and the vertex uppermost. 16. The space in any time is equal to the space described with the velocity of projection, plus, or minus the space described from rest by the action of the force, according as the body is projected in the direction of the force, or contrary to it. 17. Two balls A and B, of which B is perfectly elastic, are let fall at the same instant from two given points in the same vertical line find the point where B, after rebounding from the horizontal plane, will meet A. : 18. A chain of given length has part on a table, and part hanging over it; find the time in which it will fall off the table. 19. If a spiral tube wind round the surface of a paraboloid, standing on an horizontal plane, and make a given angle with the generating parabola, find the accelerating force on a body descending in the tube; and prove that if it descends from a point very near the vertex, it will make its successive revolutions in equal times. 20. Find the equation to the curve described by the centre of gravity of two bodies, projected with given velocities, and in given directions in the same vertical plane. 21. If r be the range, and t the time of flight of an inelastic ball on an horizontal plane, then if the same ball had an elasticity e, would be the whole space described by it, and time of motion. 22. A pendulum which vibrates seconds at Greenwich, taken to another place on the earth's surface, loses n seconds a day; compare the force of gravity at the two places. 23. The time of descent down any arc of a cycloid, from the highest point, is less than the time down the chord. 24. If A be the lowest point of a circular arc AB, and if AI AB be taken equal to and the chord AO be taken to the chord AI: √2 1, prove that the time down BO is equal to the time down the remaining arc OA. TRINITY COLLEGE, 1822. 1. IF any two forces, acting at a point, be represented in magnitude and direction by the sides of a parallelogram, the resultant is represented in magnitude and direction by its diagonal. 2. Prove, that, if several forces keep each other in equilibrium round a point, each is equal and opposite to the resultant of all the others. 3. If a, ß, y and a', B', y' be the inclinations of two lines to three rectangular axes drawn through their point of intersection; and a force F act along one of the lines: prove that its value estimated along the other is F. (cos.a. cos.a' + cos.ß. cos.ß' + cos.y. cos.y′). 4. The resultant of two parallel forces is parallel to them, equal to their sum, and its direction divides the line which joins their points of application in the inverse ratio of the forces. Construct for the resultant when the forces do not act towards the same parts. 5. Two weights are suspended from knots in a string the ends of which are fastened to two fixed points: given one of the weights, and the lengths of the suspending strings between the knots and the points where the other string produced would meet them.-Find the other weight. 6. Determine the distances of the centres of gravity of a right cone, and of a spherical surface from the vertex of each; and hence deduce the distance of the centre of gravity of a spherical sector from the centre of the sphere. 7. A given heavy spherical bowl is loaded at a certain point of its edge with a given weight: find the position in which it will rest on an horizontal plane. A right cone of given dimensions and weight stands on its vertex with its axis at a given inclination to the horizon: find the magnitude and direction of the least force, acting at the centre of its base, which will keep it in that position. |