Ex. 7. When a body is suspended by one point, the straight line joining the point of suspension and the centre of gravity is vertical. 1. If a triangle, whose sides are 3, 4 and 5, be suspended from the centre of the inscribed circle; find in what position it will rest. When a body, acted on by gravity only, is in equilibrium with one point of it resting on a surface, the straight line joining this point of contact with the centre of gravity is vertical. Therefore, if the body be a solid of revolution, the centre of gravity is always at the intersection of the axis, and the normal at the point of contact. 2. A paraboloid of revolution, whose axis equals a, and radius of base b, rests with its convex surface on a horizontal plane. Find the angle of inclination of its axis to the horizon. 3. A paraboloid laid upon a horizontal plane rests with its axis inclined to the horizon at an angle of 30°; compare the length of the axis with the latus rectum. 4. A prolate spheroid rests with its smaller end upon a horizontal plane; determine the nature of the equilibrium. 5. A solid is composed of a cone and a hemisphere, of equal bases, placed base to base. Find the ratio between the dimensions of the cone and hemisphere, in order that the whole may be at rest with any point of the spherical surface on a horizontal plane. Find the centre of gravity of the compound figure, and when this point coincides with the centre of base of hemisphere, the figure will balance as required. 6. A solid is composed of a hemisphere and a paraboloid, of equal bases, placed base to base. Find the ratio between their dimensions, in order that the whole may be at rest with any point of the spherical surface on a horizontal plane. Volume of paraboloid (volume of circumscribing cylinder). 7. A cone rests with its base upon the vertex of a given paraboloid; find the greatest ratio which the height of the cone can bear to the length of the latus rectum of the paraboloid, while the equilibrium remains stable. 8. If two hemispheres rest with the convex surface of one placed on that of the other; determine the nature of the equilibrium. If a body rest with its base on a plane, it will fall over or not according as the vertical through the centre of gravity falls without or within the base. 9. ABCD is a quadrilateral figure of which the two shorter sides AB, BC are equal, as also the two longer sides, AD, DC; and the angle ABC is a right angle. If the length of AB be given, what is Ex. 7. the greatest length of AD, that the figure may rest with the base AB on a horizontal plane, without oversetting? 10. How high can a wall 4 feet thick and inclined at an angle of 75° be built without falling? 11. The slant height of a wall is equal to 3 times its horizontal thickness: determine the inclination when it will be just supported. 12. A cube is placed on an inclined plane whose angle of elevation is 50°. Will it roll or slide? 13. What is the least angle of inclination that a plane must have to the horizon, for a prism on a regular base of n sides to roll down it? 14. A cone is placed with its base on a plane inclined at an angle of 30° to the horizon, and is prevented from sliding. Find the smallest vertical angle it can have that it may not fall over. 15. A paraboloid, of given parameter m, when prevented from sliding down a plane whose inclination is a, just stands on its base without falling over. Find the length of the axis of the paraboloid. Ex. 8. CENTRE OF PARALLEL FORCES. 1. Three parallel forces, acting at the angular points A, B, C of a plane triangle, are respectively proportional to the opposite sides a, b, c; find the distance of the centre of parallel forces from A. 2. If three parallel forces, acting at the angular points A, B, C of a given triangle, are to each other as the reciprocals of the opposite sides a, b, c; find the distance of their centre from C. GULDINUS' PROPERTIES. (1) The surface generated by a curve line, revolving about a fixed axis in its own plane =the product of the length of the curve, into the length of the path described by its centre of gravity. (2) The volume generated by a plane area revolving about a fixed axis in its own plane the product of the area, into the length of the path described by its centre of gravity. Ex. 9. = 1. A parallelogram and a triangle, on the same base and between the same parallels, revolve around the base as an axis. Compare the solids they generate. 2. Find the volume generated by the revolution of a right-angled triangle, whose sides are a and b, about the hypothenuse. 3. Determine the volume of the frustum of a right cone, generated by a trapezoid revolving about its altitude h, which joins extremities of the parallel sides a, b. Ex. 9. 4. Determine the distances of the centres of gravity of a semicircular area, and a semi-circular arc, from the diameter. 5. Find the surface of a sphere. 6. Find the centre of gravity, of the eighth part of a sphere, or of the solid generated by a quadrant of a circle revolving about one of its radii through an angle of 90°. 7. Find the distance of the centre of gravity of the area of a semi-parabola from its axis. Volume of a paraboloid=× circumscribing cylinder. 8. Find the surface and volume of the solid generated by the complete revolution of a semi-cycloid about its axis. 9. Find the surface and volume of the solid generated by the revolution of a cycloid about its base. 10. Find the volume of the solid ring generated by the revolution of an ellipse about an external axis in its own plane through an angle of 180°. 11. If any area revolve about an axis, in its own plane, and dividing it into any two parts; show that the difference between the solids generated by the parts, will be equal to the whole area multiplied by the path of its centre of gravity. MACHINES. If a, c, be the lengths of perpendiculars drawn from the fulcrum upon the directions of the forces P, W, respectively; then Pa=Wc. Ex. 10. 1. A uniform straight lever, 3 feet in length, weighs 4 lb.; what weight on the shorter arm will balance 10 lb. on the longer, the fulcrum being one foot from the end? 2. In rowing, if the oar be 12 feet long, and the rowlock 2 feet from the handle, compare the pull of the rower with the resistance of the boat. 3. One extremity of a straight lever 20 feet long (without weight) rests on a fulcrum; at what distance from the fulcrum must a weight of 112 lb. be placed, so that it may be supported by a force equivalent to 50 lb. acting at the other extremity? 4. Two weights P, Q are suspended from the extremities of the arms of a straight lever without weight, which are as 3: 5; P acts at an angle of 60°, and Q at an angle of 45°; find the ratio of P: Q. 5. A bar weighs a ounces per inch. Find its length when a given weight na, suspended at one end, keeps it in equilibrium about a fulcrum at a distance of b inches from the other end. Ex. 10. 6. Four weights, 1, 3, 7 and 5, are at equal distances on a straight lever without weight. Where must be the fulcrum on which they balance ? 7. On a uniform straight lever, weighing 6 lb., and of 6 feet in length, weights of 1, 2, 3, 4, 5 lb. are hung at respective distances of 1, 2, 3, 4, 5 feet from the extremity. Required the position of the fulcrum, about which the whole will rest. 8. If n+1 bodies, P, 2P, 3P, 4P, &c. be placed at equal distances along a straight rod without weight, and of length na; find the point on which the whole will balance. 9. If a heavy rod of uniform thickness be moveable about a fulcrum 3 feet from one end A, and 7 feet from the other end B; and a weight of 20 lb. at B be balanced by a weight of 60 lb. at A required the weight of the rod. 10. A beam, 30 feet long, balances itself on a point at one-third of its length from the thicker end; but when a weight of 10 lb. is suspended from the smaller end, the prop must be moved 2 feet towards it, in order to maintain the equilibrium. Find the weight of the beam. 11. A uniform beam, 18 feet long, rests in equilibrium upon a fulcrum 2 feet from one end; having a weight of 5 lb. at the further, and one of 110 lb. at the nearer end to the fulcrum. Find the weight of the beam. 12. A weight 3W is attached to two strings, whose lengths are AD=4 ft. BD=3 ft., at point D; the other ends of the strings being fastened to the extremities of a uniform straight lever of weight W, whose arms AC, BC are 2 and 3 ft. respectively. Find the force which, acting vertically at A, will keep the lever at rest in a horizontal position. 13. 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; if now different weights be suspended successively from the extremity A, show that the tangents of the angles, which CD makes with the vertical, will be proportional to the weights respectively. 14. Two weights are suspended from the arms of a bent lever without weight, which are inclined to the horizon at angles of 45° and 30° respectively; the first arm being 18 inches and the second 12 inches long. Find the proportion of the weights. 15. The arms of a bent lever are 3 feet and 5 feet, and inclined to each other at an angle of 150°; and at their extremities weights of 7 lb. and 6 lb. respectively are suspended. Find the inclination of each arm to the horizon, when there is equilibrium. 16. The lengths of the arms, their inclination to each other, and the weight at the extremity of the shorter arm being the same as in the last question; find what the other weight must be, in Ex. 10. order that, Ist the shorter, and 2nd the longer arm may rest in a horizontal position. 17. A uniform bent lever, when supported at the angle, rests with the shorter arm horizontal; but if this arm were twice as long, it would rest with the other arm horizontal. Find the ratio between the lengths of the arms; also the angle at which they are inclined to each other. 18. The arms of a bent lever are equal, and the weights suspended at their extremities are as I : 2; find the angle between the arms, that the arm with the less weight may be horizontal. 19. If two forces P and W sustain each other on the arms of a bent lever PCW, and act in directions PA, WA, which form the sides of an isosceles triangle PAW; show that, if AC be joined and produced to meet PW in E, P: WWE: PE. 20. AP, BW are the directions of two parallel forces P, W, which are in equilibrium on the equal arms of the bent lever ACB: draw CD perpendicular to AB, and CE parallel to AP; then P+W: P-W= tan (ACB) : tan DCE. 21. If a lever, kept at rest by weights P, W suspended from its arms a, b, so that they make angles a, ẞ with the horizon, be turned about its fulcrum through an angle 20; prove that the vertical spaces described by P and W, are as a cos (a +0) : b cos (ß—6). 22. A body weighs 10 lb. 9 oz. at one end of a false balance, and 12 lb. 4 oz. at the other end; find the real weight. 23. The whole length of the beam of a false balance is 3 ft. 9 in.; a certain body, placed in one scale, appears to weigh 9 lb.; and placed in the other appears to weigh 4 lb. Find the true weight of the body, and the lengths of the arms of the balance, supposed to be without weight. 24. One pound is weighed at the ends of a false balance, and the sum of the apparent weights is 21 lb.: what is the ratio of the lengths of the arms? 25. If in a false balance a body weighs p at one end, and q at the other; find the centre of suspension. 26. The same weight is weighed at the two ends of a false balance, I and it is observed that the whole gain is th part of the true : n weight find the distance of the fulcrum from the middle point of the balance. 27. If one of the arms of a false balance be longer than the other m by th part of the shorter: when used, the weight is put into one |