4. 4 seconds; 9. 400 feet; 176 feet. VI. 2. 104 feet per second; one second. 3. 144 feet; 96 feet per second. 136 feet per second. 5. 196 feet. 6. 4 seconds. 7. It will start upwards with the velocity which the balloon has when the string was cut; and afterwards it will 9. At the end of a second. fall. 8. 26 feet. 10. 220 feet. VII. 9. 48 feet per second. 11. 100 feet. of the weight. 13. 160 feet; 480 feet. 14. 30 miles per hour. 15. 6 the weight. 17. 16 to 9. 18. 150 feet; 48 feet per second. after 2 seconds. 19. 48 feet per second. VIII. 2. 8 feet; 8 feet per second. 4. 4 ounces; 3 ounces. 20. 225 feet; 3. 7 ounces. 5. 30 feet per second; 4. 84 pounds; 12 pounds; 60 pounds. tion is AC, and the magnitude is twice AC. 8. The direc 11. Pro duce CB through B to E, so that BE is half of BC; then the resultant is a force of 20 pounds through E parallel to AB. 12. Produce DC through C to E, so that CE is two-thirds of DC; then we get a force of 6 pounds at E along EC, and a force of 6 pounds at E at right angles to EC; and the resultant of these two is obvious. 13. They must be equal and in the same proportion to 100 pounds as the side of a square is to the diagonal. 14. Five pounds from B to C. 19. On each post the resultant pressure is 200 pounds, acting towards the centre of the circle; see Example 5. 4. X. 1. 12 pounds at a distance of 5 inches from the force of 7 pounds. 2. The force of 12 pounds must act in the direction opposite to that of the two other forces; and its distance from the smallest must be five times its distance from the other. 3. A force of 6 pounds at the centre of the square. 8 inches. 5. At B. 6. Midway between B and C. 7. 5 feet from A. 8. 3 inches. 10. At three-eighths of the length from the end where the weight of 20 pounds is fastened. 11. 12 pounds and 6 pounds. 12. 2 pounds. 15. Midway between the third corner and the middle point of the opposite side. 16. 326 inches from the top. 17. The centre of gravity of the triangle. 18. At the point 19. Find the centre of gravity of the weights 2 and 5; also of the weights 3 and 4: then the point required is midway between the two. where the axes meet. 20. Join the middle points of the three rods; the required point is the centre of gravity of the triangle thus formed. XI. 3. The rod vertical; for stable equilibrium the lighter ball must be lowest. 4. The middle point. 6. Join E to the intersection of AC and BD; and produce the straight line to meet the perimeter. 7. Let the sphere be at the point A of the triangle ABC, and the string fastened to the middle point of the side AB; let G be the centre of gravity of the triangle; and D the point midway between A and G: the direction of the string will pass through Din equilibrium, and it will be found that it will cut the side AC at right angles. 8. Midway between the centres of gravity of ABC and ADC. 9. the point where the axis of the cylinder meets the hemispherical part. 10. Let be the centre of the hexagon, At G the centre of gravity of the triangle before it is removed; join GO and produce it to B, so that OB is one-fifth of OG: then B is the centre of gravity of the remainder. 2. One pound. 3. One 4. 9 feet; 7 feet. 8. 9 pounds. 12. 10. 9 pounds. A bears 120 32 pounds; XII. 1. 2 pounds. foot from the weight of 8 pounds. 5. 32 pounds; 28 pounds 2 ounces. 9. 8 inches from the weight of 16 ounces. 11. 2240 pounds; 2400 pounds. pounds; and B bears 80 pounds. 13. 64 pounds. 14. 30 pounds. 15. Let the direction of the forces at A and B meet at 0; then AOB is an angle of 60 degrees. Bisect this angle by a straight line which meets AB at C; then C is the required fulcrum, and the pressure on it is equal to the resultant of two equal forces inclined at an angle of 60 degrees. XIII. 1. 10 pounds 9 ounces; 9 pounds. 17. 3. 32 pounds; 28 pounds 7. 14 pounds. 2. 16 to 6 pounds. ounces. 6. 3. 2 pounds; XIV. 1. 14 pounds. 2. 2 feet. 6. wheel must be 14 times that of the axle. The radius of the 5. 7 feet. 20 pounds. 4. 30 pounds. 8. 4 pounds, 5. 7 pounds. 6. Five. 7. 62 pounds. 10. 86 pounds. 2. 25 pounds. 3. 16. 7. 15 pounds. XVI. 1. 20 pounds. pounds; 60 pounds. 6. 28 pounds. 9. About 7 of an inch. times the power. XVII. 1. 1 pound. 4. 480 pounds. 4. 32 pounds. 5. 48 pounds. 8. 144 pounds. 10. The weight is about 754 XVIII. 1. 8 feet per second. 4. 11 feet per second. 2. 2 feet per second. 5. 60, 40, 30, 24,...feet per second. 7. A comes to rest; B moves in a direction equally inclined to the original directions of the two balls. 8. 9 feet; 8 feet and 18 feet per second. 9. 5 feet per second; 113 feet per second. backwards; 4 feet per second. 10. 1 foot per second XIX. 1. 10 feet per second. 3. 40 feet per second. 2. 4 seconds. 6. 824 feet per second. 7. 21 seconds. 8. 192 feet per second. 9. 477 ounces. XX. 1. 144 feet; 168 feet per second. 2. The dis tance in feet is the product of the number of seconds into 22. 3. 984 feet. 4. 400 feet per half second. 5. At the end of half a second the height is 36 feet, and the horizontal distance is 30 feet; at the end of a second the height is 64 feet, and the horizontal distance is 60 feet. 6. At starting, 100 feet per second; at the end of a second the vertical velocity is 48 feet per second, and the whole velocity is the square root of the sum of 3600 and 2304, that is the square root of 5904. 7. 2 seconds; 100 feet. 8. 5, seconds; vertical velocity, 80 feet per second downwards, horizontal velocity 60 feet. 10. Composed of a vertical velocity, and a horizontal velocity, each of 80 feet per second. 11. 36 feet; 3 seconds; 75 feet. 14. At the distance of 55 feet; the vertical velocity will be 40 feet per second downwards, and the horizontal velocity 44 feet per second. XXI. 1. 4 pounds. 2. 32 feet per second. 3. About 30 pounds. 4. 100 pounds. 5. 54 cwt. 10. 5100 seconds. XXII. 1. 9.78 inches. 2. Nearly 4 seconds. 3. 8 feet per second. 3 8 XXIV. 1. 8 feet from the end. 2. of 39.1393 inches. 3. About 20 feet. 4. About 45 feet. 5. At two-thirds of the length from the end. 6. If above, the time will be diminished; if below, it will be increased. 7. 23 feet. 8. 3 feet. 9. 7 feet. 10. 7 feet 13. 1 pounds. 11. 11 tons. 12. 24 seconds; 576 feet. 14. 2250 times the weight of the bullet; 1 of a second. 200 11 15. 240 XXV. 3. 11000 feet. 4. 016. 5. 01. XXVI. 1. 28 pounds. 3. 35 pounds. 4. 7 grains. 5. 56 of a pound. XXVII. [In obtaining answers to XXVII. and XXVIII. use the result at the end of Art. 359.] 1. About 324 pounds. 2. About 3 pounds. 3. About 6 pounds. 4. About 9 pounds. 5. 8 pounds. XXVIII. 1. About 1944 pounds. pounds. 3. About 330 pounds. pounds. 7. About 72 pounds. 8. XXIX. 2. 300000 1728 ounces. 2. About 6912 6. About 18 About 942 pounds. 3. 5 tons; for the 4. 72000 ounces; The distance from the point of application see Art. 378. 18000 ounces; 54000 ounces. 5. top of the centre of pressure of the whole is 4 feet, and of the centre of pressure of the upper half is 2 feet; then the distance of the centre of pressure of the lower half will be 4 feet 8 inches: see Example X. 4. 6. 125 × 3141.6 1728 8. 32000 9. 15 times a side 4. 36000 pounds; 2. 314-16 cubic inches. 3. Nearly 182 ounces. XXXII. 1. of a pound. 4. Very nearly 7 ounces. 7. 181 ounces. 8. About 2.4 ounces less. 9. 24 cubic feet. 3. 71680. 4. 187500 pounds. 5. 896 of a foot. 6. 69727. 7. 1953 8000 of a foot. 8. For every cubic foot of water displaced 28 ounces more would be supported in the sea than in the river; thus 20 × 20 × 112 × 16 is the number of cubic 28 |