length: find the amount of shortening produced by the pressure of 15 cwt. on a cylinder of glass a foot long and a square inch in section. 3. A bar of wrought iron is 100 feet long; the section of it is a square a side of which is a quarter of an inch; a weight of 2 tons stretches it 3 inches: find how much a weight of 5 tons will stretch a bar 80 feet long, the section being a square the side of which is half an inch. 4. Find the weight necessary to stretch the first bar of the preceding Example to of its length. 1 1000 1 1000 of its 5. A bar of iron will expand to about length as the temperature changes from the freezing point to the boiling point of water: find the weight which would produce the same amount of lengthening in the second bar of Example 3. LV. STRENGTH OF MATERIALS. 1. A bar of cast iron 20 feet long and a square inch in section is stretched by a weight of 5 tons: find how much it is lengthened. 2. If the bar in the preceding Example has a section which is a square of half an inch in side, find how much it is lengthened. 3. Find the strain that a bar of cast iron will bear if the section is a square of three quarters of an inch in side. 4. A cylindrical column of cast brass has for section a circle of one inch radius: find the weight it will bear without being crushed. 5. Find the height of a column of granite such that it would be sufficient to crush the lower parts, taking the specific gravity of granite as 26, and the endurance of granite as 5500. LVI. STRENGTH OF BEAMS. 1. A beam of oak 20 feet long, 4 inches broad, and 3 inches deep supports a weight of 252 pounds in the manner of Art. 666: find what weight would be sup ported by a beam of oak 25 feet long, 5 inches broad, and 4 inches deep. 2. A beam of oak of the same kind as in the preceding Example is supported at its ends: find what weight can be put at the middle without breaking the beam; supposing the beam 24 feet long, 3 inches broad, and 4 inches deep. 3. ́ Find the deflection of a beam of oak 20 feet long, 4 inches broad, and 5 inches deep, which carries a load of 1120 pounds at its middle point. See Art. 678. 4. A beam of oak is 12 feet long, 3 inches broad, and 3 inches deep; a weight of 315 pounds is placed at the middle find the deflection. 5. Shew that if the length and depth of a beam be changed in the same proportion while the weight and the breadth are unchanged, the deflection is unchanged. LVII. CAPILLARY PHENOMENA. 1. Find the capillary elevation in a tube of 03 of an inch in diameter for water, and for nitric acid: see Arts. 685 and 686. 2. Assuming that the capillary depression in a tube of '08 of an inch in diameter for mercury is 15 of an inch, find the capillary depression in a tube of 05 of an inch in diameter. 3. Two plates of glass are joined along a common edge so as to include a small angle; they are placed in water with this edge vertical: indicate by a diagram the form which the water will take between the two plates. 4. If instead of water the plates are placed in mercury draw the diagram. 5. The sides of a vessel containing water are composed of a material of half the specific gravity of water: shew by reasoning that we may expect the water to be exactly level near the sides. 4. 4 seconds; 6. 4 seconds. 7. 14 miles per hour. 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. 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 15. 6 17. 16 to 9. 18. 150 feet; 48 feet per hour. the weight. per second. after 2 seconds. 19. 48 feet per second. VIII. 2. 8 feet; 8 feet per second. 4. 4 ounces; 3 ounces. 7. 15 to 17. 16 10. No pressure. IX. 1. 16 pounds. 2. No force. 3. 41 pounds. 4. 84 pounds; 12 pounds; 60 pounds. tion is AC, and the magnitude is twice AC. 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. 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. 4. 8 inches. 5. At B. 7. 5 feet from A. 6. Midway between B and C. 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. 3.26 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. 20. Join the middle points of the three rods; the required point is the centre of gravity of the triangle thus formed. where the axes meet. 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 ÁB; 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 D in equilibrium, and it will be found that it will cut the side AỠ at right angles. 8. Midway between the centres of gravity of ABC and ADC. 9. At the point where the axis of the cylinder meets the hemispherical part. 10. Let be the centre of the hexagon, 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 XII. 1. 2 pounds. foot from the weight of 8 pounds. 4. 9 feet; 7 feet. 5. 32 pounds; 28 pounds 2 ounces. 8. 9 pounds. 9. 8 inches from the weight of 16 ounces. 10. 9 pounds. 11. 2240 pounds; 2400 pounds. pounds; and B bears 80 pounds. 13. 32 pounds; 64 pounds. 14. 30 pounds. 15. Let the direction 12. A bears 120 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 2 ounces. 7. 14 pounds. 2. 2 feet. 4. 300 pounds. 6. wheel must be 14 times that of the axle. XIV. 1. 14 pounds. 20 pounds. 2. 16 to 6. 6 pounds. The radius of the 3. 2 pounds; 5. 7 feet. 5. 7 pounds. XVI. 1. 20 pounds. pounds; 60 pounds. 6. 28 pounds. 9. 2. 25 pounds. 4. 32 pounds. 5. 48 pounds. 8. 144 pounds. 7. 15 pounds. About 7 of an inch. times the power. 4. 480 pounds. 10. The weight is about 754 XVIII. 1. 8 feet per second. 2. 2 feet per second. 4. 11 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; 112 feet per second. 10. 1 foot per second backwards; 4 feet 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. |