Buoyancy of Water The upward pressure of water at any point is always just enough to support the weight of the water which is above that point. If a section of water at the surface could be removed, and some object of the same weight as water put in its place, the object would remain in the same position without rising or sinking. If the object were heavier than water, then the upward pressure of the water underneath would not be sufficient to support it, and it would sink. If it were lighter than water, it would rise until its weight would just equal the weight of a quantity of water sufficient to fill the space which it still occupies beneath the surface. 1. A cubic foot of water weighs 1000 ounces. If a cubic foot of some kind of wood should weigh exactly 1000 ounces, would this wood sink or rise when placed in water ? 2. If a cubic foot of wood should weigh 500 ounces, what part of it would rise above the water in floating ? 3. What would be the specific gravity of this wood ? See Book II, p. 189. 4. If the specific gravity of a substance is , what part of it will float above water ? 5. The specific gravity of cork is about 1. What part of a piece of cork would sink beneath the surface in floating ? Buoyancy of Water 1. If a block of wood is as heavy as water, what part of the block will rise above the surface when floating ? 2. If the specific gravity of a certain kind of wood is .6, what part of the wood will rise above the surface of the water when it is floating ? 3. If a block of this wood rises 4 inches above the surface of the water, how thick is the block ? 4. A cubical block of wood measuring 12 inches each way is floating in water. Its specific gravity is g. How many cubic inches of water does the block displace? 5. How many more cubic inches would it displace, if it should sink to the level of the water ? 6. How many pounds of weight would it be necessary to place upon it to cause it to sink to the level of the water ? 7. How many cubic inches of water must be displaced. in order that the pressure of the water may support a man weighing 150 pounds ? 8. The specific gravity of ice is .92. To what depth will a block of ice a foot thick sink in fresh water? 9. Find how many inches a block of ice 52 feet thick would rise above the surface of fresh water. 10. The specific gravity of poplar wood is about .4. How many inches of a block of poplar wood 20 inches thick will float above water ? 11. The specific gravity of ash wood is about .8. If an ash block rises 2 inches above the water in floating, how thick is the block ? 12. A piece of ash timber 8 ft. long, 18 in. wide, and 18 in. thick is floating in water. How many cubic feet are below the surface of the water ? Miscellaneous Problems 1. I bought a house for $4000, and spent a sum equal to 121 % of the cost to repair it. The taxes and other expenses amount to $40 a year. What must be the annual rent in order that I may have a clear income of 5% of the total cost ? 2. The income from an investment which pays 31 % is $140. How much is invested ? 3. My property is insured at 1%. The premium is $14. For what sum is the property insured ? 4. What would be the commission at 21 % for selling 250 barrels of flour at $4.20 a barrel ? 5. In 1900 there were about 100,000 Chinese in the United States. The total population of the main body of the country was 76,131,000. What was the ratio of the number of Chinamen to the total population ? 6. Wind having a velocity of 50 miles an hour exerts a pressure of about 12 pounds to the square foot. What would be the total pressure exerted upon the side of a building 40 ft. long and 30 ft. wide ? 7. A man wishing to move a stone places the end of a bar upon the ground at a point 8 inches from the point where the bar touches the stone. The total length of the bar is 5 feet. It will require a pressure of 900 pounds to move the stone. How many pounds will it be necessary to lift upon the end of the bar ? 8. What is the exact number of days from Apr. 5th to Aug. 10th? 9. What is the exact interest of $120, reckoned according to the exact number of days, from June 5th to Oct. 8th, at 4%? A plane figure bounded by five straight lines is called a pentagon. In a regular pentagon the sides are all equal and the angles are all equal. · The sum of all the angles of the triangles into which the pentagon is divided equals the sum of all the angles of the pentagon. The sum of all the angles of any triangle is 180°. 1. How large is each angle of the regular pentagon ? 2. What kind of a triangle is ABE? EDC ? 3. What kind of a triangle is EBC? 4. Show that the triangle ABE is equal to the triangle DCE. Show that the one would exactly cover the other, if placed upon it. 5. How large is the angle at A? The angle ABC? The angle AEB? 6. How large is the angle EBC? ECD? ECB ? Take the angle ABE from the whole angle ABC. 7. In Fig. 2, if the point 0 is at the center of the regular pentagon, how do the five triangles formed compare with each other? 8. How large is the angle ABO? BAO? AOB ? 9. If the line OP is perpendicular to the line BC, how large is the angle OPB? POB ? Review Problems 1. A merchant sold a bill of goods amounting to $1200. He deducted of the bill, and then deducted t of the remainder for cash. What was the selling price? 2. A merchant deducted 1 of the list price of a bill of goods, and sold them for $ 150. What was the list price ? 3. In 1864 it took nearly $ 3 in “greenbacks” to buy what a gold dollar would buy. How much was a dollar bill really worth? 4. What was the real cost of a barrel of flour at $18 in “greenbacks” ? 5. If one of the angles of a parallelogram is 42° 40', how large is each of the other angles ? See Book II, pp. 78, 92. 6. If one of the angles of a parallelogram is 36°, and one of the angles formed by a diagonal dividing this angle is 16°, what must be the size of each of the other angles in the figure ? 7. Find the interest at 5% upon a note of $168.75, dated Jan. 1, 1899, and paid Apr. 20, 1900. 8. Find the exact interest reckoned by days at 41% on a note of $50, dated Sept. 30, 1899, and paid Feb. 4, 1900. 9. If the tax rate is 17 mills on a dollar, and my taxable property amounts to $ 3250, what will be my tax including a poll tax of $2? 10. What will be the total amount of tax raised in a town in which the taxable real estate is assessed at $ 2,345,000 and the personal property at $1,462,000, if the rate is 13 mills on a dollar, and there are 1642 poll taxes at $1.50 ? 11. If the rate of taxation is 15 mills on a dollar, and the tax on my property is $648.75, for how much property am I taxed ? |