PROBLEMS RELATING TO THE PENDULUM. 1. The length of the seconds' pendulum at London being 39.1393 inches, calculate the accelerating force of gravity. 2. A pendulum which beats seconds accurately on the earth's surface, loses 30 seconds in 24 hours when carried to the top of a mountain. Determine the mountain's height, supposing the earth's radius to be 3958 miles. 3. At what depth below the earth's surface will a seconds' pendulum beat only 59 times in a minute? 4. A pendulum loses 3 seconds per day; how much must it be shortened that it may beat seconds accurately? 5. Find the time of an oscillation of a pendulum 11 feet in length 3 miles above the earth's surface at the equator, where the length of the seconds' pendulum is 38.997 inches. 6. How may the pendulum be applied to determine the radius of the earth? 7. A seconds' pendulum is lengthened by 1 inch; find how many seconds it will lose in 12 hours. 8. If the pendulum be shortened 5 inches, what number of seconds will it gain in the same time? HYDROSTATICS. PRESSURE OF HEAVY INELASTIC FLUIDS. 1. THE whole pressure on the bottom of a pail of water, the radius of which is one foot, is 120 lbs. ; find the pressure referred to a unit of surface. 2. The pressure on the bottom of a vessel referred to a unit of surface is P, and it is found that 1 cubic foot of the fluid with which the vessel is filled weighs n lbs.; find the depth of the vessel. 3. An isosceles triangle is immersed perpendicularly in a fluid, with its vertex coinciding with the surface and its base parallel to it. How must it be divided by a line parallel to the base, so that the pressure upon the upper and lower parts respectively may be in the ratio of 1: 7? 4. A given cylinder is just immersed vertically in a given fluid; find the side of the square, upon which the pressure will be the same if it be immersed vertically with one of its sides coinciding with the surface of the fluid. 5. Determine the relation between the height and the radius of the base of a cylinder, in order that when it is just immersed vertically in a heavy fluid the pressure on the base may be equal to that on the curved surface. 6. A leaden weight is suspended by a string in a cylindrical vessel containing water; determine the additional pressure sustained by the base. 7. The sides of a hollow pyramid are isosceles triangles, the base is a rectangle having sides a and b, and the height of the pyramid is c. If the pyramid be placed with its base on a horizontal plane, and be filled with fluid, compare the pressures on the sides. 8. A cylinder is filled with fluid and laid with its axis horizontal; find the pressure on the circular base. 9. A cone is filled with fluid and laid so as to have a line in its surface horizontal; find the pressure on the circular base. 10. A given rectangle is immersed vertically in a fluid, having one side coincident with the surface. It is required to divide it by a line parallel to the surface of the fluid into two parts, the pressures on which may be in a given ratio. 11. A cylindrical vessel is filled with heavy fluid; compare the pressure on the curved surface with the weight of the fluid. 12. A hollow sphere is filled with fluid; compare the pressure on any horizontal section of the sphere with that upon any other section of the same area. 13. A hollow sphere being filled with fluid, determine those horizontal sections upon which the pressure = 3ths the weight of the fluid. Volume of sphere = 3 14. A cylinder contains some fluid; suppose the volume of the fluid, owing to a change of temperature, to be increased by one nth part, what change will take place in the pressure on the sides and base? 15. A square is immersed in a fluid, with one of its diagonals vertical; divide it by a horizontal line into two parts upon which the pressure shall be equal. 16. If an isosceles triangle be immersed in a fluid, with its base horizontal and its vertex coinciding with the surface of the fluid, how far must one side be produced in order that the pressure on the whole triangle, formed by joining its extremity with that of the other side, may be double that on the isosceles triangle? 17. A cubical vessel is filled with fluid; compare the pressures on the sides and bottom. 18. A straight line is immersed vertically in a fluid; divide it into three portions which shall be equally pressed. 19. Compare the pressures on two equal isosceles triangles just immersed in the same fluid, one with its base upwards, the other downwards. FLOATING BODIES. 1. Supposing the specific gravity of a man, of water, and of cork, to be 1.12, 1, and .24 respectively, what quantity of cork must be attached to a man weighing 150 lbs., that he may just float in the water? 2. A rod of given length, and the density of which exceeds that of water in a given proportion, hangs over a vessel of water, having one end attached to a point, at a distance above the water less than the length of the rod, about which it can move freely in a vertical plane; determine the position of equilibrium. 3. A cylinder, the specific gravity of which is .63, floats in water, specific gravity 1; determine the portion of the cylinder immersed. 4. One fourth part of a cubical solid of given dimensions, which floats in a fluid of known specific gravity, is removed by a section parallel to the surface of the fluid, when it is found to rest with the part extant equal to twice the part before immersed; determine the weight of the solid. 5. If into a cylindrical vessel containing fluid of a given kind, a body be introduced of given weight, determine the change of pressure on the sides of the cylinder, supposing the body to float. If the body does not float, is the problem determinate? 6. A sphere of 1 foot radius, composed of matter of which the specific gravity as compared with water is .35, is retained below the surface of water by a string; find the tension of the string. Given that the volume of a 7. A cylinder of known magnitude and specific gravity floats in water; if a small weight be placed upon the top of the cylinder find how much it will be depressed. 8. A string which is made of an elastic material, such that its length is increased 1 inch by every pound weight which is hung upon it, is made to support a sphere of known radius and specific gravity in water. Find how much the string will be stretched beyond its natural length. SPECIFIC GRAVITY. 1. It is found that on mixing 63 pints of sulphuric acid at 1.82 specific gravity, with 24 pints of water, one pint is lost by their mutual penetration: find the specific gravity of the compound. 2. Required the weight of an hydrometer, which sinks as deep in rectified spirits, (specific gravity .866,) as it sinks in water when loaded with 67 grains. 3. Two bodies of equal weights, when connected together, will just float; what is the relation of their specific gravities and of that of the fluid? 4. If a lighter fluid rest upon a heavier, and their specific gravities be a and b, and a body, specific gravity c, rest with one part P in the upper fluid, and the other part Q in the lower, then 5. The weight of P in water is 10 grains, of Q in air 14 grains, of P and Q connected together the weight in water |