SB centre of acceleration is changed to S' in SB, so that S'B= 5 and the absolute acceleration is diminished to one-eighth of its original value; shew that the periodic time is unaltered, and that the new minor axis is 2 of the old. ep 15. The ratio of the periodic times of two particles revolving about two centres is, and the ratio of the mean distances from their centres; compare the absolute accelerations of the centres. 16. A comet moving in a parabola, is describing sectorial areas about the sun at the same rate as a planet in a circle of which the radius is half the latus rectum of the parabola; shew that the planet will move through about 76° of longitude while the comet moves from one extremity of the latus rectum to the other. 17. A particle is describing a circle, the acceleration tending to the centre and varying inversely as the square of the distance: if the velocity at any point be increased in the ratio √3: √2, find the excentricity of the new orbit. 18. A particle is describing a parabola about the focus; shew that the velocity of the point at which the tangent cuts the tangent at the vertex varies as the square of the velocity of the point from which the tangent was drawn. 19. Assuming that the moon is retained in her orbit by the earth's attraction only, that her orbit is circular, her period 27 days, the acceleration of gravity at the earth's surface 32 feet per second, and the earth's radius 4000 miles, find the distance between the centres of the moon and earth. We subjoin here a few illustrations of the formula for the time of oscillation of a pendulum. If the length of the pendulum be 7, and the bob be made to describe a cycloidal arc, or the arc of vibration be so small that the difference between it and a cycloidal arc may be neglected, the formula for the time of a semi-vibration is It will be assumed, that above the earth's surface the force of gravity varies inversely as the square of the distance from the earth's centre; so that if g' be the value of gravity at a small height h above the earth's surface, and R the earth's radius, Also it will be assumed, that within the earth the force of gravity varies directly as the distance from the centre; so that if g" be the value of gravity at a small depth d below the earth's surface, 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 G. P. 8 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 placed with its axis horizontal; find the pressure on the circular base. 9. A cone is filled with fluid and placed 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 = 3 (the weight of 4 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? |