Page:The American Cyclopædia (1879) Volume XI.djvu/334

 MECHANICS Fio. 9. d e. zero ; which will also appear by observing that the forces a e and c e balance each other, as also do the forces 6 e and d e. The resultant of any number of forces may also be found by connecting the lines representing the forces, as shown in fig. 10. Suppose the forces to be rep- resented in quantity and direction by the lines aJ, 50, cd, and Connect the points a and, and the line a e will represent the resultant of all the forces in quantity and direction; for ac is the resultant of a b and 5 c, a d that of ac and cd, and ae that of a d and d e. The force which impels a sail ves- sel, moving with the wind off the quarter, is the re- /S sultant produced PIG. 10. by the oblique ac- tion of the wind against the sails. Let a 5, fig. 11, represent the position of the sail, and d c the direction and force of the wind. This force may be resolved into the components d/and /c, the former parallel with and the latter perpendicu- lar to the surface of the sail, and therefore the only force which is effective. But it is not acting in the direction of the keel, m To ; therefore it must be resolved into the components fg and g c, the latter of which will represent in quantity and direction the effective propelling force given by the wind, whose force is measured by d c. Centre of Gravity. The point through which the resultant of all the forces caused by attrac- tion of gravitation of the molecules of a body passes is called the centre of gravity. This point may be within the body, or, in conse- quence of its form, may be beyond it. The finding of the centre of gravity is a geometri- cal problem, but with an irregular-shaped body it can most easily be determined experimental- ly by suspending it in two positions, and find- ing the point of intersection of the two verti- cal lines which pass through the two points of suspension. This point of intersection will necessarily be the centre of gravity, for it is evident that it must reside in each of the two verticals, as each vertical is the resultant of all the gravitating forces of the body while sus- pended in any one position. In the case oJ bodies of uniform density and of geometrical FIG. 11. FIG. 12. form, the centre of gravity is readily deter- mined by geometrical principles. In a circle or sphere it coincides with the geometrical cen- tre. In a plane triangle it s at the point of intersec- tion of two lines joining the vertices of two angles with the middle of the opposite sides, as shown n fig. 12. In a cone or pyramid it is in the line joining the vertex with the centre of gravity of
 * he base, and at one fourth
 * he distance from the base. A body is said to

be in equilibrium when the centre of gravity and the point of support are in the same verti- cal line. When the point of support is above the centre of gravity, the equilibrium is said to be stable. Founded upon this is the some- times so-called paradox of maintaining a beam in a horizontal position with only one end rest- ing upon a support, as shown in fig. 13. The condition is easily understood if the beam b and the leaden ball o, with the attached bent rod, are consid- ered as forming one body whose centre of gravity is at c. When this is verti- cally below the point of support the sys- tem will be in sta- ble equilibrium. When a body has its cen- tre of gravity above the point of ^ support, but in the same vertical, it is said to be in unstable equilibrium. A distinction must be made between a state of stable equilib- rium and a merely stable condition ; for equi- librium implies a balance of force. A block, for example, may rest in a stable condition when lying upon the floor, although supported below its centre of gravity. But it cannot be said to be supported by a point; if it were, this point would need to be in a vertical with the centre of gravity. There are some cases of stable equilibrium when the centre of grav- ity is above the point of support. Thus when the body is an oblate sphe- roid, stable equilib- rium will exist when it rests upon one of its poles a or &, fig. 14, because the cen- tre of gravity occu- pies the lowest pos- sible position. Disturbing the spheroid so as to bring the axis out of the perpendicular will raise the centre of gravity, and although it carries it to one side, as from c to e', the point of support is removed still further in the same direction, as from J to d ; and there- fore gravity will bring the body back till the FIG. 13. FIG. 14.