Page:The American Cyclopædia (1879) Volume XI.djvu/335

MECHANICS 323 axis is vertical. When the point of support and the centre of gravity coincide, as in a wheel, the equilibrium is said to be indifferent, as is also the case when a sphere rests upon a horizontal plane, because the centre of gravity and point of support will always be in a vertical line. A prolate spheroid or an egg, lying on its side upon a plane, is in a state of stable equilib- rium in one direction, and in that of indifferent equilibrium in another. Supported at the pole, the case becomes one of unstable equilibrium. The vertical line which passes through the cen- tre of gravity is called the line of direction of the centre of gravity. A body will rest upon a horizontal plane only when the line of direc- tion falls within the base on which it rests; and its degree of stability or power to resist change of position depends upon the horizon- tal distance of the line of direction from the edge of the base as compared with the height of the centre of gravity above the base, or upon the length of the arc which the centre of gravity will describe when the body is raised from a horizontal position of the base to that in which the line of direction falls through the edge of the base. Thus, if a horizontal plane is rotated on one edge till its centre of gravity falls in the line of direction, it will describe the quadrant of a circle, as shown in fig. 15 ; FIG. 15. FIG. 16. while the centre of gravity of a cube requires to be moved only through an arc of 45 in or- der to bring it vertically over one edge, as shown in fig. 16. III. CENTRIFUGAL FORCE. We will consider only the case of a body re- volving in a horizontal circle, and exerting force only in the plane of the circle. Such a force is exerted when a ball is placed upon a horizontal rod, as shown in fig. 18, and ro- tary motion is produced by turning a vertical axis. Let d m, fig. 17, represent the horizontal rod, and c the centre of motion, and sup- pose the body to be placed at m. The force exerted upon it by the revolution of the rod at each moment is perpendicular to the rod and in the direction of the tangent m e. To prevent its moving in that direction, therefore, some force must be exerted to restrain it. In this case the restraining force is the tension of the bar, the body being fastened to it. This FIG. 17. force is called centripetal, and it is mani- festly precisely equal to the force with which the body tends to fly from the centre, or the centrifugal force. By its action the body is forced to move in the direction m a, and to ar- rive at a in the same time it would without restraint have arrived at e. The two forces that have produced this deflection from m e to m a are the force which is communicated by the rod, which may be represented by a b=m e, and the centripetal force, which may be repre- sented by the line m J, and which is precisely equal to the force with which the body tends to fly from the centre, or the centrifugal force. It may be demonstrated that the centrifugal force of a body moving in a horizontal circle is equal to the product of its weight multiplied into the square of its velocity, divided by the product of the radius of the circle it describes, multiplied by 0=32-16, or the constant acceler- ating increment of a falling body. This may be expressed by the equation c = rg Let n represent the number of revolutions or parts of a revolution per second, and 27rr the cir- cumference of the circle described by the body; then Zir.r.n will be its velocity. Hence x w.r.n*. Therefore the centrifugal force of a body re- volving in a horizontal circle is equal to its weight multiplied by the number of feet in the radius of the circle, and this product by the square of the number of revolutions or parts of a revolution per second, and this by 1*2275. Example : A ball weighing 10 Ibs. is whirled in a horizontal circle on a radius of 5 ft., making 15 revolutions per second; what is its cen- trifugal force? 1-2275 x 10 x 5 x 15 2 =13,8731bs. (6'936 tons). In agreement with these princi- ples are the first three of the celebrated prop- ositions of Huygens appended to his Horolo- gium Oscillatorium, which were then for the first time advanced: 1. If two equal bodies revolve in equal times in unequal circles, their centrifugal forces will be proportionate to the diameter of the circles. 2. If equal bodies revolve in equal circles with uniform but un- equal velocities, their centrifugal forces will be proportional to their diameters. 3. If two