Length Time of No.Vibr.|| Length | Time of No. Vibr. inches. vibration. per sec. inches. vibration. per sec. Centre of Gyration, Principles of Rotation. 1. The distance of R the centre of gyration, from c the centre or axis of motion, in some of the most useful cases, is exhibited below. In a circular wheel of uniform thickness c R = rad. ✔ §. In the periphery of a circle revolving } CR = rad. ✔✅ §. In a cone revolving about its vertex. . c R = √ 12 a2 + 3 μ3 its axis. . C R = r ✔ 3 In a cone. In a paraboloid In a straight lever whose arms are R and r, R = √ 3 (R+r)* 2. If the matter in any gyrating body were actually to be placed as if in the centre of gyration, it ought either to be disposed in the circumference of a circle whose radius is c R, or at two points R, R', diametrically opposite, and at distances from the centre each = C R. 3. By means of the theory of the centre of gyration, and the values of C R = g, thence deduced, the phenomena of rotation on a fixed axis become connected with those of accelerating forces for then, if a weight or other moving power p act at a radius r to give rotation to a body, weight w, and dist. of centre of gyration from axis of motion g, we shall have for the accelerating force, the expression f: = and consequently for the space descended by the actuating weight or power P, in a given time t, we shall have the usual formula introducing the above value of f. 4. In the more complex cases, the distance of the centre of gyration from the axis of motion may best be computed from an experiment. Let motion be given to the system, turning upon a horizontal axis, by a weight P acting by a cord over a pulley or wheel of radius r upon the same axis, and let s be the space through which the weight P descends in the time t, the proposed body whose weight is w turning upon the same axis with the same angular velocity: then Example. A body which weighs 100 lbs. turns upon a horizontal axis, motion being communicated to it by a weight of 10 lbs. hanging from a very light wheel of 1 foot diameter. The weight descends 2 feet in 3 seconds. Required the distance of the centre or circle of gyration from the axis of motion. Here, I take g = 32, instead of 321, and obtain an approximative result.' Whence 5. When the impulse communicated to a body is in a line passing through its centre of gravity, all the points of the body move forward with the same velocity, and in lines parallel to the direction of the impulse communicated. But when the direction of that impulse does not pass through the centre of gravity, the body acquires a rotation on an axis, and also a progressive motion, by which its centre of gravity is carried forward in the same straight line, and with the same velocity, as if the direction of the impulse had passed through the centre of gravity. The progressive and rotatory motion are independent of one another, each being the same as if the other had no existence. 6. When a body revolves on an axis, and a force is impressed, tending to make it revolve on another, it will revolve on neither, but on a line in the same plane with them, dividing the angle which they contain, so that the sines of the parts are in the inverse ratio of the angular velocities with which the body would have revolved about the said axis separately. 7. A body may begin to revolve on any line as an axis that passes through its centre of gravity, but it will not continue to revolve permanently about that axis, unless the opposite rotatory forces exactly balance one another, This admits of a simple experimental illustration. Suspend a thin circular plate of wood or metal by a cord tied to its edge, from a hook to which a rapid rotation can be given. The plate will at first turn upon an axis which is in the continuation of the cord of rotation. As the velocity augments, the plane will soon quit that axis, and revolve permanently upon a vertical axis passing through its centre of gravity, itself having assumed a horizontal position. The same will happen if a ring be suspended, and receive rotation in like manner. And if a flexible chain of small links be united at its two ends, tied to a cord and receive rotation, it will soon adjust itself so as to form a ring, and spin round in a horizontal plane. Also, if a flattened spheroid be suspended from any point, however remote from its minor axis, and have a rapid rotation given it, it will ultimately turn upon its shorter axis posited vertically. This evidently serves to confirm the motion of the earth upon its shorter axis. 8. In every body, however irregular, there are three axes of permanent rotation, at right angles to one another. These are called the principal axes of rotation: they have this remarkable property, that the momentum of inertia with regard to any of them is either a maximum or a minimum. Central Forces. Def. 1. Centripetal force is a force which tends constantly to solicit or to impel a body towards a certain fixed point or centre. 2. Centrifugal force is that by which it would recede from such a centre, were it not prevented by the centripetal force. 3. These two forces are, jointly, called central forces. 4. When a body describes a circle by means of a force directed to its centre, its actual velocity is everywhere equal to that which it would acquire in falling by the same uniform force through half the radius. 5. This velocity is the same as that which a second body would acquire by falling through half the radius, whilst the first describes a portion of the circumference equal to the whole radius. 6. In equal circles the forces are as the squares of the times inversely. 7. If the times are equal, the velocities are as the radii, and the forces are also as the radii. 8. In general, the forces are as the distances or radii of the circles directly, and the squares of the times inversely. 9. The squares of the times are as the distances directly, and the forces inversely. |