Page:The New International Encyclopædia 1st ed. v. 04.djvu/498

* CENTRE OF GRAVITY. 426 CENTRE OF OSCILLATION. by a string or l>alance it on a point, draw in the body a vertical line passing through the point of support; suspend the body by fastening the string to a ditrereiit point, or hahuR-e it with another portion of the body resting on the pivot, draw in the body a vertical line through the new point of support ; the intersection of these two lines is the centre of gravity. CENTRE OF GYRATION. It is proved in mechanics (q.v.) that the moment of inertia of any rigid body about an axis through the centre of inertia may be written .Vfc", where il is the mass of the body, h is called the 'radius of gyration' for that particular a.xis. A point in the body at a distance k from this axis, and so situated that the line joining it to the centre of inertia is pcrix'ndicular to the axis, is called the 'centre of gyration.' If the whole mass of the body were concentrated into a particle at this point, and connected by a cord without mass to the a.xis, it would have the same energy of rotation as the original body if it had the same angular motion. It is obvious that there will be dilVrrciit radii and centres of gyration for dif- ferent axes of rotaticin. CENTRE OF IMPACT. The mean point of impact, or the mean of all the hits, when a pro- jectile strikes a target a number of times. It is a point of the mean trajectory. See Ballis- tics: and GrxxKHV. CENTRE OF INERTIA (Lat. inertia, inac- tivity, literally unskillfulness). The centre of inertia of two particles whose masses are »i, and »i. is defined as follows: Draw in the plane which includes the two particles two lines OX and OY at right angles to each other; let the dis- V — X, 1 i tances of the particles from OX be i/, and j/., and from OY be sc, and x.; then the centre of inertia is such .1. point that its distances from OY' and OX are given by the equations »i, J, -(- m. J, 1 .": y = + »H + m. !h. IH^ + /Hj Tt is evident from these equations that this point coincides with the centre of gravity (q.v.) ; and that if the distance between the particles is h, the centre of inertia is upon a line joining the, TO, , two particles at a distance equal to • = n from the particle hi,. This is at once evident if the line OX is chosen to pass through both particles, and if O is made to coincide with the particle whose mass is m,. In this case i/, = y. = 0, hence i/ = ; «i- jTi = X. = /( , hence x =■ »"l -f »"3 In a perfectly similar n:anner. the position of the centre of inertia of any number of par- ticles or of a solid body made up of particles may he calculated. The physical properties of the centre of in- ertia are most interesting. They are as follows: ( I ) If a blow or a force is applied to a body in such a direction that the line of action passes through the centre of inertia, the whole body will receive a velocity in the direction of the force: there will be no rotation, and the velocity and acceleration will be exactly what they would be if the same blow or force were applied to a single particle whose mass equals that of the body. (2) If the line of action does not pass through the centre of inertia, there will be rota- tion exactly as if the centre of inertia were pivoted: but the body will also move as a whole so that the centre of inertia will descril)e the same path as it would if the line of action of the force had passed through it. These two prop- erties give a simple, self-evident method of locat- ing the centre of inertia of a body by direct experiment: Place it on a smooth table, and by trying different directions determine one such that a blow in this direction produces no rota- tion, simply translation: draw a line in the body marking this direction: locate another similar line, and the centre of inertia is where these two lines intersect. Illustrations of these two general properties of the centre of inertia are numerous. If a man falls from a building without striking the wall in his descent, his centre of inertia deseril)es a vertical line, however he twists or turns. If a hammer is thrown obliquely upward in the air. it will revolve rapidly: but one point of the hammer — viz. its centre of inertia — will describe a smooth curve, called a parabola, which a single particle would describe if it were thrown tipward in the same manner. When a bomb- shell explodes in its flight, the fragments tly oil" in different directions: but their centre of in- ertia at any instant is on a parabola, the same that it would have followed if there had been no explosi(m. See Mechanics. CENTRE OF OSCILLATION (T.at. oscilla- tio, a swinging). The perind of oscillation of a simple pendulum — i.e. of a minute particle of UKitter vibrating through a small amplitude at the end of a fine thread which is supposed to be without ^veight, is given by the formula: ^ 'J where ir = 3.1416. / is the length of the thread, and fi is the acceleration of a falling body due to gravity. Thus, the period varies as the square root of the length of the pendulum. If. however, the vibrating body is a large solid oscillating about a fi.xed axis, the period of oscillation is given by the fornuila where I is the moment of inertia around the axis of suspension. M is the mass of the body, and h is the perpendicular distance from the axis of suspension to the centre of gravity of the body. If each particle of the vibrating body were separately connected with the axis of su^ijcn-