parallelogram, regular polygon, circle, circular ring, prism, cylinder, sphere, spheroid, middle frustums of spheroid, etc. Of a triangle: On a line drawn from any angle to the middle of the opposite side, at a distance of one third of the line from the side; or at the intersection of such lines drawn from any two angles. Of a trapezium or trapezoid: Draw a diagonal, dividing it into two triangles. Draw a line joining their centres of gravity. Draw the other diagonal, making two other triangles, and a line joining their centres. The intersection of the two lines is the centre of gravity required. Of a sector of a circle: On the radius which bisects the arc, 2cr÷ 37 from the centre, c being the chord, the radius, and the arc. Of a semicircle. On the middle radius, .4244r from the centre. Of a quadrant: On the middle radius, .6002r from the centre. Of a segment of a circle: c3 12a from the centre. c chord, a = area. Of a parabolic surface: In the axis, 3/5 of its length from the vertex. Of a semi-parabola (surface): 3/5 length of the axis from the vertex, and % of the semi-base from the axis. Of a cone or pyramid: In the axis, 14 of its length from the base. Of a paraboloid: In the axis, % of its length from the vertex. Of a cylinder, or regular prism; In the middle point of the axis. Of a frustum of a cone or pyramid: Let a = length of a line drawn from the vertex of the cone when complete to the centre of gravity of the base, and a' that portion of it between the vertex and the top of the frustum; then distance of centre of gravity of the frustum from centre of gravity of its 3a's a base= 4(a+aa'+a'2)° For two bodies, fixed one at each end of a straight bar, the common centre of gravity is in the bar, at that point which divides the distance between their respective centres of gravity in the inverse ratio of the weights. In this solution the weight of the bar is neglected. But it may be taken as a third body, and allowed for as in the following directions: For more than two bodies connected in one system: Find the common centre of gravity of two of them; and find the common centre of these two jointly with a third body, and so on to the last body of the group. Another method, by the principle of moments: To find the centre of gravity of a system of bodies, or a body consisting of several parts, whose several centres are known. If the bodies are in a plane, refer their several centres to two rectangular co-ordinate axes. Multiply each weight by its distance from one of the axes, add the products, and divide the sum by the sum of the weights: the result is the distance of the centre of gravity from that axis. Do the same with regard to the other axis. If the bodies are not in a plane, refer them to three planes at right angles to each other, and determine the mean distance of the sum of the weights from each of the three planes. MOMENT OF INERTIA. == The moment of inertia of the weight of a body with respect to an axis is the algebraic sum of the products obtained by multiplying the weight of each elementary particle by the square of its distance from the axis. If the moment of inertia with respect to any axis = I, the weight of any element of the body=w, and its distance from the axis = r, we have I = (wr2). The moment of inertia varies, in the same body, according to the position of the axis. It is the least possible when the axis passes through the centre of gravity. To find the moment of inertia of a body, referred to a given axis, divide the body into small parts of regular figure. Multiply the weight of each part by the square of the distance of its centre of gravity from the axis. The sum of the products is the moment of inertia. The value of the moment of inertia thus obtained will be more nearly exact, the smaller and more numerous the parts into which the body is divided. MOMENTS OF INERTIA OF REGULAR SOLIDS.-Rod, or bar, of uniform thickness, with respect to an axis perpendicular to the length of the rod, W = weight of rod, 21 = length, d = distance of centre of gravity from axis. Thin circular plate, axis in its } 1 = W (+d2); own plane, r = radius of plate. (2) 2 Circular ring, axis perpendicular } I = W 7-12 to its own plane, r and r' are the exterior and interior radii of the ring. Cylinder, axis perpendicular to} 1 = W the axis of the cylinder, 8 r = radius of base, 21 = length of the cylinder. By making d = 0 in any of the above formulæ we find the moment of inertia for a parallel axis through the centre of gravity. The moment of inertia, Zwr2, numerically equals the weight of a body which, if concentrated at the distance unity from the axis of rotation, would require the same work to produce a given increase of angular velocity that the actual body requires. It bears the same relation to angular acceleration which weight does to linear acceleration (Rankine). The term moment of inertia is also used in regard to areas, as the cross-sections of beams under strain. In this case I = Ear2, in which a is any elementary area, and r its distance from the centre. (See under Strength of Materials, p. 247.) Some writers call Emr2 = Σwr2 +g the moment of inertia. CENTRE AND RADIUS OF GYRATION. The centre of gyration, with reference to an axis, is a point at which, if The entire weight of a body be concentrated, its moment of inertia will remain unchanged; or, in a revolving body, the point in which the whole weight of the body may be conceived to be concentrated, as if a pound of platinum were substituted for a pound of revolving feathers, the angular velocity and the accumulated work remaining the same. The distance of this point from the axis is the radius of gyration. If W = the weight of a body, I wr2 = its moment of inertia, and k = its radius of gyration, The moment of inertia = the weight X the square of the radius of gyration. To find the radius of gyration divide the body into a considerable number of equal small parts-the more numerous the more nearly exact is the result, then take the mean of all the squares of the distances of the parts from the axis of revolution, and find the square root of the mean square. Or, if the moment of inertia is known, divide it by the weight and extract the square root. For radius of gyration of an area, as a cross-section of a beam, divide the moment of inertia of the area by the area and extract the square root. The radius of gyration is the least possible when the axis passes through the centre of gravity. This minimum radius is called the principal radius of gyration. If we denote it by k and any other radius of gyration by k', we have for the five cases given under the head of moment of inertia above the following values: Principal Radii of Gyration and Squares of Radii of Gyration. (For radii of gyration of sections of columns, see page 249.) Thin rectangular tube: sides b, h, Thin circ.plate: rad.r,diam.h,ax. diam. form thickness, or cylinder of any .289 1/412 +b2 .289 Wh2 + 4'2 7071 VR2+12 .289 √72+3(R2 + 1·2) .289 1/12+6R2 (b2+c2)÷3 412 +b2 12 (h2+h'2)÷12 h2 +6 h2 h +3b 12h+b 1/4r2 = h2+16 (h2+h'2)÷16 12 7.2 12 +4 .63257 2/5r2 Paraboloid: r rad. of base, rev. Ellipsoid: semi-axes a, b, c; revolving on axis 2a.... Spherical shell: radii R, r, revolving on its diam... Same: very thin, radius r... Solid cone: rrad. of base, rev. on axis.. --- 2.3 CENTRES OF OSCILLATION AND OF PERCUSSION. Centre of Oscillation.-If a body oscillate about a fixed horizontal axis, not passing through its centre of gravity, there is a point in the line drawn from the centre of gravity perpendicular to the axis whose motion is the same as it would be if the whole mass were collected at that point and allowed to vibrate as a pendulum about the fixed axis. This point is called the centre of oscillation. The Radius of Oscillation, or distance of the centre of oscillation from the point of suspension = the square of the radius of gyration ÷ distance of the centre of gravity from the point of suspension or axis. The centres of oscillation and suspension are convertible. If a straight line, or uniform thin bar or cylinder, be suspended at one end, oscillating about it as an axis, the centre of oscillation is at % the length of the rod from the axis. If the point of suspension is at the length from the end, the centre of oscillation is also at 2 the length from the axis, that is, it is at the other end. In both cases the oscillation will be performed in the same time. If the point of suspension is at the centre of gravity, the length of the equivalent simple pendulum is infinite, and therefore the time of vibration is infinite. For a sphere suspended by a cord, r = radius, h distance of axis of motion from the centre of the sphere, h' distance of centre of oscillation 2 1.2 from centre of the sphere, l = radius of oscillation h+h'h+: 5 If the sphere vibrate about an axis tangent to its surface, h = r, and 1 = r Lengths of the radius of oscillation of a few regular plane figures or thin· plates. suspended by the vertex or uppermost point. 1st. When the vibrations are flatwise, or perpendicular to the plane of the figure: In an isosceles triangle the radius of oscillation is equal to 4 of the height of the triangle. In a circle, 5% of the diameter. In a parabola, 5/7 of the height. 2d. When the vibrations are edgewise, or in the plane of the figure: In a circle the radius of oscillation is 34 of the diameter. In a rectangle suspended by one angle, % of the diagonal. In a parabola, suspended by the vertex, 5/7 of the height, plus % of the parameter. In a parabola, suspended by the middle of the base, 4/7 of the height plus 1⁄2 the parameter. Centre of Percussion. The centre of percussion of a body oscillat ing about a fixed axis is the point at which, if a blow is struck by the body, the percussive action is the same as if the whole mass of the body were con centrated at the point. This point is identical with the centre of oscillation. THE PENDULUM. A body of any form suspended from a fixed axis about which it oscillates by the force of gravity is called a compound pendulum. The ideal body concentrated at the centre of oscillation, suspended from the centre of suspension by a string without weight, is called a simple pendulum. This equivalent simple pendulum has the same weight as the given body, and also the same moment of inertia, referred to an axis passing through the point of suspension, and it oscillates in the same time. The ordinary pendulum of a given length vibrates in equal times when the angle of the vibrations does not exceed 4 or 5 degrees, that is, 2° or 21⁄2° each side of the vertical. This property of a pendulum is called its isochronism. The time of vibration of a pendulum varies directly as the square root of the length, and inversely as the square root of the acceleration due to gravity at the given latitude and elevation above the earth's surface. If T the time of vibration, = length of the simple pendulum, g = accel. eration tion g is constant and T∞ √ĩ. If ʼn be constant, 1 = 32.16, T=π V since is constant, T g Τα If T be constant, gT221; 1 ∞ g; g = At a given loca then for any location From this equation the force of gravity at any place may be determined if the length of the simple pendulum, vibrating seconds, at that place is known. At New York this length is 39.1017 inches = 3.2585 ft., whence g = 32.16 ft. At London the length is 39.1393 inches. At the equator 39.0152 or 39.0168 inches, according to different authorities. Time of vibration of a pendulum of a given length at New York t being in seconds and 1 in inches. Length of a pendulum having a given time of vibration, l = ta × 39.1017 inches. The time of vibration of a pendulum may be varied by the addition of a weight at a point above the centre of suspension, which counteracts the lower weight, and lengthens the period of vibration. By varying the height of the upper weight the time is varied. To find the weight of the upper bob of a compound pendulum, vibrating seconds, when the weight of the lower bob, and the distances of the weights from the point of suspension are given: W = the weight of the lower bob, w the weight of the upper bob; D= the distance of the lower bob and d≈ the distance of the upper bob from the point of suspension, in inches. Thus, by means of a second bob, short pendulums may be constructed to vibrate as slowly as longer pendulums. By increasing w or d until the lower weight is entirely counterbalanced, the time of vibration may be made infinite. Conical Pendulum.-A weight suspended by a cord and revolving at a uniform speed in the circumference of a circular horizontal plane whose radius is, the distance of the plane below the point of suspension being h, is held in equilibrium by three forces-the tension in the cord, the centrifugal force, which tends to increase the radius r, and the force of gravity acting downward. If v = the velocity in feet per second, the centre of gravity of the weight, as it describes the circumference, g= 32.16, and r and h are taken in feet, the time in seconds of performing one revolution is If t = 1 second, h= .8146 foot = 9.775 inches. The principle of the conical pendulum is used in the ordinary fly-ball governor for steam-engines. (See Governors.) CENTRIFUGAL FORCE. A body revolving in a curved path of radius = R in feet exerts a force, called centrifugal force, F, upon the arm or cord which restrains it from moving in a straight line, or flying off at a tangent." If W weight of the body in pounds, N = number of revolutions per minute, v = linear velocity of the centre of gravity of the body, in feet per second, g = 32.16, then If n = number of revolutions per second, F= 1.2276 WRn3. (For centrifugal force in fly-wheels, see Fly-wheels.) VELOCITY, ACCELERATION, FALLING BODIES. Velocity is the rate of motion, or the distance passed over by a body in a given time. If s space in feet passed over in t seconds, and v = second, if the velocity is uniform, velocity in feet per If the velocity varies uniformly, the mean velocity v = 1 +2, in which 2 v1 is the velocity at the beginning and v, the velocity at the end of the time t. Acceleration is the change in velocity which takes place in a unit of time. Unit of acceleration = a 1 foot per second in one second. For uniformly varying velocity, the acceleration is a constant quantity, and |