CENTRES OF GRAVITY.

To ascertain the Centre of Gravity of a Solid of Revolution.

The centre of gravity is upon the axis of revolution, and it is necessary to determine only its distance from some particular point, as, for example, the vertex, or the intersection of the axis by a base, or the origin of co-ordinates, A, Fig. 7. As before, Let AX, the axis of x, be the axis of revolution, etc. (See page 286.)

Let G be the centre of gravity of the solid, and AG′=g.
The general formula is

If y or x and d x be eliminated by means of the equation of the curve which generates the surface, and the integration be effected between the limits x-a and x=b, or y=r and y=r', the distance of the centre of gravity, G', from the origin of co-ordinates is determined.

EXAMPLE. To ascertain the Centre of Gravity of a Cylinder with a Circular Base. (See Fig. 8, page 287.)

The equation is y=r=r'; whence

EXAMPLE 2. To ascertain the Centre of Gravity of the Frustrum of a Cone with a Circular Base. (See Example 2, p. 287.)

Whence, To ascertain the distance of the centre of gravity from the vertex of the cone,

Multiply together the sum of the radi of the two bases, the sum of their squares, and the altitude, and divide the product by the difference of the cubes of those radii.

Corollary 1. If the distance from the greater base (or b G ́) be required,

2. For the entire cone, r=0, and

g=h, or g'=h; that is,

The distance of the centre of gravity from the vertex is equal to the altitude; or, from the base, is equal to the altitude.

EXAMPLE 3. To ascertain the Centre of Gravity of a Spherical Segment. (See Example 3, page 288.)

Equation of curve, y2=R2-x2; whence x dx=—y dy; and pf-y3 dy g= V

/2

2

́ 4 h [ { (r2 2 + r2) +}h2] ̄2 h[r22+r2+} h2]2 Hence, To ascertain the distance of the centre of gravity of the segment from the centre of the sphere,

Take the difference of the 4th powers of the radii of the bases as a dividend; and for the divisor, multiply the sum of the squares of the radii and the square of the altitude by twice the altitude; the quotient is the distance required.

The centre of gravity is between the centre of the sphere and the lesser base.

Corollary. For a segment with a single base, r=0; and

EXAMPLE 4. To ascertain the Centre of Gravity of a Frustrum of a Prolate or Oblate Spheroid.

The distance from the centre of the spheroid is

oblate spheroid, A and B representing semi-transverse axes, d and d' respectively the distances from the centre of the spheroid to the base and end of the frustrum. If both these are on the same side of the centre, the upper sign is used, but if they are on different sides, the lower sign is used.

EXAMPLE 5. To ascertain the Centre of Gravity of a Segment of a Prolate or Oblate Spheroid.

Frustrum of Hyperboloid of Revolution.

Distance from centre of hyperboloid=2

(d′+d)(d22+d2-2 a2)

d'+d'd+d2-3a2'

a=semi-transverse axis, d-distance from centre to base of seg

ment.

Segment of Hyperboloid of Revolution, Fig. 12, p. 247. Distance from centre of hyperboloid (point of intersection

of the diameters ta and d f)=}

(d+a)2
2a+d'

a and d as before,

and d'=distance from centre to base of frustrum.

Frustrum of a Paraboloid of Revolution.

Distance from vertex of paraboloid=

d'2+d'd+d2 d'+d

d

representing height of paraboloid, and d' the distance between the

frustrum and vertex.

For the mensuration of SPHERICAL TRIANGLES and PYRAMIDS, SPIRALS, EPI-CYCLES, EPI-CYCLOIDS, CARDIOIDS, HELICOIDS, PELI-COIDS, etc., etc., see Loomis's Analytical Geometry and Calculus; Davies and Peck's Dictionary of Mathematics; Docharty's Geometry; Hackley's Geometry.

For a Glossary and Explanation of Geometrical Figures, see Davies and Peck, and the Library of Useful Knowledge, vols. i. and ii.