SECT. 6. Symmetrical Solid about its Axis.

(1) To find the radius of gyration of a homogeneous sphere

about a diameter.

Let x, x + dx, be the distances of the circular faces of a thin circular slice of the sphere, at right angles to the diameter, from the centre, and let y be the radius of the section. Then, p denoting the density of the sphere, the moment of inertia of this slice about the diameter will be equal to

and therefore the moment of inertia of the whole sphere, a being its radius, will be equal to

{{#p[** y*dx = \πp [ ** (a2 — a3)* dx = fs#pa3.

-a

-α

But the mass of the sphere is equal to πpa3; hence

k2 = fa2.

Euler; Theoria Motus Corporum Solidorum, p. 198.

(2) To find the radius of gyration of a right cone about its axis.

If a denote the radius of the base of the cone,

(3) To find the radius of gyration of a hollow sphere about a diameter.

If a, b, be the external and internal radii,

(4) To find the radius of gyration of a solid cylinder about

its axis.

If a denote the radius of the cylinder,

k2 = 1, a2.

Euler; Ib. p. 200.

(5) To find the moment of inertia of a sphere about a diameter, the density varying as the nth power of the distance from the centre.

=

If μ the density at a unit of distance from the centre, and a the radius, the moment of inertia is equal to

(6) To find the radii of gyration of an ellipsoid about its axes. If h, k, l, be the radii of gyration about the axes 2a, 2b, 2c, respectively,

Poisson; Traité de Mécanique, Tom. 2, p. 47.

(7) If the density at any point of a right circular cone be proportional to the shortest distance of the point from the conical surface, to find the radius of gyration about the axis. If a be the radius of the base of the cone, and k the radius of gyration,

(8) If a plane closed figure symmetrical on both sides of a line AA', or the annulus intercepted between two non-intersecting plane closed figures symmetrical on both sides of AA', revolve round any line BB, parallel to AA' and lying in the plane of the figure or annulus, but not intersecting it, prove that the moment of inertia of the generated solid, with respect to BB', is equal to m (h2 + 3k3); where m is the mass of the solid, h the radius of the cylinder generated by the revolution of AA', and k the radius of gyration of the generating area with respect to AA'.

Townsend: Quarterly Journal of Mathematics, Vol. x. p. 203.

SECT. 7. Moment of Inertia of a Solid not Symmetrical with respect to the Axis of Gyration.

(1) To find the radius of gyration of a solid cylinder about an axis perpendicular to its own through its middle point.

Ꮖ

Let x be the distance of any thin circular slice of the cylinder from the middle point of its axis; do the thickness of the slice; P the density of the cylinder, b its radius, and 2a its length. Then, the moment of inertia of the slice about any diameter being equal to

its moment of inertia about the axis of gyration in the present problem will be equal to

πpb2dx. (x2 + }b2).

Hence, Mk denoting the moment of inertia of the whole cylinder about the proposed axis, we have

and therefore, M being equal to 2πpab2, we have

k2 = }a2 + 472.

Euler; Theoria Motus Corporum Solidorum, p. 196.

(2) To determine the moment of inertia of an ellipsoid about the diagonal of the inscribed parallelepiped of maximum volume. If a, B, y, be the angles made by the diagonal with the axes of x, y, z,

x, y, 2, being the co-ordinates of an extremity of the diagonal.

Also, M being the mass of the ellipsoid, the moments of inertia about the axes of x, y, z, are

But, when the parallelepiped is a maximum, we may easily

(3) The vertex of a cone is at the centre of a sphere and the base of the cone is an area on the sphere environed by a circle to find the moment of inertia of the cone about a straight line, through its vertex, perpendicular to its axis.

Let Oz (fig. 178) be the axis of the cone; Ox, Oy, two lines through the vertex O at right angles to Oz; PN a perpendicular from P, any point in the cone, upon the plane x Oy: join ON, and draw NM at right angles to Oy: join PO, PM. Let m = an element of the cone at the point P, Mk2= the required moment of inertia about Oy.

=

Let a the radius of the sphere, PƆ=r, 23 = the vertical angle of the cone, POz = 0, NOx = $.

hence

Mk2 [*[*[*r* (sin e cos" + sin" e coso 4) d0 dp dr

00

=а (4-3 cos B-cos3 B).

If ẞ=π, or the cone become a sphere,

Mk

Μ = πα"; but M= πα":

hence k2=
= fa2.

(4) To find the radius of gyration of a right cone about an axis through its vertex at right angles to its geometrical axis. If a = the altitude of the cone, and c = the radius of the base, k2 = 2 (4a2 + c2).

(5) To find the radius of gyration of a right cone about an axis at right angles to the axis of the cone and passing through its centre of gravity.

If a be the altitude of the cone, and c the radius of its base; then

80

k2 = 3 (a2+4c").

Euler; Ib. p. 197.

(6) To find the radius of gyration of a circular right cone about a generating line.

If a be the altitude of the cone and c the radius of its base,

Griffin: Solutions of the Examples on the motion of a Rigid Body, p. 9.

(7) To find the radius of gyration of a double convex lens about its axis, and about a diameter to the circle in which its