4. An isosceles triangle floats in a fluid with its vertex downwards; supposing the triangle to experience a slight angular displacement about a line perpendicular to its plane, to find the period of its small angular oscillations.

Let ABC (fig. 64) be the triangle, with its axis CD vertical; let G, H, be the centres of gravity of the triangle and of the fluid displaced respectively.

The time of oscillation is equal to

where k = the radius of gyration of the triangle about a normal line through G, I = the moment of inertia of the line of floatation about a line through D at right angles to the plane of the triangle, b GH, and V the volume of the fluid displaced. Let AD: = C= BD, CD = h.

=

=

Let I' denote the moment of inertia of the triangle about an axis through Cat right angles to its plane; then

I'ch.k+ch. (h)2 = ch (k2 + h2)

....

(2a + y2) dxdy = ["(2x3y' + }y”) da,

; and therefore

2c

3h3

1 +

x3 dx

3h2

(3h2 + c2)

dx

= }ch (3h? + c).

.(1).

Let c' half the line of floatation: then we shall have

=

Again, p denoting the density of the triangle and p' of the fluid,

and therefore b (= HG) = CG - CH = 3h {

.....

(5).

.(6).

Hence, substituting in the formula for the time of oscillation, the values of k, I, V, b, given in (2), (5), (4), (6), we get

Encycl. Metrop. Mixed Sc., vol. 1. p. 194.

5. A square lamina of uniform density and thickness is floating on a fluid with two sides vertical: supposing the lamina to experience a slight angular displacement about a line perpendicular to its plane, to find the length of a simple

pendulum which shall vibrate isochronously with the consequent angular oscillations.

If a represent the length of a side of the lamina, p the density of the fluid, and p' of the lamina; then, L denoting the length of the required pendulum,

Daniel Bernoulli: De Motibus oscillatoriis corporum humido insidentium; Comment. Acad. Petrop. 1739, p. 106.

THE END.

METCALFE AND PALMER, FRINTERS, CAMBRIDGE.

BY THE SAME AUTHOR.

I.

A Collection of Problems in Illustration of the Principles of Theoretical Mechanics. By WILLIAM WALTON, B.A., Trinity College, Cambridge.

11.

A Treatise on the Application of Analysis to Solid Geometry, commenced by D. F. GREGORY, M.A., late Fellow and Assistant Tutor of Trinity College, Cambridge; concluded by WILLIAM WALTON, M.A., Trinity College, Cambridge.

III.

A Treatise on the Differential Calculus.

By WILLIAM WALTON, M.A., Trinity College.

IV.

Examples of the Processes of the Differential and Integral Calculus, collected by D. F. GREGORY, M.A., Fellow of Trinity College; Second Edition, edited by WILLIAM WALTON, M.A., Trinity College.