ON THE MOTION OF PENDULUMS. 21
a solid of revolution of large, or even moderately large, dimensions be suspended by a fine wire coinciding with the axis of revolution, and made to oscillate by the torsion of the wire, the effect of the fluid may be calculated with a very close degree of approximation by regarding each element of the surface of the solid as an element of an infinite plane oscillating with the same linear velocity*.
For example, let a circular disk of radius a be suspended horizontally by a fine wire attached to the centre, and made to oscillate. Let r be the radius vector of any element of the disk, measured from its centre, 0 the angle through which the disk has turned from its mean, position. Then in equation (13), we must put V=rd0/dt, whence
The area of the armulus of the disk comprised between the radii r and r + dr is kirrdr, both faces being taken, and if G be the whole
r
moment of the force of the fluid on the disk, G = - 4?r l«r*Tsdr,
J o
whence
Let My* be the moment of inertia of the disk, and let n^ be what n would become if the fluid were removed, so that — n*My*6 is the moment of the force of torsion. Then when the fluid is present the equation of motion of the disk becomes
or, putting for shortness
which gives, neglecting /32,
0=d,€-n^sm(nt+a) .................. (15),
where n = nl (1 — /3).
* [That is, of course, on the supposition that the oscillations are not excessively slow.]