CHAPTER XII.
PLANE REPRESENTATION OF DYNAMICAL PROBLEMS CONCERNING A BODY
WITH TWO DEGREES OF FREEDOM*.
133. The Kinetic Energy.
If a rigid body of mass twist about a screw 0, with the twist velocity
0, then the kinetic energy of the body may be written in the form
Muefft,
where ua is a linear magnitude appropriate to the screw 0 (§ 89).
Ihe function is the arithmetic mean between the square of the radius
°1 gyration and the square ot the pitch, for the kinetic energy of the body
when twisting about 0 is the sum of two parts : one, the kinetic energy of
the rotation; the other, of the translation. The energy of the rotation
is simply
this being in accordance with the definition of the radius of gyration pe.
Ihe kinetic energy due to the translation is, of course,
whence the total kinetic energy is
^AJ0- (pg2 + pf),
and therefore
M»3 = 1 (pe + pe)-
134. Body with two Degrees of Freedom.
Ihe movements are under these circumstances restricted to twists about
the screws of a cylindroid, and we shall now examine the law of distribution
* Royal Irish Academy, Cunningham Memoirs, No. 4, p. 19 (1886); see also Proceedings of the
Royal Irish Academy, 2nd Series, Vol. iv. p. 29 (1883).