CHAPTER XXIII.
VARIOUS EXERCISES.
324. The Co-ordinates of a Rigid Body.
We have already explained (§ 302) how nine co-ordinates define a rigid
body sufficiently for the present theory. One set of such co-ordinates with
respect to any three rectangular axes may be obtained as follows.
Let the element dm have the co-ordinates x, y, z, then causing the
integrals to extend over the whole mass, we compute the nine quantities
fa dm = Mxa ; fa dm = My,;. fa d m. = Mz0;
fazdm = Ml2 ; fazdm = fay dm = Mlfa;
fay'2 + z'2) dm = Mp2; fax'2 + z2) dm = Mp.2; fax2 + y2)dm = Mp.2.
The nine quantities x„, y„, z0, I2, I2, ls2, p2, p2, p2 constitute an adequate
system of co-ordinates of the rigid body.
If 0lt 02,... 0e be the canonical co-ordinates of a screw about which twists
a rigid body whose co-ordinates are x0, y0, z0, I2, fa, fa, pi2, p2, p2 with respect
to the associated Cartesian axes, then the kinetic energy is Mue202, where M
is the mass, 0 the twist velocity, and where
vfa = a8#/ + a202 + b2032 + b‘202 + c2052 + c2062
+ bx0 (6>s - 04) (05 + 06) - cx0 (06 - 0e) (0S + 04)
+ cy0 (0„ - 06) (0! + 02) - ay0 (fa - 02) (0, + 0e)
+ az„ (0t — 02) (03 + 04) - bz0 (Øg - 04) (0X + 02)
+ i (P12 - a2) (0, + Ø.fa + I (p22 - b2) (03 + Øfa + I (p32 - c2) (0, + 0fa
- (2 (03 + 04) (05 + Øg) - k2 (Øg + 0g) (0! + 02) - ls2 (0, + 02) (03 + fa).
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