Since I~=Ii., I~=o, we deduce 100=3/4Ma2, ~ =4MaZ; hence the value of the squared radius of gyration isfora diameter 3/4ai, and for the axis of symmetry 3/4af.
Which we shall meet with presently as the ellipsoid of gyration at G.
In the case of an axial moment, the square root of the resulting mean square is called the radius of gyration of the system about the axis in question.
From its axis (0), if the radius of gyration about a longitudinal axis through G, aiid 0 the inclin - ation of OG to the vertical, FIG.
The radius of gyration of the section is 2a 2.
This is called the ellipsoid of gyration at 0; it was introduced into the theory by J.
If M be the total mass, k the radius of gyration (~ ii) about the axis, we have sin 0, (3)
R is called the radius of gyration of the body with regard to an axi:
After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area Acpm, and this is equated to the kinetic energy e of the shot, in foot-tons, (I) e d2 I + p, a in which the factor 4(k 2 /d 2)tan 2 S represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and S represents the angle of the rifling; this factor may be ignored in the subsequent calculations as small, less than I %.
The square of the radius of gyration with respect to a diameter is ia2.
The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes.
The squares of the radii of gyration about the principal axes at P may be denoted by k,i+k32, k,f + ki2, k12 + k,2 hence by (32) and (35), they are rfOi, r2Oi, r20s, respectively.
It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.
If k be the radius of gyration about p we find k2 =2Xarea AHEDCBAXONap, where a$ is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole 0 from this line.
If K be the radius of gyration about a parallel axis through G, we have kf=K2+h2 by If (16), and therefore i=h+K1/h, whence GO.GP=K2.