BCD : zCDA : ADAB : LABC, G is at the centre of the inscribed sphere.
If we have a continuous distribution of matter, instead of a system of discrete particles, the summations in (6) are to be replaced by integrations. Examples will be found in textbooks of the calculus and of analytical statics. As particular cases: the masscentre of a uniform thin triangular plate coincides with that of three equal particles at the corners; and that of a uniform solid tetrahedron coincides with that of four equal particles at the vertices. Again, the masscentre of a uniform solid right circular cone divides the axis in the ratio 3 : I ; that of a uniform solid hemisphere divides the axial radius in the ratio 3 : 5.
It is easily seen from (6) that if the configuration of a system of particles be altered by ' homogeneous strain " (see ELASTICITY) the new position of the masscentre will be at that point of the strained figure which corresponds to the original masscentre.
The formula (2) shows that a system of concurrent forces represented by m, . OP1, M2. OP2, .. . m°.OP„ will have a
resultant represented by E(m).OG. If we imagine 0 to recede to infinity in any direction we learn that a system of parallel forces proportional to m2, . . . in,., acting at P,, P2 . . . . P° have a resultant proportional to E(m) which acts always through a point G fixed relatively to the given masssystem. This contains the theory of the " centre of gravity " (§§ 4, 9). We
may note also that if P2, . . . P°, and P,', P2', . . . P°'
represent two configurations of the series of particles, then
Z(m . PP')=E(m).GG', (8)
where G,G' are the two positions of the masscentre. The
forces m,.P,PI', m2•P2P2', . .. m°.P°P°', considered as localized vectors, do not, however, as a rule reduce to a single resultant.
We proceed to the theory of the plane, axial and polar quadratic moments of the system. Thy axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h,, /12, . . . h° be the perpendicular distances of the particles from any fixed plane, the sum Z(mh2) is the quadratic moment with respect to the plane. If p,, p2, ... Pa be the perpendicular distances from any given axis, the sum 2;(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis. If r,, r,, . . . ra be the distances from a fixed point, the sum E(mr2) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments
We note that
Iv== h+L, Izn = Iz+h, I.0 = Is+Iv, and
Io=Is+Iy+I.= 2(Is.+Izs+Isb)• (13)
In the case of continuous distributions of matter the summations in (9), (to), (II) are of course to be replaced by integrations. For a uniform thin circular plate, we find, taking the origin at its centre, and the axis of z normal to its plane, l,, = IMa2, where M is the mass and a the radius. Since Is = Ig, Ix =o, we deduce Iss =1M a2, Isv=IMa2; hence the value of the squared radius of gyration is for a diameter and for the axis of symmetry Za2. Again, for a uniform solid sphere having its centre at the origin we find Io=lMa2, Is=I„=Iz=IMa2, Iy,=Iss=Isv=IMa2; i. e. the square of the radius of gyration with respect to a diameter is ia2. The method of homogeneous strain can be applied to deduce the corresponding results for an ellipsoid of semiaxes a, b, c. If the coordinate axes coincide with the principal axes, we find Is =IMO, I9=IMb2, Is =Mc2, whence Irz=(b'+c2), &c.
If 4(x, y, z) be any homogeneous quadratic function of x, y, z,
we have
E{md,(x, y, z)}= E}mb(z+E, y+n, z+3')}
=z1m4,(7, y, z)}+z{m(,,j, 3)}, _ (14)
since the terms which are bilinear in respect to x, y, z, and t, rI, vanish, in virtue of the relations (7). Thus
Is = I+I(m)x2, (15)
Iv:= I78+I(m)•(y2{z2), (16)
with similar relations, and
lo = lc+ 1(m). OG2. (17) 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 formula (17) is due to J. L. Lagrange; it may be written
I(m . OP2) =1(m . GP2)+OG2, (18)
1(m) (m)
and expresses that the mean square of the distances of the particles from 0 exceeds the mean square of the distances from G by OG2. The masscentre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make 0 coincide with P2, . . . P„ in succession, we obtain
o +m2 . P1P22+ ... +mn . P,P„2 = $(m . GP2) +1(m) . GP,2,
m, . P21'12+ o +...+mn.P2P„2=a.(m.GP2)+1(m).GP22, (19)
m, . PnP,2fm2. PnPz2+ ... + o = x(m . GP2) +1(m) . GPn2.
If we multiply these equations by m,, m2, . . . m°, respectively, and add, we find
EE (mrm.. P,P.2) = I (m) . E(m . GP2), (20) provided the summation E1 on the left hand be understood to include each pair of particles once only. This, theorem, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances. For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R2=I a2, if a be the length of an edge.
Another type of quadratic moment is supplied by the deviationmoments, or products of inertia of a distribution of matter. Thus the sum M(m.yz) is called the " product of inertia " with respect to the planes y=o, z=o. This may be expressed in terms of the product of inertia with respect to parallel planes through G by means of the formula (14);
Viz.:— .) =E(m.,l')+Z(m) . 3j2 (21)
(6)
(it) (12)
The quadratic moments with respect to different planes through a fixed point 0 are related to one another as follows. The moment with respect to the plane
ax+µy+vz =o, (22)
where X, v are directioncosines, is
Etm(xx1,Y+vz)2}=E(mx2) X2+Fi(my2) . µ2+2(mz2) .v2
+2E(myz) . µvF2E(mzx) . vX+21(mxy) . Xµ, (23) and therefore varies as the square of the perpendicular drawn from 0 to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22). If the coordinate axes coincide with the principal axes of this quadric, we shall have
E(myz)=o, (mzx)=o, (mxy)=o; (24) and if we write
(mx2)=Mat, I(my2)=Mb2, m(mz2) =Mc2, (25) where M=1(m), the quadratic moment becomes M(a2X2+b2µ2.. c2v2), or Mp2, where p is the distance of the origin from that tangent plane of the ellipsoid
a2+b2+c2 = 1, (26)
which is parallel to (22). It appears from (24) that through any assigned point 0 three rectangular axes can be drawn such that the product of inertia with respect to each pair of coordinate planes vanishes; these are called the principal axes of inertia at O. The ellipsoid (26) was first employed by J. Binet (1811), and may be called " Binet's Ellipsoid " for the point O. Evidently the quadratic moment for a variable plane through 0 will have a
stationary " value when, and only when, the plane coincides with a principal plane of (26). It may further be shown that if Binet's ellipsoid be referred to any system of conjugate diameters as coordinate axes, its equation will be
x'2 y'2 z'2
a'2 b'2 c'2 _ I'
k2 a2+k2 _ y2 z2 b2 r k2 c2=1
,
and the quadrics corresponding to different values of k2 will be confocal. If we write
k2 = a2+b2+c2+0,
b24 e2 = a2, c2 +a2 = FN , a2+ b2 =y2
the equation (31) becomes
a2+0 6'2y+B+72zz_0=1'
for different values of 0 this represents a system of quadrics confocal with the ellipsoid
x2 y2 z2 ail+YR2+y2 = 1
which we shall meet with presently as the " ellipsoid of gyration " at G. Now consider the tangent plane w at any point P of a confocal, the tangent plane w' at an adjacent point N', and a plane w" through P parallel to w'. The distance between the planes w' and co" will be of the second order of small quantities, and the quadratic moments with respect to w' and co" will therefore be equal, to the first order. Since the quadratic moments with respect to co and co' are equal, it follows that co is a plane of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes of inertia at P are the normals to the three confocals of the system
(33) which pass through P. Moreover if x, y, z be the coordinates of P, (33) is an equation to find the corresponding values of 0; and if BI, 02, 03 be the roots we find
91 +B2 +Oa = r2 – a2 _ 02 — y2, (35) where r2=x2+y2+z2. The squares of the radii of gyration about the principal axes at P may be denoted by k22+k32, k32+ k,2, k,2+ k22; hence by (32) and (35) they are r2—0I, r2—B2, r2—03, respectively.
To find the relations between the moments of inertia about different axes through any assigned point 0, we take 0 as origin. Since the square of the distance of a point (x, y, z) from the axis
x_Y= (36) x
is x2+y2+z2—(Xx} py+vz)2, the moment of inertia about this axis is
I =E[m{ (X2+µ2+v2) (x2+y2+z2) – (Xx+ µy+vz)21] =Aa2+Bµ2+Cv2—2Fµv2Gv?—2H]µ,
provided
A=f.{m(y2+z2)}, B=Z{m(z2+x2)}, C=z{m(x2+y2)},
F=E(myz), G=Z(mzx), H=Z(mxy);
i.e. A, B, C are the moments of inertia about the coordinate axes, and F, G, H are the products of inertia with respect to the pairs of coordinate planes. If we construct the quadric
Ax2+By2+Cz2—2Fyz—2Gzx2Hxy=Met, (39) where a is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction X, µ, v is found by putting x, y, z=Ar, µr, vr. Hence, by comparison with (37),
I = M e4/r2• (40) The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. When referred to its principal axes, the equation of the quadric takes the form
Axe+By2+Cz2=Me. (41) The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at 0, as already defined in connexion with the theory of plane quadratic moments. The new A, B, C are called the principal moments of inertia at O. Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at O. Since, by (12), B+C>A, &c., the sum of the two lesser principal moments must exceed the greatest principal moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the ellipsoid cannot be less than I/,12.
If we write A=Ma2, B=M(32, C=My2, the formula (37), when referred to the principal axes at 0, becomes
I M (a2X2+xµ2+72v2 = Mpg,
if p denotes the perpendicular drawn from 0 in the direction (X, v) to a tangent plane of the ellipsoid
x2 y2 z2
a2+—132 +7—2 =
(27)
provided
(mx'2)=Ma'2, Z(my'2)Mb'2, (mz'2)=Mc'2;
also that
E(my'z')=o, 2(mz'x')=o, (mx'y')=o. (28)
Let us now take as coordinate axes the principal axes of inertia at the masscentre G. If a, b, c be the semiaxes of the Binet's ellipsoid of G, the quadratic moment with respect to the plane Xx + µy + vz =o will be M(a2X2 + b2µ2 + c2v2), and that with respect to a parallel plane
ax+µy+vz = p
will be M(a2X2+b2µ2+c2v2+p2), by (15). This will given value Mkt, provided
p2 = (k2  a2)X2+ (k2  b2)122+(k2  c2)v2.
Hence the planes of constant quadratic moment Mkt will envelop the quadric
(29) have a
(30)
(31)
(32)
(34)
(37) (38)
(42)
(43)
(33) This is called the ellipsoid of gyration at 0; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having 0 as centre.
If A =B = C, the momental ellipsoid becomes a. sphere; all axes through 0 are then principal axes, and the moment of inertia is the same for each. The masssystem is then said to possess kinetic symmetry about O.
If all the masses lie in a plane (z=o) we have, in the notation of (25), c2=o, and therefore A =Mb2, B=Ma2, C=M(a2+b2), so that the equation of the momental ellipsoid takes the form
b2x2+a2 y2+(a2+b2) z2=e4• (44) The section of this by the plane z=o is similar to
x2
a2 +by2
2 =1, (45)
which may be called the momental ellipse at O. It possesses the property that the radius of gyration about any diameter is half the distance between the two tangents which are parallel to that diameter. In the case of a uniform triangular plate it may be shown that the momental ellipse at G is concentric, similar and similarly situated
to the ellipse which touches the sides of the triangle at their middle points.
The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of 2 (mx . x) or E (mx2). The construction of a second funicular may be dispensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective
particles are indicated by the a corresponding segments of a line in the forcediagram, drawn parallel to p. The funicular ZABCD . . corresponding to any pole 0 is constructed for a system of forces acting parallel to p through the positions of the particles and proportional to the respective masses; and its successive sides are produced to meet p in the points H, K, L, M , .. As explained in § 5, P the moment of the first par
ticle is represented on a cer
tain scale by HK, that of the
second by KL, and so on.
The quadratic moment of the
first particle will then be
represented by twice the area
End of Article: BCD 

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