Page:EB1911 - Volume 17.djvu/991

is the sum of the projections of the several vectors, the equation (2) gives

x̄＝Σ(mx)/Σ(m),

(5)

if x̄ be the projection of OG→. Hence if the Cartesian co-ordinates of P₁, P₂, . . . P_n relative to any axes, rectangular or oblique be (x₁, y₁, z₁), (x₂, y₂, z₂), . . ., (x_n, y_n, z_n), the mass-centre (x̄, ȳ, z̄) is determined by the formulae

x̄ =	Σ(mx)	,   ȳ =	Σ(my)	,   z̄ =	Σ(mz)	.
	Σ(m)		Σ(m)		Σ(m)

(6)

If we write x = x̄ + ξ, y = ȳ + η, z = z̄ + ζ, so that ξ, η, ζ denote co-ordinates relative to the mass-centre G, we have from (6)

Σ(mξ) = 0,   Σ(mη) = 0,   Σ(mζ) = 0.

(7)

One or two special cases may be noticed. If three masses α, β, γ be situate at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′ in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then divide AA′ so that α·AG = (β + γ) GA′. It is easily proved that

α : β : γ = ΔBGA : ΔGCA : ΔGAB;

also, by giving suitable values (positive or negative) to the ratios α : β : γ we can make G assume any assigned position in the plane ABC. We have here the origin of the “barycentric co-ordinates” of Möbius, now usually known as “areal” co-ordinates. If α + β + γ = 0, G is at infinity; if α = β = γ, G is at the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find

α : β : γ : δ = tetⁿ GBCD : tetⁿ GCDA : tetⁿ GDAB : tetⁿ GABC,

and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If α + β + γ + δ = O, G is at infinity; if α = β = γ = δ, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB : ΔABC, 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 mass-centre 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 mass-centre of a uniform solid right circular cone divides the axis in the ratio 3 : 1; 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 mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.

The formula (2) shows that a system of concurrent forces represented by m₁·OP→₁, m₂·OP→₂, . . . m_n·OP→_n will have a resultant represented by Σ(m)·OG→. If we imagine O to recede to infinity in any direction we learn that a system of parallel forces proportional to m₁, m₂,... m_n, acting at P₁, P₂ . . . P_n have a resultant proportional to Σ(m) which acts always through a point G fixed relatively to the given mass-system. This contains the theory of the “centre of gravity” (§§ 4, 9). We may note also that if P₁, P₂, . . . P_n, and P₁′, P₂′, . . . P_n′ represent two configurations of the series of particles, then

Σ(m·PP→′) = Σ(m)·GG→′,

(8)

where G, G′ are the two positions of the mass-centre. The forces m₁·P₁P₁′→, m₂·P₂P₂→′, . . . m_n·P_nP_n′→, 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. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h₁, h₂, . . . h_n be the perpendicular distances of the particles from any fixed plane, the sum Σ(mh²) is the quadratic moment with respect to the plane. If p₁, p₂, . . . p_n be the perpendicular distances from any given axis, the sum Σ(mp²) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis. If r₁, r₂, . . . r_n be the distances from a fixed point, the sum Σ(mr ²) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments by the total mass Σ(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole. 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. If we take rectangular axes through any point O, the quadratic moments with respect to the co-ordinate planes are

I_x = Σ(mx²),   I_y = Σ(my²),   I_z = Σ(mz²);

(9)

those with respect to the co-ordinate axes are

I_yz = Σ {m (y² + z²)},   I_zx = Σ {m (z² + x²)},   I_xy = Σ {m (x² + y²)};

(10)

whilst the polar quadratic moment with respect to O is

I₀ = Σ {m (x² + y² + z²)}.

(11)

We note that

I_yz = I_y + I_z,   I_zx = I_z + I_x,   I_xy = I_x + I_y,

(12)

and

I₀ = I_x + I_y + I_z = 1/2 (I_yz + I_zx + I_xy).

(13)

In the case of continuous distributions of matter the summations in (9), (10), (11) 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, I₀ = 1/2Ma², where M is the mass and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = 1/2Ma², I_xy = 1/2Ma²; hence the value of the squared radius of gyration is for a diameter 1/4a², and for the axis of symmetry 1/2a². Again, for a uniform solid sphere having its centre at the origin we find I₀ = 3/5Ma², I_x = I_y = I_z = 1/5Ma², I_yz = I_zx = l_xy = 3/5Ma²; i.e. the square of the radius of gyration with respect to a diameter is 2/5a². The method of homogeneous strain can be applied to deduce the corresponding results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide with the principal axes, we find I_x = 1/5Ma², I_y = 1/5Mb², I_z = 1/5Mc², whence I_yz = 1/5M (b² + c²), &c.

If φ(x, y, z) be any homogeneous quadratic function of x, y, z, we have

Σ {mφ (x, y, z)} = Σ {mφ (x̄ + ξ, ȳ + η, z̄ + ζ)} = Σ {mφ (x̄, y, z)} + Σ {mφ (ξ, η, ζ)},

(14)

since the terms which are bilinear in respect to x̄, ȳ, z̄, and ξ, η, ζ vanish, in virtue of the relations (7). Thus

I_x = Iξ + Σ(m)x²,

(15)

I_yz = Iηζ + Σ(m) · (ȳ² + z̄²),

(16)

with similar relations, and

I_O = I_G + Σ(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

Σ(m · OP2)	=	Σ(m · GP2)	+ OG2,
Σ(m)		Σ(m)

(18)

and expresses that the mean square of the distances of the particles from O exceeds the mean square of the distances from G by OG². The mass-centre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make O coincide with P₁, P₂, . . . P_n in succession, we obtain

0	+ m2·P1P22	+ . . .	+ mn·P1Pn2	= Σ(m · GP2) + Σ(m) · GP12,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
m1·P2P12	+   0	+ . . .	+ mn·P2Pn2	= Σ(m · GP2) + Σ(m) · GP22,
. . . . . . . . .
m1·PnP12	+ m2·PnP22	+ . . .	+   0	= Σ(m · GP2) + Σ(m) · GPn2.

(19)

If we multiply these equations by m₁, m₂ . . . m_n, respectively, and add, we find

ΣΣ (m_r m_s · P_r P_s²) = Σ (m) · Σ (m · GP²),

(20)

provided the summation ΣΣ 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 = 3/8a², if a be the length of an edge.

Another type of quadratic moment is supplied by the deviation-moments, or products of inertia of a distribution of matter. Thus the sum Σ(m·yz) is called the “product of inertia” with respect to the planes y = 0, z = 0. 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.:—

Σ (m · yz) = Σ (m · ηζ) + Σ (m) · ȳz̄

(21)