/n be the perpendicular distances from any given axis, the sum ~(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis.
To find the relations between the moments of inertia about different axes through any assigned point 0, we take 0 as origin.
Consequently the inertia to overcome in moving the cylinder r=b, solid or liquid, is its own inertia, increased by the inertia of liquid (a2+b2)/(a2,..b2) times the volume of the cylinder r=b; this total inertia is called the effective inertia of the cylinder r =b, at the instant the two cylinders are concentric.
Denoting the effective inertia of the liquid parallel to Ox by aW' the momentum aW'U = 4)0W' (24) _ U i -AO' 25) in this way the air drag was calculated by Green for an ellipsoida pendulum.
D/dI.(Mu) =X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation.
2 V I - a /al ' Y' I-a /al ' and the effective inertia of the liquid in the interspace Ao+2A1 W, =1 a13 +2a3W'.
Another type of quadratic moment is supplied by the deviationmoments, or products of inertia of a distribution of matter.
With #=o, the stream is parallel to xo, and 4)=m ch (n-a)cos = - Uc ch (n-a) sh n cos /sh (n-a) (22) over the cylinder n, and as in (12) § 29, =-Ux =-Uc ch n cos t, (23) for liquid filling the cylinder; and _ th n (14) 01 th (7 7 - a) ' over the surface of n; so that parallel to Ox, the effective inertia of the cylinder n, displacing M' liquid, is increased by M'thn/th(n-a), reducing when a= oo to /If' th n = M' (b/a).
But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W' the weight of fluid medium displaced.
Eschenhagen 2 first designed a set of magnetographs in which this idea of small moment of inertia was carried to its useful limit, the magnets only weighing 1 .
A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes.
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.
In that year, though the Church was under no direct threat of attack, owing to the inertia of the emperor Philip the Arabian, the atmosphere was full of conflict.
The advantages of using small magnets, so that their moment of inertia may be small and hence they may be able to respond to rapid changes in the earth's field, were first insisted upon by E.
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 co-ordinate planes vanishes; these are called the principal axes of inertia at 0.
Mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction.
If by the attachment of another body of known moment of inertia I, the period is altered from T to -r, we have T=21r,/l(I+I)(K~.
The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.
In the absence of a medium the inertia of the body to transtion is the same in all directions, and is measured by the (3) But the change of the resultant momentum F of the medium as.
The effective angular inertia of the body in the medium is now required; denote it by C 1 about the axis of the figure, and by C2 about a diameter of the mean section.
Same is true of physical quantities such as potential, temperature, &c., throughout small regions in which their variations are continuous; and also, without restriction of dimensions, of moments of inertia, &c. Hence, in addition to its geometrical applications to surfaces of the second order, the theory of quadric functions of position is of fundamental importance in physics.
It is to be remembered that all force is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as force of inertia, accelerating force, moving force, once classical, are proscribed.
The point G determined by (I) is called the mass-centre or centre of inertia of the given system.
Thus the sum ~(m.yz) is called the product of inertia with respect to the planes y=o, z=o.
The moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.
The fact that the moment of inertia of the magnet varies witli the temperature must, however, be taken into account.
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.:
Since the quadratic moments with respect to w and of are equal, it follows that w is a plane 01 stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes 01 inertia at P are the normals to the three confocals of the systen (3,~) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of 0; and if Of, 02, 03 be the roots we find Oi+O2+81r1a2$-7, (35)
The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?
When Montanus proposed to summon all true Christians to Pepuza, in order to live a holy life and prepare for the day of the Lord, there was nothing whatever to prevent the execution of his plan except the inertia and lukewarmness of Christendom.
Then the deviation y= DE of the neutral axis of the bent beam at any point D from the axis OX is given by the relation d 2 y Ml dx 2 = EI' where M is the bending moment and I the amount of inertia of the beam at D, and E is the coefficient of elasticity.