# Inertia Sentence Examples

- The new A, B, C are called the principal moments of
**inertia**at 0. - 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 connection with the theory of plane quadratic moments. - The moment of
**inertia**about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. - Provided the
**inertia**of the snrin~ itself be neglected. - 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. - They both protested against the political and ecclesiastical
**inertia**of their native state, and adopted the doctrines of freedom and reason. - /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. - Is x+y2+z(Xx+uy+vz), the moment of
**inertia**about this axis is =AX2+Bu+Cv12F~w2Gv?.2HXu, (37) provided A=~m(y+z)), B=~{m(z1+x)j, C=~m(x+y)}, ~ ~ 8 F=~(myz), G=~(mzx), H=~(mxy); 5 ~ - 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. - 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. - 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'. - 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 . - 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. - Similarly, the
**inertia**parallel to Oy and Oz is NW' - 1 B W', B C (b2 +-X, c 2 ab and A +C abc/ZP, Ao For a sphere a=b=c, Ao= Bo=Co =, 'a' = Q = = z, (9) U from (II), (16) so that the effective**inertia**of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration W - W', g (30) W+ aW Thus a spherical air bubble, in which W/W' is insensible, will begin to rise in water with acceleration 2g. - 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. - 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. - Another type of quadratic moment is supplied by the deviationmoments, or products of
**inertia**of a distribution of matter. - 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. - 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. - 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. - 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). - 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. - 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. - 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. - On the other hand, under the influence of the mechanics of his day, which had hardly distinguished between
**inertia**, or the inability of a body to change itself, and resistance or the ability of bodies to oppose one another, he concluded that, as**inertia**is passive, so is resistance, and refused to recognize that in collision the mutual resistance of moving bodies is a force, or active power, of changing their movements in opposite directions. - The fact that the moment of
**inertia**of the magnet varies witli the temperature must, however, be taken into account. - 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. - 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) - To find the relations between the moments of
**inertia**about different axes through any assigned point 0, we take 0 as origin. - 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 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. - It is usually accurate enough in deflection calculations to take for I the moment of
**inertia**at the centre of the beam and to consider it constant for the length of the beam. - But on the modern theory, which includes the play of electrical phenomena as a function of the aether, there are other considerations which show that this number io 2 is really an enormous overestimate; and it is not impossible that the co-efficient of ultimate
**inertia**of the aether is greater than the co-efficient of**inertia**(of different kind) of any existing material substance. - When the atoms are in motion these strain-forms produce straining and unstraining in the aether as they pass across it, which in its motional or kinetic aspect constitutes the resulting magnetic field; as the strains are slight the coefficient of ultimate
**inertia**here involved must be great. - W, or 1w, if I denote the moment of
**inertia**(~ II) about the axis. - = constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the ellipsoid A = o, and moving bodily with it, the effective
**inertia**of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the ellipsoid is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the ellipsoid A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular**inertia**of the liquid, outside and inside the ellipsoid X = o. - Even with the particles retarding the motion of the aether, the same will be true if, to counterbalance the increased
**inertia**, suitable forces are caused to act on the aether at all points where the**inertia**is altered. - Consider, for example, a submarine boat under water; the
**inertia**is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) .