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Take a point in the axis for the point —* ahout which the moments are measured, and let L be the perpendicular distance from A of the resultant R. Now, as in Art. 106. it appears that the sum of the moments of the effective forces about A is
.-. RL=/^2V (80)
To determine the value of R let it be observed that the effective force -fmtfl on any particle mA, acting in a direction nxmx, perpendicular to the distance Am, from the axis A, may be resolved into two others, parallel to the two rectangular axes Ay and Ax, each of which is equal to the product of this effective force, whose direction is nxmv and the cosine of the inclination of ?«,w, to the corresponding axis. Now the inclination of mlnl to Ax is the same as the inclination of Aw, to Ay, since these two last lines are perpendicular to the two former. The cosine of this incliua
AN v tion equals therefore —.—'or—, if AN, = ?/,. Similarly ^ Awi, p, J
AM" $ the cosine of the inclination of n,m, to Ay equals . 'or —, if
I Pi AM, = x,. The resolved parts in the directions of Ay and
Ar of the effective force -fin.p, are therefore -fm,p.-1, and
9 9 Pi
^/'"iPi;1. or ^/'"i^i and £a»i*i
9 rt 9 9
Similarly the resolved parts in the directions of Ax and
Ay of the effective force upon m% are fm.^^ and -fm^cv and
so of the rest
The sums X and Y of the resolved forces in the directions
of Ax and Ay respectively (Art. 11.) are therefore
p"Wi+yfmi!/2 + gfmi93+ • • • . =X,
Now if G be the distance AG of the centre of gravity from A, G= VG^ + GJ,
.-. R = ^/MG (82).
Substituting in equation (82) the value off from equation (78), we have
And substituting in equation (80) for R its value from equation (82),
J 9 9
where L is the distance of the point of application of the resultant of the effective forces, from the axis.
Now let i be the inclination of the resultant R to the axis Ax,
.-. (Art. 11.), RcosJ=X, Rsin. 0 = Y,
.•. tan. 0 = ^ > Dut by equations (81),
Y G, AG, \r>r<
.-. tanfl = tan. AGG„ .-. 0 = AGG,.
The inclination of the resultant R to Ax is therefore equal to the angle AGG,, but the perpendicular to AG is evidently inclined to Ax at this same angle. Therefore the direction of the resultant R is perpendicular to the line AG, drawn from the axis to the centre of gravity. Moreover its magnitude and the distance of its point of application from A have been before determined by equations (83) and (8*).
The Centre Of Percussion.
109. It is evident, that if at a point of the body through which the resultant of the effective forces upon it passes, there be opposed an obstacle to its motion, then there will be produced upon that obstacle the same effect as though the whole of the effective forces were collected in that point, and made to act there upon the obstacle, so that the whole of these forces will take effect upon the obstacle, and there will be no effect of these forces produced elsewhere, and therefore no repercussion upon the axis. It is for this reason that the point O in the resultant, where it cuts the line AG drawn from the axis to the centre of gravity, is called the Centre Of Percussion. Its distance L from A is determined by the equation
which is obtained from equation (84) by writing MKS for I (Art. 80.), K being the radius of gyration. If at the centre of percussion the body receive an impulse when at rest, then since the resultant of the effective forces thereby produced will have its direction through the point where the impulse is communicated, it follows that the whole impulse will take effect in the production of those effective forces, and no portion be expended on the axis.
The Centre Of Oscillation.
110. It has been shown (Art. 98.) that in the simple pendulum, supposed to be a single exceedingly small element of matter suspended by a thread without weight, the time of each oscillation is dependent upon the length of this thread, or the distance of the suspended element from the axis about which it oscillates. If therefore we imagine a number of such elements to be thus suspended at different distances from the same axis, and if we suppose them, after having been at first united into a continuous body, placed in an inclined position, all to be released at once from this union with one another, and allowed to oscillate freely, it is manifest that their oscillations will all be performed in different times. Now let all these elements again be conceived united in one oscillating mass. All being then compelled to perform these oscillations in the same time, whilst all tend to perform them in different times, the motions of some are manifestly retarded by their connexion with the rest, and those of others accelerated, the former being those which lie near to the axis, and the others those more remote; so that between the two there must be some point in the body where the elements cease to be retarded and begin to be accelerated, and where therefore they are neither accelerated nor retarded by their connexion with the rest; an element there performing its oscillations precisely in the same time as it would do, if it were not connected with the rest, but suspended freely from the axis by a thread without weight. This point in the body, at the distance of which from the axis a single particle, suspended freely, would perform its oscillations precisely in the same time that the body does, is called the Centre Of Oscillation.
'Hie centre of oscillation coincides with the centre of percussion.
111. For (by equation 79) the increment of angular velocity per second/ of a body revolving about an horizontal axis, the forces impressed upon it being the weights of
its parts only, is represented by the formula #—=—sin. 0,
where 0 is the inclination to the vertical of the line AG, drawn from the axis to its centre of gravity. But (by equation 84-), L= j-r^' where L is the distance AO of the centre M(jr
of percussion from the axis,
(86), .•. fh=g sin. 0.
Now it has been shown (Art. 98.), that the impressed moving force on a particle whose weight is to, suspended from a thread without weight, inclined to the vertical at an angle 0, is represented by w sin. 0; moreover \if represent the increment of velocity per second on this particle, then —f is
9 the effective force upon it. Therefore by D'Alembert's piin
wrin.8 = H/, .-./=£ sin. 0, .-. /=/L.
Now /L is the increment of velocity at the centre of percussion, and f is that upon a single particle suspended freely at any distance from the axis. If such a particle were therefore suspended at a distance from the axis equal to that