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(equation 18), MG,=m,y,+m, Y2 + m2 Y2 + ・・・ .9
MG,=m,x,+m ̧x, +m ̧x,+ . . . .;

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Now (Art. 11.), R= √X2+Y', therefore

R: = "ƒM √G,'+G,'.

Now if G be the distance AG of the centre of gravity from A, G = √G,2+G22,

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Substituting in equation (82) the value of ƒ from equation (78,) we have

R=

MG≤Pp..... (83).

I

And substituting in equation (80) for R its value from equation (82),

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where L is the distance of the point of application of the resultant of the effective forces from the axis.

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..tan. tan. AGG,, ..=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 (84).

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 else where, 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

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which is obtained from equation (84) by writing MK' 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 ome 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.

The centre of oscillation coincides with the centre

percussion.

of

111. For (by equation 79) the increment of angular velo city per second ƒ of a body revolving about an horizontal axis, the forces impressed upon it being the weights of its parts only, is represented by the forMG

mula g sin. 4, where is the inclination to the ver

I I

tical of the line AG, drawn from the axis to its

I

centre of gravity. But (by equation 84), L= where L MG' is the distance AO of the centre of percussion from the axis,

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Now it has been shown (Art. 98.), that the impressed moving force on a particle whose weight is w, suspended from a thread without weight, inclined to the vertical at an angle, is represented by w sin. ; moreover if f represent.

the increment of velocity per second on this particle, then

w

fis the effective force upon it. Therefore by D'Alembert's principle,

w sin.

го

f", :f=g sin., :.ƒ'=fL.

Now fL 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 of the centre of percussion, since it would receive, at the same distance from the axis, the same increments of velocity per second that the centre of percussion does, it would manifestly move exactly as that point does, and perform its oscillations in the same time that the body does. Therefore, &c.

112. The centres of suspension and oscillation are reci

procal.

Let O represent the centre of oscillation of a body when suspended from the axis A; also let G be its centre of gravity. Let AOL, AG=G, OG=G1; also let the radius of gyration about A be represented by K', and that about G by . Therefore (equation 59), K2=G2+k2;

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Now let the body be suspended from O instead of A; when thus suspended it will have, as before, a centre of oscillation. Let the distance of this centre of oscillation from O be L,,

:. by equation (87), L‚=G,+G,'

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Since then the centre of oscillation in this second case is at the distance L from O, it is in A; what was before the centre of suspension has now therefore become the centre of oscillation. Thus when the centre of oscillation is converted into the centre of suspension, the centre of suspension is thereby converted into the centre of oscillation. This is what is meant, when it is said that the centres of oscillation and suspension are reciprocal.

PROJECTILES.

113. To determine the path of a body projected obliquely

in vacuo.

Suppose the whole time, T seconds, of the flight of the

K

H

body to any given point P of its path, to be divided into equal exceedingly small intervals, represented by AT, and conceive the whole effect of gravity upon the projectile during each one. of these intervals to be collected into a single impulse at the termination of that interval, so that there may be communicated to it at once, by that single impulse, all the additional velocity which is in reality communicated to it by gravity at the different periods of the small time AT.

Let AB be the space which the projectile would describe, with its velocity of projection alone, in the first interval of time; then will it be projected from B at the commencement of the second interval of time in the direction ABT with a velocity which would alone carry it through the distance BK AB in that interval of time; whilst at the same time it receives from the impulse of gravity a velocity such as would alone carry it vertically through a space in that interval of time which may be represented by BF. By reason of these two impulses communicated together, the body will therefore describe in the second interval of time the diago nal BC of the parallelogram of which BK and BF are adja

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