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of a pendulum vibrating seconds of mean time in the latitude of London, in a vacuum at the level of the sea, Fah. thermometer being at 62°, and the barometer at 30 inches.

OF THE CENTRES OF PERCUSSION, OSCILLATION, AND GYRATION.

53. THE Centre of Percussion of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest as it were in equilibrio, without acting on the centre of suspension.

54. The Centre of Oscillation is that point, in a vibrating body, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension.

55. The Centre of Gyration, is that point, in which, if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself.

56. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different, unconnected bodies, each revolving about a centre, the angular velocity is as the absolute velocity directly, and the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal.

PROP. XXVIII.

57. To find the centre of percussion of a body, or system of bodies.

LET the body revolve about an axis passing through any point in the line SGO, passing through the centres of gravity and percussion, & and O. Let MN be the section of the body, or the plane in which the axis SGO moves. And conceive all the particles of the body to be reduced to this plane, by perpendiculars let fall from them to the plane; a supposition which will not affect the centres G, O, nor the angular motion of the body.

S

M

B

Let A be the place of one of the particles, so reduced; join SA, and draw AP perpendicular to AS, and Aa perpendicular to SGO: then AP will be the direction of A's motion, as it revolves about S; and the whole mass being stopped at O, the body A will urge the point P forward, with a force proportional to its quantity of matter and velocity; or to its matter and distance from the point of suspension S; that

is, as A. SA; and the efficacy of this force in a direction perpendicular to SO, at the point P, is as A. Sa, by similar triangles; also, the effect of this force on the lever, to turn it about O, being as the length of the lever, is as A. Sa. PO =A. Sa. SO — SP = A. Sa . SO — A. Sa. SP = A. Sa. SO — A. SA'. In like manner, the forces of B and C, to turn the system about 0,

are as,

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But, since the forces on the contrary sides of O destroy one another, by the definition of this force, the sum of the positive parts of these quantities, must be equal to the sum of the negative parts,

that is, A. Sa. SO + B. Sb. SO + C. Sc. SO, &c. =

A. SAB. SB + C. SC3, &c. ;

and hence S0 =

A. SAB. SB2 + C. SC &c.
A. Sa + B. Sb + C. Sc &c. '

the centre of percussion below the axis of motion.

which is the distance of

And here it must be observed that, if any of the points a, b, &c. fall on the contrary side of S, the corresponding product A. Sa, or B. Sb., &c. must be made negative.

Corol. 1. Since, by cor. 8, pr. 15, A+B+ C, &c. or the body b × the distance of the centre of gravity, SG, is = A. Sa + B. Sb + C. Sc, &e which is the denominator of the value of SO; therefore the distance of the cenA. SA' + B. SB2 + C . SC* &c. SG X body b

tre of percussion is SO =

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and, by cor. 5, pr. 13, the sum of the last terms is nothing, namely,

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2SG. Ga + 2SG. Gb + 2SG, Gc, &c. = 0; therefore the sum of the others, or A . SA2 + B . SB3, &c.

or=

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= A + B, &c. . SG2 + A. GA' + B. GB' + C . GC, &c. b. SG2 + A. GA' + B. GB2 + C . GC, &c which being substituted in the numerator of the foregoing value of SO, gives, b. SG2 + A. GA2 + B . GB2 + &c.

SO =

or SO SG +

b. SG

A . GA' + B. GB2 + C . GC2, &c.
b. SG

Corol. 3. Hence, the distance of the centre of percussion always exceeds the distance of the centre of gravity, and the excess is always

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is always the same constant quantity, wherever the point of suspension S is placed; since the point G, and the bodies A, B, &c. are constant. Or GO is always reciprocally as SG, that is, GO is less, as SG is greater; and conse quently the point rises upwards, and approaches towards the point G, as the

voint S is removed to the greater distance; and they coincide when SC is infinite. But when S coincides with G, then GO is infinite, or O is at an infinite distance

PROP. XXIX.

59. If a body A, at the distance SA from an axis passing through S, be made to revolve about that axis by any force f acting at P in the line SP, perpen dicular to the axis of motion; it is required to determine the quantity or matter of another body Q, which, being placed at P, the point where the force acts, it shall be accelerated in the same manner, as when A revolved at the distance SA; and, consequently, that the angular velocity of A and Q about S, may be the same in both cases.

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of the force f, acting at P, on the body at A; that is, the force f acting at P, will have the same effect on the body A, as the

SP

force f, acting directly at the point A. But as ASP reSA

volves altogether about the axis at S, the absolute velocities of the points A and S, or of the bodies A and Q, will be as the

radii SA, SP of the circles described by them. Here then we have two bodies

SP
SA

A and Q, which being urged directly by the forces ƒ and f, acquire velocities which are as SP and SA. But the motive forces of bodies are as their mass and velocity; therefore

f:f:: A. SA: Q. SP, and SP' : SA' :: A: Q =

SA
SP

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SP SA which therefore is the mass of matter which, being placed at P, would receive the same angular motion from the action of any force at P, as the body A receives. So that the resistance of any body A, to a force acting at any point P, is directly as the square of its distance SA from the axis of motion, and reciprocally as the square of the distance SP of the point where the force acts.

Corol. 1. Hence the force which accelerates the point P, is to the force f. Sp of gravity, as to 1, or as ƒ. SP' to A . SA'.

A.SA'

Corol. 2. If any number of bodies, A, B, C, be put in motion, about a fixed axis passing through S, by a force acting at P; the point P will be accelerated in the same manner, and consequently the whole system will have the same angular velocity, if, instead of the bodies A, B, C, placed at the distances SA, SB, SC, SA SB A, SP SP these being collected into the point P. And hence, the moving force being ƒ, and the matter moved being A. SAB. SB2 + C. SC' SP

there be substituted the bodies

SC

B, SP

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C;

S

; therefore the accelerating force is

f. SP

A . SAa ‍+ B . SB3 + C. SC2; which is to the accelerating force of gravity as f. SP to A. SA' + B. SB2 + C. SC2

Corol. 3. The angular velocity of the whole system of bodies, is as
f. SP

A. SA' + B. SB'+ C. SC For the absolute velocity of the point P, is as
the accelerating force, or directly as the motive force ƒ, and inversely as the
A. SA', &c.
SP

mass

: but the angular velocity is as the absolute velocity directly and the radius SP inversely; and therefore the angular velocity of P, or of the

ƒ.SP whole system, which is the same thing, is as A. SA' + B. SB2 + C . SC2.

PROF. XXX.

60. To determine the centre of oscillation of any compound mass or body MN, or of any system of bodies A, B, C.

LET MN be the plane of vibration, to which let all the matter be reduced, by letting fall perpendiculars from every particle, to this plane. Let G be the

centre of gravity, and O the centre of oscillation; through the axis S draw SGO, and the horizontal line Sq; then from every particle A, B, C, &c. let fall perpendiculars Aa, Ap, Bb, Bq, Cc, Cr, to these two lines; and join SA, SB, SC; also, draw Gm, On perpendicular to Sq. Now, the forces of the weights A, B, C, to turn the body about the axis, are A. Sp, B. Sq, - C. Sr; and therefore, by cor. 3, prop. 29, the angular motion generated by all these forces is A. Sp+ B. Sq - C. Sr

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A . SA2 + SB2 + C.SC”

Also, the angular velocity which any particle p,

placed in O, generates, in the system, by its weight, is

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Sm

P. SO

or

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because of the similar triangles SGm, SOn. But, by the problem,

SG. SO' the vibrations are performed alike in both cases, and therefore these two expressions must be equal to each other, that is,

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And hence SO = X

Sm A. SA + B. SB2 + C. SC
SG A . Sp + B . Sq — C . Sr ·
But, by cor. 2, prop. 15, the sum A. Sp+ B.
A.

Sm; therefore the distance SO =

Sq— C. Sr = (A + B + C) SA' + B . SB2 + C SC SG. (A+B+C)

A. SAB. SB2 + C. SC

A. Sa + B. Sb + C. Sc by prop. 16, the distance of the centre of oscillation O, below the axis of suspension; where any of the products A. Sa, B. Sb, must be negative, when a, b, &c. lie on the other side of S; which is the same expression as that for the distance of the centre of percussion, found in prop. 29.

Hence it appears, that the centres of percussion and of oscillation are in the very same point. And therefore the properties in all the corollaries there found for the former, are to be here understood of the latter.

Corol. 1. If m be any particle of a body and r its distance from the axis of motion S; also, G, O, the centres of gravity and oscillation. Then the distance of the centre of oscillation of the body, from the axis of motion, is moment of inertia

SO=

Σ (mr2)
SG.zm

=

SG.mass

Corol. 2. If b denote the matter in any compound body, whose centres of gravity and oscillation are G and O; the body P, which being placed at P, where the force acts as in the last proposition, and which receives the same motion from that force as the compound body b, is P =

SG. SO
SP

.b.

A SAB. SB + C. SC'.

SP

For, by corol. 2, prop. 29, this body P = But, by corol. 1, prop. 28, SG. SO . b = A . SA” + B . SB' + C. SC3; therefore P=

SG. SO
SP2

b.

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61. SCHOLIUM.-By the integral calculus, the centre of oscillation, for a regular body, will be found from cor. 1. But for an irregular one; suspend it at the given point; and hang up also a simple pendulum of such a length, that, making them both vibrate, they may keep time together. Then the length of the simple pendulum, is equal to the distance of the centre of oscillation of the bo ly, below the point of suspension.

62. Or it will be still better found thus: suspend the body very freely by the given point, and make it vibrate in small arcs, counting the number of vibrations it makes in any time, as a minute, by a good stop watch; and let that number of vibrations made in a minute be called n: then shall the 140850 distance of the centre of oscillation, be SO =

nn

inches. For, the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 inches, and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time; therefore,

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the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion.

63. The foregoing determination of the point, into which all the matter of a body being collected, it shall oscillate in the same manner as before, only respects the case in which the body is put in motion by the gravity of its own particles, and the point is the centre of oscillation: but when the body is put in motion by some other extraneous force, instead of its gravity, then the point is

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