285. If a Body A, at the Distance sa from an axis passing through s, be made to revolve about that axis by any Force acting at P in the Line Sr, Perpendicular to the Axis of Motion: It is required to determine the Quantity or Matter of another Body &, which being placed at r, 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 a about s, may be the Same in Both Cases.

SP

SA

By the nature of the lever, sa: SP

f .f, the effect of the force f, acting at P,

on the body at A; that is, the force facting at P, will have the same effect on the body a, as

SP

the force-f, acting directly at the point 4.

SA

A

But as Asp revolves 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 circle described by them. Here then we have two bodies a and Q, which being urged

SP

SA

A

directly by the forces f and -f, acquire velocities which are as SP and SA. And since the motive forces of bodies are as their mass and velocity: therefore

SP2

which therefore expresses the mass of matter which, being placed at P, would receive the same angular motion from the action of any force at r, as the body a receives. So that the resistance of any body A, to a force acting at any point e, 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.

286. Corol. 1. Hence the force which accelerates the point r, is to the force of gravity, as f. SP2 to 1, or as f. sp2 to A SA2

A SA2.

287. Corol. 2. If any number of bodies A 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,

placedat the distances SA, SB, sc, there be substituted the bodies

SP

SB2
A, B,
SP

SC3
Sp2

C c; these being collected into the point P.

And hence, the moving force being f, and the matter moved
A SA2+B. SB +c, sc2

SP

; theref.

being
is the accelerating force; which therefore is to the accelerat-
ing force of gravity, as ƒ sp2 to A. SA2+B. SB2+c. Sc2.
288. Corol. 3, The angular velocity of the whole system
of bodies, is as
For the absolutė

A. SA2+B. SB +c. Sc2

velocity of the point P, is as the accelerating force, or di-
rectly as the motive force f, and inversely as the mass
A. SAS &c.
but the angular velocity is as the absolute velo-

SP2

་

city directly, and the radius, SP inversely; therefore the an
gular velocity of r, or of the whole system, which is the same
ƒ. SB-
thing is as

SA2 +B. SB2+c. sc2

PROPOSITION LVI.

289. To determine the Centre of Oscillation of any Compound Mass or Body MN, or of any System of Bodies A, B, C, &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 oscilla-
tion; through the axis s T
draw seo, and the horizon-
tal line sq; then from every
particle, A, B, C, &c. let fall
perpendiculars aα, ap, bb, вq;
cc, cr, to these two lines; and
join sa, SB, SC; also, draw M
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.
$q,-c sr; therefore, by
cor. 3, prop. 55, the angular

motion generated by all these forces is

A. Sp + B. Sq - C .sr

2

A. SA2 +B. SB2 ++c.sc3

Also; the angular veloc. any particle p, placed in o, gene

rates in the system, by its weight, issn or

sn,

or

sm

P.S03 S02 SG. SO because of the similar triangles sem, son. But, by the problem, the vibrations are performed alike in both cases, and therefore, these two expressions must be equal to each other, A. Sp+B. sq-c、 sr A. SA2+ B. SB2 + c . Sc2 SA2 + B. SB3 + c. sc2

that is,

sm

SG.SO

sm A

X

SG

A sp + B sq C sr

; and hence

But, by cor. 2, pr. 41, the sum A. sp+ B. sq-c. sr=(A+B +c) sm; therefore the distance so=

A SA2 + B. SB2 + C. Sca

SG. (A + B + c)

SA2 + B SB2 C. Sc2

A sa + B. sb + c ..sc by prop. 42, which is the distance of the centre of oscillation o, below the axis of suspension; where any of the products a. sα, B. sb, must be negative, when a, b, &c. lie on the other side of s. So that this is the same expression as that for the distance of the centre of percussion, found in prop. 54.

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.

290. Corol. 1. If p be any particle of a body, b, and d 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

sum of all the pd2

SG X the body b

291. Corol. 2. If b denote the matter in any compound body, whose centres of gravity and oscillation are & 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 r

For, by corol. 2, prop. 54, this body p is = A. SA2 B. SB2+ c. sc2

*

SCHOLIUM,

292. By the method of Fluxions; 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 body, below the point of suspension."

293. 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 =

inches.

For, the length of the pendulum vibrating seconds, or 60 times
in a minute, being 391 inches; and the lengths of pendulums
being reciprocally as the square of the number of vibrations
602 × 391
made in the same time; therefore n2 : 602 :: 391

140850

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.

294. 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 different from the former, and is* ·
called the Centre of Gyration; which is determined in the
following manner:

PROPOSITION LVII.

295. To determine the Centre of Gyration of a Compound Body
or of a System of Bodies.

LET R be the centre of gyration, or
the point into which all the particles A,
B, C, &c. being collected, it shall receive
the same angular motion from a force f
acting at P, as the whole system re-
ceives.

Now, by cor. 3, pr. 54, the angular
velocity generated in the system by the
f.SP
force f, is as

and

A SA2 + B. SB2 &c.

by the same, the angular velocity of the system placed in R, ទ

centre of gyration below the axis of motion.

G

296. Corol. 1. Because a SA2+ B. SB2 &c. = sà. so, b, where & is the centre of gravity, o the centre of oscillation, and b the body A + B + C &c.; therefore sn2=SG. so; that is, the distance of the centre of gyration, is a mean proportional between those of gravity and oscillation.

B:

c

297. Córol. 2. If p denote any particle of a body b, at d dis

tance from the axis of motion; then SR2

PROPOSITION LVIII.

2

sum of all the pd2.

body b

298. To determine the velocity with which a Ball moves, which
being shot against a Ballistic Pendulum, causes it to vibrate
through a given Angle..

The Ballistic Pendulum is a heavy block

of wood MN, suspended vertically by a strong
horizontal iron axis at s, to which it is con-
nected by a firm iron stem. This problem
is the application of the last proposition, or
of prop. 54, and was invented by the very
ingenious Mr. Robins, to determine the ini-
tial velocities of military projectiles; a cir-
cumstance very useful in that science; and
it is the best method yet known for deter-
mining them wth any degree of accuracy,

Let