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

PROP. XXXI.

64. To determine the centre of gyration of a compound body, or of a sys tem of bodies.

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

Now, by cor 3, prop. 29, the angular velocity ge nerated in the system, by the force f, is as

f. SP

; and, by the same, the an

A. SAB. SB', &c.
gular velocity of the system placed in R, is

f. SP

(A + B + C, &c.). SR2; then, by making these two expressions equal to each other, we have

SR = √

A. SAB. SB'+ C. SC

A+B+C

for the distance of the centre of gyration below the axis of motion.

R

0

Corol. 1. Because A. SA'+ B. SB3, &c. = SG. SO. b, where G is the centre of gravity, O the centre of oscillation, and b the body A + B + C &c.; therefore SR' = SG. SO; that is, the distance of the centre of gyration is a mean proportional between those of gravity and oscillation.

Cor. 2. If m denote any particle of a body at distance from the axis (mr) moment of inertia fridm

of motion; then SR2 =

Στ

=

mass

=

PROP. XXXI?

65. 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 K, to which it is connected by a firm iron stem. This problem is the application of the last proposition, or of prop. 29, and was invented by the very ingenious Mr Robins, to determine the initial velocities of military projectiles; a circumstance very useful in that science; and it is the only method yet known for determining them with any degree of accuracy.

Let G, S, O be the centres of gravity, gyration, and oscillation, as determined by the foregoing propositions;

and let P be the point where the ball strikes the face of the pendulum, tas

momentum of which, or the product of its weight and velocity, is expressed by the force f, acting at P, in the foregoing propositions.

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the velocity of the ball,

u = the velocity of the point of impact P,

c = chord of the arc described by the point O.

By Prop. 30, if the mass p be placed all at S, the pendulum will receive the KS2 same motion from the blow in the point P; and as KP': KS' :: p: KP · p or

por

go

P, (prop. 29) the mass which being placed at P, the pendulum will still receive the same motion as before. Here then are two quantities of matter,

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namely, b and p, the former moving with the velocity v, and striking the latter at rest; to determine their common velocity u, with which they will jointly proceed forward together after the stroke. In which case, by the law of the impact of non-elastic bodies, we have p+bb:: vu, and there

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22

bii+gop fore v bii point P, and the known dimensions and weights of the bodies.

u the velocity of the ball in terms of u, the velocity of the

I ut now to determine the value of u, we must have recourse to the angle through which the pendulum vibrates; for when the pendulum descends down again to the vertical position, it will have acquired the same velocity with which it began to ascend, and, by the laws of falling bodies, the velocity of the centre of oscillation is such, as a heavy body would acquire by freely falling through the versed sine of the arc described by the same centre 0. Eut the chord of that arc is c, and its radius is o; and, by the nature of the circle, the chord is a mean proportional between the versed sine and diameter, therefor 20: cc: the versed sine of the arc described by O. Then by the laws

CC

20'

Сс

2a

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of falling bodies, √1671⁄2 : √ 20:32: C, the velocity acquired by the point O in descending through the arc whose chord is c, where a = 161⁄2 feet:

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Then, by substituting this value for u, the velocity of the ball, before found,

becomes v

bii + yop
bio

2a

So that the velocity of the ball, is di

rectly as the chord of the arc described by the pendulum in its vibration. 66. SCHOLIUM. In the foregoing solution, the change in the centre of oscillation is omitted, which is caused by the ball lodging in the point P. But the allowance for that small change, and that of some other small quantities, may be seen in my Tracts, where all the circumstances of this method are treated at full length.

67. For an example in numbers of this method, suppose the weights and dimensions to be as follow: namely,

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Therefore 656.56 × 2·1337, or 1401 feet, is the velocity, per second, with which the ball moved, when it struck the pendulum.

OF HYDROSTATICS.

1. HYDROSTATICS is the science which treats of the pressure, or weight, and equilibrium of water, and other fluids, especially those that, are non-elastic, 2. A fluid is elastic, when it can be reduced into a less bulk by compres sion, and which restores itself to its former bulk again when the pressure is removed; as, air. And it is non-elastic, when it is not compressible or expan sible; as, water, &c.

PROP. I

3. If any part of a fluid be raised higher than the rest, by any force, and then left to itself; the higher parts will descend to the lower places, and the fluid will not rest, till its surface be quite even and level.

FOR, the parts of a fluid being easily moveable every way, the higher parts will descend by their superior gravity, and raise the lower parts, till the whole coms to rest in a level or horizontal plane.

Corol. 1. Hence, water which communicates

with other water, by means of a close canal or pipe, will stand at the same height in both places. Like as water in the two legs of a syphon.

Corol. 2. For the same reason, if a fluid gravitate towards a centre; it will dispose itself into a spherical figure, the centre of which is the centre of force. Like as the sea in respect of the earth.

PROP. II.

4. When a fluid is at rest in a vessel, the base of which is parallel to the horizon; equal parts of the base are equally pressed by the fluids.

FOR, upon every equal part of the base there is an equal column of the fluid supported by it. And, as all the columns are of equal height, by the last proposition, they are of equal weight, and therefore they press the base equally; that is, equal parts of the base sustain an equal pressure.

266. Corol. 1. All parts of the fluid press equally at the same depth.

For, if a plane parallel to the horizon be conceived to be drawn at that depth; then, the pressure being the same in any part of that plane, by the proposition, therefore the parts of the fluid, instead of the plane, sustain the same pressure at the same depth.

267. Corol. 2. The pressure of the fluid at any depth, is as the depth of the fluid.

For the pressure is as the weight, and the weight is as the height of the fluid.

PROP. III.

5. When a fluid is pressed by its own weight, or by any other force; at any point it presses equally, in all directions whatever.

THIS arises from the nature of fluidity, by which it yields to any force in any direction. If it cannot recede from any force applied, it will press against other parts of the fluid in the direction of that force. And the pressure in all directions will be the same. For if it were less in any part, the fluid would move that way, till the pressure were equal every way.

Corol. 1. In a vessel containing a fluid; the pressure is the same against the bottom, as against the sides, or even upwards, at the same depth.

Corol. 2. Hence, and from the last pro

position, if ABCD be a vessel of water, and there be taken, in the base produced, DE to represent the pressure at the bottom; joining AE, and drawing any parallels to the base, as FG, HI; then shall FG represent the pressure at the depth AG, and HI the pressure at the depth AI, and so on; because the parallels, FG, HI, ED, by sim. triangles, are as the depths, AG, AI, AD; which are as the pressures, by the proposition.

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And hence the sum of all the FG, HI, &c. or area of the triangle ADE, is as the pressure against all the points G, I, &c. that is, against the line AD. But as every point in the line CD is pressed with a force as DE, and that thence the pressure on the whole line CD is as the rectangle ED. DC, while that against the side is as the triangle ADE or AD. DE; therefore the pressure on the horizontal line DC, is to the pressure against the vertical line DA, as DC to DA. And hence, if the vessel be an upright rectangular one, the pressure on the bottom, or whole weight of the fluid, is to the pressure against

one side, as the base is to half that side. And therefore the weight of the fluid is to the pressure against all the four upright sides, as the base is to half the upright surface. And the same holds true also in any upright vessel, whatever the sides be, or in a cylindrical vessel. Or, in the cylinder, the weight of the fluid, is to the pressure against the upright surface, as the radius of the base is to double the altitude.

Moreover, when the rectangular prism becomes a cube, it appears that the weight of the fluid on the base, is double the pressure against one of the upright sides, or half the pressure against the whole upright surface.

Corol. 3. The pressure of a fluid against any upright surface, as the gate of a sluice or canal, is equal to half the weight of a column of the fluid these base is the surface pressed, and its altitude the same as the altitude of shat surface.

For the pressure on a horizontal base equal to the upright surface, is equal that column; and the pressure on the upright surface is but half that on the base, of the same area.

So that, if b be the breadth, and d the depth of such a gate or upright surface; then the pressure against it, is equal to the weight of the fluid whose magnitude is bď2 = §AB. AD'.

If the fluid be water, a cubic foot of which weighs 1000 ounces, or 62} pounds; and if the depth AI be 12 feet, the breadth AB 20 feet; then the content, or AB. AD is 1440 feet; and the pressure is 1,440,000 ounces, or 90,000 pounds, or 40 tons weight nearly.

PROP. IV.

6. The pressure of a fluid, on the base of the vessel in which it is contained, is as the base and perpendicular altitude; whatever be the figure of the vesse

that contains it.

A

B

If the sides of the base be upright, so that it be a prism of a uniform width throughout, then the case is evident; for then the base supports the whole fluid, and the pressure is just equal to the weight of the fluid.

b B

But if the vessel be wider at top than bottom; then the bottom sustains, or is pressed by, only the part contained within the upright lines aC, bD; because the parts ACa, BDb are supported by the sides AC, BD; and those parts have no other effect on the part abDC than keeping it in its position, by the lateral pressure against aC and b1), which does not alter its perpendicular pressure downwards. And thus the pressure on the bottom is less than the weight of the contained fluid And if the vessel be widest at bottom; then the bottom is still pressed with a weight which is equal to that of the whole upright column ABDC. For, as the parts of the fluid are in equilibrio, all the parts have an equal pressure at the same depth; so that the parts within Cc and d D press equally as those in cd, and therefore equally the same

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AB

b

e d

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