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the intersection of these lines, will likewise pass through the centre of gravity. The same thing may be effected by laying the body on a table, till it is just ready to fall off, and then marking a line upon it by the edge of the table: this done in two positions of the body will in like manner point out the position of the centre of gravity.

When it is proposed to find the centre of gravity of the arch of a bridge, or any other structure, let it be laid down accurately to scale upon pasteboard; and the figure being carefully cut out, its centre of gravity may be ascertained by the preceding process.

9. If on any plane passing through the centre of gravity of a body, perpendiculars be let fall from each of its moleculæ, the sum of all the perpendicular distances on one side of the plane will be equal to the sum of all those on the other side. And a similar property obtains with regard to the common centre of gravity of a system of bodies.

10. The position, distance, and motion of the centre of gravity of any body is a medium of the positions, distances, and motions of all the particles in the body.

11. The common centre of gravity, or of position, of two bodies divides the right line drawn between the respective centres of the two bodies in the inverse ratio of their masses.

12. The centre of gravity of three or more bodies may, hence, be found, by considering the first and second as a single body equal to their sum and placed in their common centre of gravity, determining the centre of gravity of this imaginary body, and a third. These three again being conceived united at their common centre, we may proceed, in like manner, to a fourth and so on, ad libitum.

Or, if в, B', B", &c. denote the masses of any bodies, D, D', D', &c. the perpendicular distances of their respective centres of gravity from any line or plane: then, the distance, ▲, of their common centre of gravity from any line or plane, B D+B' D'+B" D" &c.

is found by this theorem: viz. ▲ =

B+ B+ B" &c.

13. If the particles or bodies of any system be moving uniformly and rectilineally, with any velocities and directions whatever, the centre of gravity is either at rest, or moves uniformly in a right line.

Hence if a rotatory motion be given to a body and it be then left to move freely, the axis of rotation will pass through the centre of gravity: for, that centre, either remaining at rest or moving uniformly forward in a right line, has no rotation.

Here too it may be remarked, that a force applied at the centre of gravity of a body, cannot produce a rotatory mo

14. The centre of gravity of a right line, or of a parallelogram, prism, or cylinder, is in its middle point; as is also that of a circle, or of its circumference, or of a sphere, or of a regular polygon; the centre of gravity of a triangle is somewhere in a line drawn from an angle to the middle of the opposite side; that of an ellipse, a parabola, a cone, a conoid, a spheroid, &c. somewhere in its axis. And the same of all symmetrical figures.

15. The centre of gravity of a triangle is the point of intersection of lines drawn from the three angles to the middles of the sides respectively opposite it divides each of those lines into two portions in the ratio of 2 to 1.

B

16. In a Trapezium. Divide the figure into two triangles by the diagonal A c, and find the centres of gravity E and F of these triangles; join E F, and find the common centre G of these two by this proportion, A B C ADCFG: E G, Or A B C D ADC EF: EG. Or, divide the figure into two triangles by a diagonal B D. find their centres of gravity; the line which joins them will intersect E F in G, the centre of gravity of the trapezium.

17. In like manner, for any other plane figure, whatever be the number of sides, divide it into several triangles, and find the centre of gravity of each; then connect two centres together, and find their common centre as above; then connect this and the centre of a third, and find the common centre of these; and so on, always connecting the last found common centre to another centre, till the whole are included in this process; so shall the last common centre be that which is required.

18. The centre of gravity of a circular arc is distant from the centre a fourth proportional to the arc, the radius, and the chord of the arc.

19. In a circular sector, the distance from the centre of

2 cr

the circle is ; where a denotes the arc, c, its chord, and

r the radius.

3 a

20. The centres of gravity of the surface of a cylinder, of a cone, and of a conic frustrum, are respectively at the same distances from the origin as are the centres of gravity, of the parallelogram, triangle, and trapezoid, which are vertical sections of the respective solids.

21. The centre of gravity of the surface of a spheric segment is at the middle of its versed sine or height.

22. The centre of gravity of the convex surface of a spherical zone, is in the middle of that portion of the axis of the sphere which is intercepted by the two bases of the zone.

23. In a cone, as well as any other pyramid, the distance from the vertex is of the axis.

24. In a conic frustum, the distance on the axis from the

centre of the less end, is 1 h.

3 R2 Rr+r2:
R2 + Rr + r2

where h the

height, R and r the radii of the greater and less ends.

25. The same theorem will serve for the frustum of any regular pyramid, taking R and r for the sides of the two ends. 26. In the paraboloid, the distance from the vertex is axis. 27. In the frustum of the paraboloid, the distance on the 2 R2+r2:

axis from the centre of the less end, ish.

2

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where

h the height, R and the radii of the greater and less ends.

Many other results are given in the first volume of my Mechanics. The preceding are selected as the most useful. The centre of gravity of the human body is always near the same place, viz. in the pelvis, between the hips, the ossa pubis, and the low

er part of the backbone. Elevating the arms or the legs will elevate the centre of gravity a little: still, it is always so placed that the limbs may move freely round it, and this centre moves much less than if it were in any other part of the body. If a man walked upon wooden legs,the centre of gravity of his body would describe portions of circles, as

Α

B

A B. If a man with two wooden legs were to run, the centre of gravity would describe portions of parabolas, as c D. But the flexibility of the joints and muscles of the human legs serves to take away the angles from these curves, and give a softer undulation, as E F.

The centre of gravity of a human body, is not precisely in the same place as that of the statue of a man: for, in the former, the substance is not, throughout, of the same density, in the latter it is.

SECTION III.-Mechanical Powers.

1. The simple machines of which the more complex machines are constituted, and which, indeed, are often employed separately, are called Mechanical Powers.

2. Of these we usually reckon six viz. the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. To these, however, is sometimes added the funicular machine, being that which is formed by the action of several powers, at different points of a flexible cord.

3. Weight and Power, when regarded as opposed to each other, signify the body to be moved, or the resistance to be overcome, and the body of force by which that is accomplished. They are usually represented by their initial letters, w and P.

Levers.

1. A lever is an inflexible bar, whether straight or bent, and supposed capable of turning upon a fixed, unyielding point called a fulcrum.

2. When the fulcrum is between the power and the weight, the lever is said to be of the first kind.

When the weight is between the power and the prop, the lever is of the second kind.

When the power is between the weight and the prop, or fulcrum, the lever is of the third kind.

The hammer lever, or the operation of a hammer in drawing a nail, is sometimes considered as a fourth kind.

F

F

F

B

P

B

W

W

W

B

3. In all these cases when there is an equilibrium, it is indicated by this general property, that the product of the weight into the distance at which it acts, is equal to the product of the power into the distance at which it acts: the distances being estimated in directions perpendicular to those in which the weight and power act respectively. Thus, in each of the three preceding figures,

P. AF=W. BF,

or the power and weight are reciprocally as the distances at which they act.

And if, in the first figure, for example, the arm AF were

4 times F B, 4 lbs. hanging at в would be balanced by 1 lb. at A. If AF were 5 times F B, 1 lb. at a would balance 5 lbs. at B; and so on.

4. If several weights hang upon a lever, some on one side of the fulcrum, some on the other, then there will be an equilibrium, when the sum of the products of the weights into their respective distances on one side, is equal to the several products of weights and distances on the other side.

5. When the weight of the lever is to be taken into the account, proceed just as though it were a separate weight suspended at its centre of gravity.

6. When two, three, or more levers act one upon another in succession, then the entire mechanical advantage which they supply, is found by taking, not the sum, but the product of their separate advantages. Thus, if the arms of three levers, acting thus in connexion, are as 3 to 1, 4 to 1, and 5 to 1, then the joint advantage is that of 3 x4x5 to 1, or 60 to 1 so that 1 lb. would, through their intervention, balance 60.

7. In the first kind of lever the pressure upon the fulcrum =P+w: in the other two it is pw.

8. Upon the foregoing principles depends the nature of scales and beams for weighing all bodies. For, if the distances be equal, then will the weights be equal also; which gives the construction of the common scales. And the Roman statera, or steel-yard, is also a lever, but of unequal arms or distances, so contrived that one weight only may serve to weigh a great many, by sliding it backwards and forwards to different distances upon the longer arm of the lever. the common balance, or scales, if the weight of an article when ascertained in one scale is not the same as its weight in the other, the square root of the product of those two weights will give the true weight.

In

9. From numerous examples of the power and use of the lever, one which shows its manner of application in the printing-presses of the late Earl Stanhope may be advantageously introduced.

In the adjoining figure, let ABCD be the general frame of the press, connected by the cross pieces N O, D C. E is a centre connected with the frame by the bars EN, ER, E O. To this centre are affixed a bar K D, and a lever E F, to which the hand is applied when the press is used.

A

N

D

R

L

KEM

H

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