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203. Hence we can easily compute the force of any machine turned by a screw. Let the annexed figure represent a press driven by a screw, whose threads are each a quarter of an inch asunder; and let the screw be turned by a handle of 4 feet long, from A to B; then, if the natural force of a man, by which he can lift, pull, or draw, be 150 pounds; and it be required to determine with what force the screw will press on the board at D, when the man turns the handle at a and B, with his whole force. Then the diameter AB of the power being 4 feet, or 48 inches, its circumference is 48 x 3.1416 or 1504 nearly; and the distance of the threads being of an inch; therefore the power is to the pressure, as 1 to 605}; but the power is equal to 150lb; theref. as 1 : 6031 :: 150 : 90480; and consequently the pressure at d is equal to a weight of 90480 pounds, independent of friction.
204. Again, if the endless screw AB be turned by a ha dle Ac of 20 inches, the threads of the screw being distant half an inch each; and the screw turns
I a toothed wheel ę, whose pinion turns another wheel F, and the pinion M of this another wheel G, to the pinion or barrel of which is hung a weight w; it is required to determine
hat weight the man will be able to raise, working at the handle c; supposing the diameters of the wheels to be 18 inches, and those of the pinions and barrel 2 inches; the teeth and pi. nions being all of a size.
Here 20 x 3•1416 x 2 = 125.664, is the circumference
of the power:
And 125.664 to 1, or 251.328 to 1, is the force of the screw alone.
Also, 18 to 2, or 9 to 1, being the proportion of the wheels to the pinions; and as there are three of them, therefore 93 to 1), or 729 to 1, is the power gained by the wheels.
Consequently 251.328 x 729 to 1, or 1832183 to 1 nearly, is the ratio of the power to the weight, arising from the advantage both of the screw and the wheels.
But the power is 1501b; therefore 150 x 183218%, or 27482716 pounds, is the weight the man can sustain, which is equal to 12269 tons weight.
But the power has to overcome, not only the weight, but also the friction of the screw, which is very great, in some cases equal to the weight itself, since it is sometimes sufficient to sustain the weight, when the power is taken off.
ON THE CENTRE OF GRAVITY. 205. THE Centre of GRAVITY of a body, is a certain point within it, on which the body being freely suspended, it will rest in any position; and it will always descend to the lowest place to which it can get, in other positions.
PROPOSITION XXXVIII. 206. If a Perpendicular to the Horizon, from the Centre of
Gravity of any Body, fall Within the Base of the Body, it will rest in that Position ; but if the Perpendicular
fall Without the Base, the Body will not rest in that Position, but will tumble down.
FOR, if co, be the perp: from the centre of gravity C, within the base : then the body cannot fall over towards E A; because, in turning on the point A, the centre of gravity
А C would describe an arc which would rise fromcto E; contrary to the nature of that centre, which only rests when in the lowest place. For the same reason, the body will not fall towards D. And therefore it will stand in that position.
But if the perpendicular fall without the base, as cb; then the body will tumble over on that side: because, in turning on the point a, the centre c descends by describing the descending arc ce.
207. Corol. 1. If a perpendicular, drawn from the centre of gravity, fall just on the extremity of the base; the body may stand; but any the least force will cause it to fall that way. And the nearer the perpendicular is to any side, or the narrower the base is, the easier it will be made to fall, or be pushed over that way; because the centre of gravity has the less height to rise : which is the reason that a globe is made to roll on a smooth plane by any the least force. But the nearer the perpendicular is to the middle of the base, or the broader the base is, the firmer the body stands.
208. Corol. 2. Hence if the centre of gravity of a body be supported, the whole body is supported. And the place of the centre of gravity must be accounted the place of the body; for into that point the whole matter of the body may be supposed to be collected, and therefore all the force also with which it endeavours to descend.
209. Corol. 3. From the property which the centre of gravity has, of always descending to the lowest point, is derived an easy mechanical method of finding that centre.
Thus, if the body be hung up by any point A, and a plumb line AB be hung by the same point, it will pass through the centre of gravity; because that centre is not in the lowest point till it fall in the plumb line. Mark the line ab on it. Then hang the body up by any other point D, with a plumb line De, which will also pass through the centre of gravity, for the same reason as before; and therefore that centre must be at c where the two plumb lines cross each other.
210. Or, if the body be suspended by two or more cords GF, GH, &c, then a plumb line from the point G will cut the body in its centre of gravity c.
211. Likewise, because a body rests when its centre of gravity is supported, but not else; we hence derive another easy method of finding that centre mechanically. For, if the body be laid on the edge of a prism, or over one side of a table, and moved backward and forward till it rest, or balance itself; then is the centre of gravity just over the line of the edge. And if the body be then shifted into another position, and balanced on the edge again, this line will also pass by the centre of gravity; and consequently the intersection of the two' will give the centre itself.
212. The common Centre of Gravity c of any two Bodies A, B,
divides the Line joining their Centres, into two Parts, which
That is, AC : BC :: B; A.
FOR, if the centre of gravity c be supported, the two bodies A and B will be supported, and will rest in equilibrio. But,
Ć by the nature of the lever, when two bodies are in equilibrio about a fixed point c, they are reciprocally as their distances from that point; therefore A :B:: CB : CA.
213. Corol. 1. Hence AB : AC :: A + B : B; or, the whole distance between the two bodies, is to the distance of either of them from the common centre, as the sum of the bodies is to the other body.
214. Corol. 2. Hence also, CA . A = 6B . B; or the two products are equal, which are made by multiplying each body by its distance from the centre of gravity.
215. Corol. 3. As the centre c is pressed with a force equal to both the weights A and B, while the points A and B are each pressed with the respective weights A and B. Therefore, if the two bodies be both united in their common centre c, and only the ends A and B of the line AB be supported, each will still bear, or be pressed by the same weights A and B as before. So that, if a weight of 100lb. be laid on a bar at C, supported by two men at A and B, distant from c, the one 4 feet, and the other 6 feet; then the nearer will bear the weight of 60lb, and the farther only 40lb. weight.
216. Corol. 4. Since the effect of any body to turn
D a lever about the fixed point c, is as that body and as its distance from that point; therefore, if c be the common centre of gravity of all the bodies A, B, D, E, F, placed in the straight line AF; then is CA . A + CB . B = CD.D+ CE . E + CF.F; or, the sum of the products on one side, equal to the sum of the products on the other, made by mul. tiplying each body by its distance from that centre. And if several bodies be in equilibrium on any straight lever, then the prop is in the centre of gravity. 217. Corol. 5. And thought
BG the bodies be not situated in
f a straight line, but scattered about in any promiscuous manner,the same property as in the last corollary still hold true, if perpendiculars to any line whatever af be drawn through the several bodies, and their common centre of gravity, namely, that ca . A + cb = cd. D + ce. E f cf. F. For the bodies have the same effect on the line af, to turn it about the point c, whether they are placed at the points a, b, d, e, f, or in any part of the perpendiculars Aa, Bb, dd, Ee, sf.
218. If there be three or more Bodies, and if a Line be drawn
from any one Body D to the Centre of Gravity of the rest c; then the Common Centre of Gravity E of all the Bodies, divides the line cd into two parts in E, which are Reciprocally Proportional as the Body D to the Sum of all the other Bodies.
That is, CE 1 ED ::D: A + B &c. For, suppose the bodies A and B to be collected into the common
9 centre of gravity C, and let their sum be called s. Then, by the last prop. CE : ED::D:s or A + B &c.
219. Corol. Hence we have a method of finding the common centre of gravity of any number of bodies ; namely, by first finding the centre of any two of them, then the centre of that centre and a third, and so on for a fourth, or fifth, &c.