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power is increased in the proportion of the radius of the screw to the radius of the power, or length of the handle, or as their circumferences. Therefore, finally, the power is to the pressure, as the distance of the threads, is to the circumference described by the power. 85. Corol. When the screw is put in motion; then the power is to the weight which would keep it in equilibrio, as the velocity of the latter is to that of the former ; and hence their two momenta are equal, which are produced by multiplying each weight or power by its own velocity. So that this is a general property in all the mechanical powers, namely, that the momentum of a power is equal to that of the weight which would balance it in equilibrio; or that each of them is reciprocally proportional to its velocity.
86. 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 summo 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 × 3.1416 or 150; nearly; and the distance of the threads being 4 of an inch; therefore the power is to the pressure, as 1 to 603; ; but the power is equal to 150lb ; theref. as 1 : 603} :: 150 : 90480; and consequently the pressure at D is equal to a weight of 90.480 pounds, independent of friction.
87. Again, if the endless screw AB be turned by a handle Ac of 20 inches, the threads of the screw being distant half an inch each ; and the screw turns a toothed wheel E, whose pinion L 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 what weight the man will be able to raise, working at the handle c ; supposing
the diameters of the wheels
88. Upon the same principle the advantage of any other combination of the mechanical powers may be computed : allowance, however, being always to be made for stiffness of cords, friction, and other causes of resistance.
ON THE CENTRE OF GRAVITY.
89. The CENTRE of GRAvity of a body, or of a system of bodies, is a certain point within it, or connected with it, on which the body being freely suspended, it will rest in any
position, and that centre will always tend to descend to the lowest place to which it can get, when it is not the point of suspension.
90. Prop. 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 out of the base, the body will not rest in that position, but will fall down.
For, if cB be the perp. from the centre of gravity c, within the base : then the body cannot fall over towards A ; because, in turning on the point A, the centre of gravity c would describe an arc which would rise from C to E; contrary to the nature of that centre, which only rests permanently 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 out of the base, as cb ; then the body will fall over on that side : because, in turning on the point a, the centre c descends by describing the descending arc ce.
91. 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.
92. 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 may, in many inquiries, 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.
93. Corol. 3. From the property which the centre of gravity has, of tending to descend 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 B in the plumb line. Mark the line AB on Ö it. Then hang the body up by any other & point D, with a plumb line DE, which will A. also pass through the centre of gravity, for the same reason as before ; and therefore that centre must be at c where B the two plumb lines cross each other.
94. 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 Z-
95. 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 indicate the place of the centre itself. The place of the centre of gravity may be investigated, from its analogy to the centre of parallel forces; but the following method is adopted here, as in some respects easier of comprehension. 96. Prop. The common centre of gravity c of any two bodies A, B, divides the line joining their respective centres, into two parts, which are reciprocally as the bodies. 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, 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.
97. 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.
98. Corol. 2. Hence also, ca. A = ce. B; or the two products are equal, which are made by multiplying each body into its distance from the centre of gravity.
99. 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 weight 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. This should be noted as a principle of extensive application.
100. Corol. 4. Since the effect of any body to turn -4 so a lever about the fixed X B C D. E. F. 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 + ch. 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 multiplying each body into its distance from that centre. And if several bodies be in equilibrio on any straight lever, then the prop is in the centre of gravity.
101. Corol. 5. And though the bodies be not situated in a straight line, but scattered aboutin any promiscuous manner, the same property as in the last corollary still holds true, if perpendiculars to any line whatever afbe drawn through the several bodies, and their common centre of gravity, namely, that ca. A + ch. B = ca. D + ce. E + cf. F. For