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the splitting and raising the largest and hardest rocks, the raising and lifting the largest ship, by driving a wedge below it, which a man can do by the blow of a mallet; and thus it appears that the small blow of a hammer, on the back of a wedge, is incomparably greater than any mere pressure, and will overcome it.

OF THE SCREW.

47. THE Screw is one of the six mechanical powers, chiefly used in pressing or squeezing bodies close, though sometimes also in raising weights.

The screw is a spiral thread or groove cut round a cylinder, and everywhere making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder is to the distance between two threads of the screw; as is evident by considering, that, in making one round, the spiral rises along the cylinder the distance between the two threads.

PROP. XI.

48. The force of a power applied to turn a Screw round, is to the force with which it presses upwards or downwards, setting aside the friction, as the distance between two threads is to the circumference where the power is applied.

THE Screw being an inclined plane, or half wedge, whose height is the distance between two threads, and its base the said circumference; and the force in the horizontal direction, being to that in the vertical one, as the lines perpendicular to them, namely, as the height of the plane, or distance of the two threads, is to the base of the plane, or circumference at the place where the power is applied; therefore the power is to the pressure, as the distance of two threads is to that circumference.

Corollary. 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.

49. SCHOLIUM.-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 that the screw is 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 re

quired 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. The diameter AB of the power being 4 feet or 48 inches, its circumference is 48 X 31416 or 150% nearly; and the distance of the threads being of an inch; therefore the power is to the pressure, as 1 to 603}: but the power is equal to 150 lb.; therefore as 1: 603 :: 150 90,480; and conse

quently the pressure at D is equal to a weight of 90,480 pounds, independer. of friction.

50. 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 turn 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 to be 18 inches, and those of the pinions and barrel 2 inches; the teeth and pinions being all of a size.

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Here 20 x 31416 X 2 = 125.364, is the circumference of the power.

And 125 664 to, 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 of both the screw and the wheels.

But the power is 150 pounds; and therefore 150 X 1832183, 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.

EXAMPLES ON THE PRINCIPLES OF THE MECHANICAL POWERS.

ON THE LEVER.

51. Ex. 1. The arms of a bent lever are to each other as 4 to 5, and are inclined at an angle of 135°. The lever rests upon a fulcrum at its angular point, and weights are suspended from the extremities of the two arms, such that the shorter arm rests in a horizontal position; what is the ratio of the weights? Ans. 8 5/2 or 1:·8838835.

Ex. 2. The difference of the lengths of the arms of a lever is (a) inches; the same weight weighs (w) pounds at one end, and (w) ounces at the other; find the lengths of the arms.

a

Ans. and 3

4a 3'

Ex. 3. A lever three feet in length weighs 61b.; what weight on the shorter arm will balance 12lb. on the longer, the fulcrum being one foot from the end?

Ans. 27lb.

Ex. 4. The compound lever DK is composed of three levers of the first kind, DA, AB, BK, acting upon one another. The arms DC, CA of the first lever are respectively 8 and 6 inches; those of the second, AO, OB, are 12 and 2, and those of the third, BH, HK are 16 and 3; find the ratio of P, the power at D, to W, the weight suspended at K.

Ans. P: W:: 3 : 128.

Ex. 5. Suppose AB is a squared beam, or lever of oak, 30 feet long, each end being one foot square; what weight W at the end A would keep it in a horizontal position on a fulcrum C, 3 feet from that end, each cubic foot of the beam weighing 54lb.? Ans. 6480lb.

Ex. 6. AB is a uniform straight lever, 20 feet in length, and weighing 40lb.; and HBK, a flexible chain of the same length, and weight 130lb., is laid upon the lever in such a manner that it is kept in equilibrium on a fulcrum C, which is five feet from the end B; how much of the chain overhangs the end B?

30

Ans. 20

13

√26, or 8.233032 feet.

ON THE WHEEL AND AXLE.

52. Ex. 1. In a combination of four wheels and axles, each of the radii of the wheels is to each of the radii of the axles as 5 to 1; what power will balance a weight of 1875 pounds? Ans. 3 pounds.

Ex. 2. A power of 61b. keeps in equilibrium a weight of 240lb., by means of a wheel and axle: the diameter of the axle is 6 inches; what is the radius of the wheel? Ans. 10 feet.

Ex. 3. In a combination of wheels and pinions, the circumference of each pinion is applied to the circumference of the next wheel, and the ratios of the radii of the wheels and pinions are 2: 1, 22: 1; 23: 1, and so on. Find the number of wheels, when the power is to the weight as 1 : n.

43

Ans. The number of wheels may be found from the quadratic equation
2 log n
x2 + x =
where x number of wheels.
log 2

ON THE PULLEY.

Ex. 1. What power will sustain 40 pounds over five moveable pulleys?
Ans. 14lb.

Ex. 2. In a system of pulleys, where each pulley has a separate string passing over it, and fastened to the weight, P: W:: 1:63; what is the number of moveable pulleys?

Ans. 5.

Ex. 3. In the same system, the number of moveable pulleys is 3, and the weight of each pulley 2lb.; what weight will a power of 60lb. support?

Ans. 922lb.

ON THE INCLINED PLANE.

54. Ex. 1. A power of 1lb. acting parallel to a plane supports a weight of 2lb.; what is the inclination of the plane? Ans. 30°.

Ex. 2. Two weights are fastened to the ends of a thread which moves freely over a pulley, and the thread makes angles and 6 with the horizon when at rest; also one of the weights which is on a smooth plane is double of the other which hangs vertically; what is the inclination of the plane?

Ans. cot = 2 sec B

tan 6, where = augle of inclination.

Ex. 3. A weight of 40 pounds acting parallel to the length, sustains another of 56 pounds on an inclined plane whose base is 340 feet; find the height and length of the plane.

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55. Ex. 1. The distance between two contiguous threads of a screw is 2 inches, and the arm to which P is applied is 20 inches; find the ratio of P to W when there is an equilibrium. Ans. P: W:: 1: 62.832.

Ex. 2. What must be the distance between the threads in a screw, that a man exerting a force of 50lb. at the end of an arm 18 inches in length, may press with a force of ten tons? Ans. 25245 inches, or 4in., nearly.

OF THE CENTRE OF GRAVITY.

56. THE Centre of Gravity of a body, is a certain point within it, upon which the body being freely suspended, it will rest in any position; and it will descend to the lowest place to which it can get, in other positions.

PROP. XII.

57. 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.

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But if the perp. 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 centre arc Cc.

Corollary. 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 it stands.

Corollary. 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 with which it endeavours to descend.

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