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its opposite force B. Consequently the three forces, A, B, c, are proportional to DE, CE, CD, the three lines parallel to the directions in which they act.
31. Corol. 2. Because the three sides co, cE, DE, are proportional to the sines of their opposite angles E, D, c ; therefore the three forces, when in equilibrio, are proportional to the sines of the angles of the triangle made of their lines of direction ; namely, each force proportional to the sine of the angle made by the direction of the other two.
32. Corol. 3. The three forces, acting against, and keeping one another in equilibrio, are also proportional to the sides of any other triangle made by drawing lines either perpendicular to the directions of the forces, or forming any given angle with those directions. For such a triangle is always similar to the former, which is made by drawing lines parallel to the directions; and therefore their sides are in the same proportion to one another.
33. Corol. 4. If any number of forces be kept in equilibrio by their actions against one another; they may be all reduced to two equal and opposite ones.—For, any two of the forces may be reduced to one force acting in the same plane ; then this last force and another may likewise be reduced to another force acting in their plane : and so on, till at last they be all reduced to the action of only two opposite forces ; which will be equal, as well as opposite, because the whole are in equilibrio by the supposition.
34. Corol. 5. If one of the forces, as c, be a weight, which is sustained by two strings drawing in the directions DA, DB : then the force or tension of the string AD, is to the weight c, or tension of the string DC, as DE to DC ; and the force or tension of the other string BD, is to the weight c, or tension of cd, as ce to CD.
35; Corol. 6. Since in any triangle coe we have, by the principles of trigonometry,
DC* = DE” + Ec" -- 2DE . Eccos. DEc, it follows, that if F, f, be two forces that act simultaneously in directions, which make an angle A, then we may find the tmagnitude of the resultant, R, by the equation R = y (F" +f + 2*fcos. A).
36. Remark.-The properties, in this proposition and its corollaries, hold true of all similar forces whatever, whether they be instantaneous or continual, or whether they act by percussion, drawing, pushing, pressing, or weighing; and are of the utmost importance in mechanics and the doctrine of forces.
37. If three forces, whose directions concur in one point, are represented by the three contiguous edges of a parallelopiped, their resultant will be represented, both in magnitude and direction, by the diagonal drawn from the point of concourse, to the opposite angle of the parallelopiped.
The demonstration of this is left for the exercise of the student.
38. PRop. To find the resultant of several forces concurring in one point, and acting in one plane. 1st. Graphically.—Let, for example, four forces, A, B, C, D, act upon the point P, in magnitudes and directions represented by the lines PA, PB, Pc, pd. From the point A draw c Ab parallel and equal to f c’ PB ; from b draw be parallel and equal to pc ; from / c draw cd parallel and equal AZ-B- & to PD ; and so on, till all the forces have thus been o C brought into the construc- P43 tion. Then join Pd, which a D 3 y § will represent both the magnitude and the direction of the required resultant. This is, in effect, the same thing as finding the resultant of two of the forces A and B; then blending that resultant
with a third force c.; their resultant with a fourth force D ; and so on.
2d By computation. Drawing the lines Aa, Ab', &c. respectively parallel and perpendicular to the last force pd we have dó = Aa + bb'+ cc = A sin. APD + B sin. BPD+c sin. CPD. rö-ra-i-a8+8)+y}=A cos. App-HB cos. BPD+ccos.cpd+d
tan drö = o - - - - - Pd = v(p3' + dā’) = Pösec. drö. The numerical computation is best effected by means of a table of natural sines, &c.
39. Remark. Connected with this subject is the doctrine of moments; for an elucidation of which, however, the student should consult some of the books written expressly on mechanics, as those by Marrat, Gregory, or Poisson.
THE MECHANICAL POWERS, &c.
40. WEIGHT and Power, when opposed to each other, sigs nify the body to be moved, and the body that moves it; or the patient and agent. The power is the agent, which moves, or endeavours to move, the patient or weight. 41. Machine, or Engine, is any mechanical instrument contrived to move bodies. And it is composed of the mechanical powers. 42. Mechanical powers, are certain simple instruments, commonly employed for raising greater weights, or overcom. ing greater resistances, than could be effected by the natural strength without them. These are usually accounted six in number, viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw. 43. Centre of Motion, is the fixed point about which a body moves. And the Axis of Motion, is the fixed line about which it moves. 44. Centre of Gravity, is a certain point, on which a body being freely suspended, it will rest in any position.
OF THE LEVER.
45. A Lever is any inflexible rod, bar, or beam, which serves to raise weights, while it is supported at a point by a fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds of levers.
47. A Lever of the Second kind has the weight between the power and the prop. Such as oars, rud- 9–W. #P ders, cutting knives that 4 are fixed at one end, &c.
48. A Lever of the Third kind has the power between the weight and the prop. Such as tongs, the bones and muscles of
animals, a man rearing a iP ladder, &c. 4 3 l C
49. A Fourth kind is some
times added, called the Bended X C Lever. As a hammer drawing a nail.
50. In all these instruments the power may be represented by a weight, which is its most natural measure, acting downward; but having its direction changed, when neces. sary, by means of a fixed pulley.
51. Prop. When the weight and power keep the lever in equilibrio, they are to each other reciprocally as the distances of their lines of direction from the prop. That is, p : w :: cd : ce; where cd and cE are perpendicular to wo and Ao, the directions of the two weights, or the weight and power w and A. For, draw cf parallel to Ao, and cB parallel to wo: Also, join co, which will be the direction of the pressure on the prop c ; for there cannot be an equilibrium unless the directions of the three forces all meet in, or tend to, the same point, as o. Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle cho or cro, drawn in the direction of those forces ; therefore - - - - - P : W - : CF : Fo or CB, But, because of the parallels, the two triangles cof, cEB are equiangular, therefore - - - CD : CE. : : CF : CB. Hence, by equality, - - P : W* : : CD : CE. That is, each force is reciprocally proportional to the distance of its direction from the fulcrum. Another proof might easily be made out from art. 25, on parallel forces; but it will be found that this demonstration will serve for all the other kinds of levers, by drawing the lines as directed.
52. Corol. 1. When the angle A is = the angle w, then is cd : CE. : : cw : CA:: P : w. Or when the two forces act perpendicularly on the lever, as two weights, &c.; then, in case of an equilibrium, d coincides with w, and E with P ; consequently then the above proportion becomes also P : w :: cw : cA, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces.
53. Corol. 2. If any force P be applied to a lever at A ; its effect on the lever, to turn it about the centre of motion c, is as the length of the lever cA, and the sine of the angle of direction cAE. For the perp. CE is as cA X sin. Z A.
54. Corol. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power into the distance of its direction, is equal to the product of the weight into the distance of its direction. That is, p X ce = w X cD.
55. Corol. 4. If the lever, with the weight and power fixed to it, be made to move about the centre C ; the momentum of the power will be equal to the momentum of the weight ; and their velocities will be in reciprocal proportion to each other. For the weight and power will describe circles whose radii are the distances co, cE ; and since the circumferences or spaces described are as the radii, and also as the velocities, therefore the velocities are as the radii cd, ce; and the momenta, which are as the masses and velocities, are as the masses and radii; that is, as P × ce and w X cd, which are equal by cor. 3.
56. Corol. 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly ; then, of these three, the power, weight, and pressure on the prop. any one is as the distance of the other two.
57. Corol. 6. If A B C T) E. several weights P, Q, A | R, s, act on a straight lever, and keep it in equilibrio; then the sum of the products P Q. TR. S on one side of the prop. will be equal to the sum on the other side, made by multiplying each weight by its distance; namely, (p x Ac) + (a X BC) = (R × Dc) + (s X Ec),