Now considering AD as a lever, fulcrum D, (see fig. on p. 25), that is acted on by the forces AB, AC applied at A, since we have Force in the line AB: force in line AC :: AB: AC :: CD: BD, since BC is a parallelogram, :: DN: DM, .. the two forces acting on the lever AD are inversely as the perpendiculars from the fulcrum on their lines of action, and therefore the lever will be kept at rest about D by them; wherefore the lever will also be kept at rest by the Resultant of those forces, which produces the same effect as they do when they act at the same point and at the same instant. This Resultant therefore must act in the line AD, for it keeps the lever at rest, which it could not do were it to act at A and make any angle with the lever AD on either side of it. Q. E. D. (2) Again: Having shewn that the resultant of AB and AC acts in the line of the diagonal, next to prove that the diagonal represents it in magnitude as well as in direction. Produce DA, and suppose a force AE to be taken equal and opposite to the Resultant of AB and AC. The effects of AB and AC will now be counteracted by AE, and the point A, which is acted on by the three forces AB, AC, and AE, will remain at rest. Whatever be the effect therefore that is produced by the joint action of AE and AC, it is counteracted by AB; that is, AB must be equal and opposite to the Resultant of AE and AC. Complete the parallelogram EC, and draw the diagonal AF. By the first part of the Prop. AF is the line which the resultant of AE and AC acts in; therefore the force AB must be equal and opposite to that resultant, because it destroys its effect; and therefore AF must be in the same straight line with AB. = : Wherefore EC and FD are parallelograms; and since AE FC AD, the Resultant of AB and AC (which is equal and opposite to AE) will be properly represented in magnitude by AD the diagonal of the parallelogram of which AB and AC are the sides. Wherefore, "If the adjacent sides, &c." Q. E. d. 29. PROP. IX. If three forces represented in magnitude and direction by the three sides. of a triangle [when taken in order], act on a point, they will keep it at rest. Let the sides AB, BC, CA, taken in order *, of the triangle ABC, repre sent in magnitude and direction (see Art. 15), three forces that act on A the point 4; They will keep 4 at rest. Complete the parallelogram BD. B * By the expression "taken in order" is meant, that if ABC be the triangle and AB be one of the forces, BC (and not CB) is the next, and CA (not AC) is the third; so that in describing the forces we proceed regularly round the triangle, from A to B, from B to C, and from C to A again. Then AD is parallel and equal to BC, and will therefore properly represent in magnitude and in line of action (see Art. 15) the force which acts at the point in the direction BC. Now the forces AB and AD acting at A will produce a resultant AC. Prop. vIII. If therefore a force CA act at A, the force AC will be counteracted, and the point A will remain at rest; Wherefore, if three forces, represented in magnitude and direction by AB, AD, CA,-(or, which is the same thing, if they be represented by the three sides AB, BC, CA taken in order of the triangle ABC)—act on a point A, they will keep it at rest. Q. E. D. 27 CHAPTER IV. MECHANICAL POWERS. 30. THE MECHANICAL POWERS are certain instruments by which a lesser weight, called the Power, may be made to balance a greater, called the Weight. We are now going to investigate the properties of three of them, which bear the names of the Wheel and Axle, the Pulley, and the Inclined Plane. They will be supposed to be rigid bodies and without weight.* 31. THE WHEEL AND AXLE consists of a cylinder (or Axle) AB that terminates in another cylinder (or Wheel) CD of greater base. The two cylinders have a com mon axis EF which is supported E in a horizontal position. The (Power) P is a heavy body that hangs freely by a string coiled C B W round the Wheel, and the Weight (W) is another body hanging by a string that is coiled round the Axle and tending to turn the machine round in an opposite direction. PROP. X. There is an equilibrium on the Wheel and Axle when the Power is to the Weight as the radius of the Axle is to the radius of the Wheel. The other Mechanical Powers are the Lever, the Toothed Wheel, the Wedge, and the Screw. A Suppose the strings that the Power and the Weight hang by to be in the planes of the circles HD and AG, which are those ends of the two cylinders that lie in the same plane. Since the cylinders have a common axis, their circular ends have a common center; let O be it, and join O with M and N the points in which the strings leave (and therefore touch) the circumferences of the circles. P M B N W OM and ON are therefore perpendicular to MP and NW the lines in which the Power and the Weight act. Now since P and W acting perpendicularly on the arms OM and ON balance each other, we have, supposing MON a lever moveable round O as fulcrum, P: W:: ON: OM :: radius of axle: radius of wheel. When therefore this ratio of the Power to the Weight is fulfilled on the Wheel and Axle, there will be equilibrium. Q. E. D. [COR. Since MO and NO are both perpendicular to vertical lines and also pass through the same point 0, MON is a horizontal straight line.] 32. DEF. THE PULLEY consists of a grooved wheel moveable round an axis whose ends are fixed in a frame called the Block. When the Pulley is used as a Mechanical Power, a string, which is fastened at one end, is passed round part of the circumference of the wheel, and the Power is applied at the other end of the string. The Weight is attached to the Block. |