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forces, plane, direction, parallel, decomposed, lines and perpendicular

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FORCE, Direction Te, the straight line which it tends to make a body describe.

FoncEs, Composition of. If two forces he conceived to act on a material point, it is evident that if they both act in the same direction, they will mutually increase each other's effect ; but if they act in opposite directions, the point will move only in consequence of their difference, and it would remain at rest if the forces were equal. If the directions of the two forces make an angle with each other, the resulting force will take a mean direction ; and it Can be demonstrated geometrically, that if. reckoning from the point of intersection of the two directions of the forces, we take on these directions straight lines to represent them, and then form a parallelo gram with such lines, its diagonal will their result ing force, both as to its direction and magnitude. The resulting force thus determined, which likewise represents the velocity of the moving point, may therefore be substituted as a force equivalent to the two component flirees ; and reciprocally, for any force whatever, we may substitute any two forces. which according to this rule would compose it. Hence we see that any force whatever may be decomposed into any two forces, parallel to two axes situated in the same plane, and perpendicular to each other. To effect this, it is only necessary to draw from the first extremity of the line representing, the three, to other lines parallel to the axis. and to form with such lines a rectangle, whose diagonal will he the force required to be decomposed. The two sides of this rectangle, or parallelogram, will represent the forces into which the given force may be decomposed, parallel to such axis. It' the force be inclined to a plane in position, a line in its direction may be taken to represent it. having one of its extremities on the surface of the plane. and the perpendicular falling from the other extremity will be the primitive force decomposed in the direction perpendicular to the plane. The straight line, which in the plane joins the other extremity of the line representing the force with the perpendicular (or the orthographic projection of the line of the plane) will repre sent the primitive force decomposed, parallel to the plane.

This second partial force may itself be decomposed into two others, parallel to two axes in the same plane, perpendicu lar to each other. Thus we see that every force may be decomposed into three others, parallel to three axes perpen dicular to each other; which axes are termed rectangular co-ordinates.

Hence we have a very simple mode of obtaining the resulting force of any number of fm-ces supposed to act on a material point. This method was first adopted by Maclaurin. followed by La Grange, in the .11t;clianique .zinalgtigne, and also by La Place in the Nechanique Celeste, By decompos ing each of these forces into three others, parallel to the given axes in position, and perpendicular to each other, we have all the forces parallel to the same axis reduced to one single force, which latter will be equal to the sum of the forces act ing in the same direction. minus the sum of those acting in a contrary direction : so that the point will he acted on by three forces perpendicular to ea•lbother. From the point of intersection, or origin of the eo•ordinates. take three right lines to represent them in each of their directions, and on such lines form a rectangular parallelopipedon, and the diagonal of this solid will represent the quantity and direc tion of the resulting force of all the forces acting on the point.

The principle of the composition of forces is of the most extensi% e utility in mechanies. and is in itself sufficient for dete)mining the law of equilibrium in every case. Thus, if we successively compose all the forces, taking than by two's, and then take th6 result as a new fintee, we obtain one that is equivalent to all the rest, and which, in case of equilibrium, nmst equal 0. when the system under consideration has no fixed point ; lint it' the conditions of the problem insist on an immoveable point, the resulting force must necessarily pass through it.

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