If two or more forces are acting simultane-ously at a point ,the same effect can be produced by a single force of the proper size and direction. This single force, which is equivalent to the action of two or more forces, is called the resultant. Putting component forces together to find the resultant force is called composition of forces. (See fig. 2-1-2.) The vectors representing the forces must be added to find the resultant. Because a vector represents both magnitude and direction, the method for adding vectors differs from the procedure used for scalar quantities (quantities having only magnitude and no direc-tion). To find the resultant force when a force of 5 pounds and a force of 10 pounds are applied at a right angle to point A, refer to figure 2-1-2.

Figure 2-1-2.Composition of two right-angle forces.

The resultant force may be found as follows: 

Represent the given forces by vectors AB and AC drawn to a suitable scale. At points B and C draw dashed lines perpendicular to AB and AC, respec-tively.

From point A, draw a line to the point of intersection X, of the dashed lines. Vector AX represents the resultant of the two forces. Thus, when two mutually perpendicular forces act on a point, the vector representing the resultant force is the diagonal of a rectangle. The length of AX, if measured on the same scale as that for the two original forces, is the resultant force; in this case approximately 11.2 pounds. The angle gives the direction of the resultant force with respect to the horizontal.

Mathematically, the resultant force of per-pendicular forces can be found by using the Pythagorean theorem which deals with the solution of right triangles. The formula is This states that the hypotenuse, side "C" (our unknown resultant force) squared is equal to the sum of side "a" (one of our known forces) squared and side "b" (another of our known forces) squared.

If we substitute the known information in figure 2-1-2 we have the following:

Unknown resultant force 5 lb or the known force on one side of our right triangle, side BX (same as side AC) 10 lb or the known force on the other side of our right triangle, side AB

Setting up the equation we have:

To find the resultant of two forces that are not at right angles, the following graphic method may be used (See fig. 2-1-3.)

Let AB and AC represent the two forces drawn accurately to scale. From point C draw a line parallel to AB and from point B draw a line parallel to AC. The lines intersect at point X.

Figure 2-1-3.-Graphic method of the composition of forces.

The force AX is the resultant of the two forces AC and AB. Note that the two dashed lines and the two given forces make a parallelogram ACXB. Arriving at the resultant in this manner is called the parallelogram method. The resultant force and direction of the resultant is found by measuring the length of line AX and determining the direc-tion of line AX from the figure drawn to scale. This method applies to any two forces acting on a point whether they act at right angles or not. Note that the parallelogram becomes a rectangle for forces acting at right angles. With a slight modification, the parallelogram method of addi-tion applies also to the reverse operation of sub-traction.

Consider the problem of subtracting force AC from AB. (See fig. 2-1-4.) First, force AC is reversed in direction giving -AC (dashed line). Then, forces -AC and AB are

Figure 2-1-4.-Parallelogram method of subtracting forces.

added by the parallelogram method, giving the resulting AX, which in this case is the difference between forces AB and AC. A simple check to verify the results consists of adding AX to AC; the sum or resultant should be identical with AB.

Application of Vectors and Resultant Forces

The methods presented for computing vectors and resultant forces are the simplest and quickest methods for the Aerographers Mate. There are other more complex methods described in Mathematics, Vol. 1, NAVED-TRA 10069-D and Vol. II, NAVEDTRA 10071-B.

The primary purposes of using vectors and resultant forces are for computing radiological fallout patterns and drift calculations for search and rescue operations.

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