Composition and Resolution
by Vector Methods
OBJECT: To study the composition and resolution of vector quantities.
METHOD: Concurrent forces acting on a body are used as examples of vector quantities. These forces are represented by vectors. The resultant and equilibrant of several sets of such known forces are determined by both graphical and analytical methods. These results are tested on a force table as a check on the first condition for the equilibrium of a rigid body.
THEORY: Measurable quantities may be classified as either (1) scalar quantities or (2) vector quantities. A scalar quantity has magnitude only, but a vector quantity has both magnitude and direction. For example, ...view middle of the document...
A vector is the line segment whose length represents the magnitude of a vector quantity and whose direction is that of the vector quantity. The sense along the line is indicated by an arrow. For example, a force of 100lb. acting at an angle of 30° above the horizontal may be represented by the line OA. Fig. 1, which is 5 units long and has the correct direction. Each unit of length thus represents 20lb.
When vectors do not have the same line of action, their vector sum is not their algebraic sum but a geometric sum.
This geometric sum may be determined by either graphical or analytical methods. Graphical methods are simple and direct but are limited in precision to that obtainable by drawing instruments. Analytical methods have no such inherent limitations. In this experiment both graphical and analytical methods will he applied to forces as examples of vector quantities, but the same methods apply to all vector quantities.
The vector sum, or resultant, of a set of forces is the single force that will have the same effect, insofar as motion is concerned, as the joint action of the several forces.
Vector Summation by Graphical Methods: As an example of vector addition let us consider the case of two forces acting on a body in such a direction that the forces are concurrent, that is their lines of action, if projected would intersect at a point. The vectors OA and OB representing two such forces are shown in Fig. 2. Their vector sum or resultant R, is found by constructing a parallelogram having the two vectors as sides and drawing the concurrent diagonal, as shown in Fig. 3. This diagonal vector R represents in magnitude and direction the single force that is equivalent to the origina1 pair, that is their vector sum. When the resultant of more than two vectors is to be obtained graphically a polygon method is used. This is illustrated in Fig. 4. The vector A is first constructed by the use of a chosen scale and reference direction. Then, from the head of A, the vector B is drawn. It is clear that the vector M is the resultant of vectors A and B, since M would be the concurrent diagonal of a parallelogram if such a parallelogram had been drawn, as was done in Fig. 3. Similarly, it follows that the vector R is the resultant of M and C or of A, B, and C. When the resultant of several forces is required this method is simpler than the parallelogram method. It should be noted that when the parallelogram method is used, the arrows, with their tails together, all radiate from a common point. But in the polygon method the tail of the second arrow coincides with the head of the first,...