Graphic Methods of Coplanar Force Resolution

The Parallelogram of Forces Method is one of the graphical methods developed to find the resultant of a coplanar force system. Two or more concurrent forces can be replaced by a single resultant force that is statically equivalent to these forces.

The illustration shows two vectors and their resultant. The resultant force is shown as the dashed vector. In order to resolve these forces graphically, one must first extend the lines of action of two concurrent forces until they intersect. This intersection is known as the point of origin for the system. Both forces, as well as the resultant, must ALL act either away from or toward the point of origin.

Animation illustrating how to resolve vectors with the Parallelogram method

The resultant can be represented graphically by the diagonal of the parallelogram formed by using the two force vectors to determine the length of the sides of the parallelogram. The magnitude of the resultant can be accurately measured as the scaled length of the diagonal. The resultant MUST go through the point of intersection of its components!!!

(Remember: graphical solutions depend upon the accuracy of the drawing. The length of each vector should be carefully scaled to equal the magnitude of the force).

More than two non-parallel forces can be combined by successively eliminating one of the forces. Combine any two of the forces into their resultant by the parallelogram method. Combine this resultant with any of the remaining forces (or with the resultant of any of the remaining forces) until all of the forces are included. One must remember that the vectors can only be translated (or moved) along their lines of action. Two vectors (or Forces) cannot be combined (or resolved) until both of them are meeting head-to-head or tail-to-tail!

The resolution of the this system is a single vector that has a magnitude of approximately 4k with a direction of up and to the left. Try it yourself!!!

The Triangle of Forces Method is another graphical method developed to find the resultant of a coplanar force system. Since the opposite sides of a parallelogram are equal, a force triangle may also be found instead of using the parallelogram method. This method is quite useful because it can be successivly applied to any number of concurrent forces.

To calculate the resultant of the force system shown above, move force A so that it's tail meets the head of force B. Now forces A and B form a "Head-to-Tail" arrangement. The resultant R is found by starting at the tail of B (the point of intersection of forces A and B) and drawing a vector which terminates at the head of the transposed A. Note that if force B had been transposed instead of force A, the resultant would have started from the tail of A and terminated at the head of force B. Again, this process could be repeated for any number of force vectors.

The resultant is described by the vector's magnitude and direction. These are determined by scaling the length and angle respectively. The accuracy of these values depends upon the accuracy of the graphics.

More than two non-parallel, non-concurrent forces can be combined by successively eliminating one of the forces. Combine any two of the forces into their resultant by the triangle method, and then extend that resultant until it intersects the line of action of another force. One continues this process until all forces have been included. In this way, each one of the forces is successivly combined with the resultant of the previous triangle. One cannot simply continue to add the vectors head-to-head or tail-to-tail because the resulting lines of action would then be incorrect!

Animation illustrating how to resolve vectors with the Triangle method

This illustration of a fixed jib crane allows one to read the forces as they meet at the tip of the boom. The diagram indicates the forces acting in the crane: red for tension and blue for compression.

The graphical methods of force decomposition could be used to determine the magnitude of the forces within the crane. In this case the two components for each of the structural elements are shown. All of the vectors are representational. They are not drawn to scale. The actual magnitudes are simply determined IF one would actually draw ALL of the vectors to scale and then measure the results.

### Questions for Thought

Could one determine components that are not related to the x and y axis for the crane? why might one need to do this?