WORLD FOR CIVIL

Example 1 - Concurrent, Coplanar Forces

In this relatively simple structure, we have a weight supported by two cables, which run over pulleys (which we will assume are very low friction) and are attached to 100 lb. weights as shown in the diagram. The two cords each make an angle of 50^o with the vertical. Determine the weight of the body. (The effect of the pulleys is just to change the direction of the force, it may be considered to not effect the value of the tensions in the ropes.)

If we examine the first diagram for a moment we observe this problem may be classified as a problem involving Concurrent, Coplanar Forces. That is, the vectors representing the two support forces in Cable 1 and Cable 2, and the vector representing the load force will all intersect at one point, just above the body. When the force vectors all intersect at one point, the forces are said to be Concurrent. Additionally, we note that this is a two-dimensional problem, that forces lie in the x-y plane only. When the problem involves forces in two dimensions only, the forces are said to be Coplanar.

(Notice in this problem, that since the two supporting members are cables, and cables can only be in tension, the directions the support forces act are easy to determine. In later problems this will not necessarily be the case, and will be discussed later.)

To "Solve" this problem, that is to determine the weight supported by forces (tensions) in cable 1 and cable 2, we will now follow a very specific procedure or technique, as follows:

1. Draw a Free Body Diagram (FBD) of the structure or a portion of the structure. This Free Body Diagram should include a coordinate system and vectors representing all the external forces (which include support forces and load forces) acting on the structure. These forces should be labeled either with actual known values or symbols representing unknown forces. The second diagram 2 is the Free Body Diagram of point just above the weight where with all forces come together.

2. Resolve (break) forces not in x or y direction into their x and y components. Notice for Cable 1, and Cable 2, the vectors representing the tensions in the cables were acting at angles with respect to the x-axis, that is, they are not simply in the x or y direction. Thus the forces Cable 1, Cable 2, we must be replaced with their horizontal and vertical components. In the third diagram, the components of Cable 1 and Cable 2 are shown.

3. Apply the Equilibrium Conditions and solve for unknowns. In this step we will now apply the actual equilibrium equations. Since the problem is in two dimensions only (coplanar) we have the following two equilibrium conditions: The sum of the forces in the x direction, and the sum of the forces in the y direction must be zero. We now place our forces into these equations, remembering to put the correct sign with the force, that is if the force acts in the positive direction it is positive and if the force acts in the negative direction, it is negative in the equation.
or, -100 cos 40^o + 100 cos 40^o = 0 (Just as we would expect, the x-forces balance each other.)

or, 100 sin 40^o + 100 sin 40^o - weight of body = 0
In this instance, it is very easy to solve for the weigth of the body from the y-equation; and find:
Weight of body = 128.56 lb.