0.13 Phy1100: force -- vector solutions for coplanar forces concurrent  (Page 7/13)

The parallelogram rule

As I told you earlier, any two forces meeting at a point (we're considering the ring to be a point) can be replaced by a singleresultant force, which is represented by the diagonal of a parallelogram whose two adjacent sides represent the two forces.

If we can figure out what the resultant force is, we can figure out the answer to the above questions regarding the tensions and the load being borne by each of the uppercords.

We should be able to figure out what the resultant force is with a thought experiment based on our life experiences.

Tie a fourth cord to the ring

Pretend that you tie another cord to the top of the ring and loop it over the pipe somewhere between the tie points of the other two cords. (This cord isindicated by the object labeled "slack cord" in the image in the file named Phy1100b1.svg and also in the image in Figure 5 .)

Then pull very slowly on that fourth cord until both of the other cords go slack. At thatpoint, the fourth cord, (which is somewhat analogous to the resultant of the forces in the other two cords) has assumed the entire 10-newton load.

Is the ring in equilibrium?

Did the ring and the mass move sideways when the other two cords went slack. If so, the equilibrium of the system was disrupted. Relax the tension on the fourthcord, move it slightly sideways, and repeat the experiment. Continue this process until you find a location where the ring doesn't move sideways when thefourth cord assumes the total load.

Only one location will suffice

You can probably agree, from prior experience on the swing set at the playground as a child, that the only location at which the fourth cord canassume the total load without the ring moving sideways is when the fourth cord passes over the pipe directly above the ring and the mass. Therefore, we knowthat the resultant of the forces being exerted by the original two upper cords must be on a vertical line directly above the ring and the mass.

The diagonal

We also know on the basis of the above rule that the length of the resultant force is proportional to the length of thediagonal of a parallelogram whose two adjacent sides represent the two forces that are being resolved.

The resultant force

The resultant force is indicated by a heavy dashed vector pointing upward in the image in Phy1100c1.svg. Two sides of the parallelogram are represented bylighter dashed lines. The other two sides of the parallelogram are represented by heavy solid vectors, one at 45 degrees relative to the horizontal and theother at 60 degrees relative to the horizontal.

The length of the resultant force vector

Let's think a bit about the required length of the resultant force vector. The above experiment resolves the force system into only two forces acting onthe ring. (The original two upward force vectors are eliminated from the picture when those two cords go slack.) One force is directed up and the other force is directed down.

The vector sum must be zero for equilibrium

We also know that the vector sum of all the forces acting on a point must be zero in in order for the point to be in equilibrium. Since there are the onlytwo forces acting on the ring, the only way that they can sum to zero is for them to be equal in magnitude and opposite in direction from one another.