Expanded problem from this GD thread.

A flagpole 50' high is tilting slightly.  An engineer (whom we have paid enough that we will accept without question his answer), has told us that a side force of 500 pounds at the tip of the flagpole will provide enough torque to hold the flagpole vertical at the base (this flagpole will not bend or buckle, but will only pivot at its base because, of course it will).  Our group of intrepid mathematicians has determined that a guy wire planted 9 feet from the base and attached at the tip will hold the flagpole plumb if tensioned to approximately 3390 pounds.

The ground anchor point has already been placed and cannot be moved - it is 9 feet from the base of the flagpole, level with the base of the flagpole, and in the correct direction for applying the force necessary to hold the flagpole level.

Question:  Can we change the attachment point of the guy wire to the flagpole to allow a lesser tension on the cable?  If so, what is the optimal position on the flagpole to attach the cable, and what will the minimum required tension be?  

Mike

Note - this is a purely theoretical mathematic/geometry problem - the flagpole will not bend or buckle under load, can't be cut, etc.

No.
The force at the top creates a moment at the base of the pole to make it straight. If you lower the anchor point on the pole you need to increase the force to compensate for the reduced length of the moment arm.

True.  The shorter the moment arm, the less force will be required.


You won't halve the force by doubling the height, however.  Given a theoretically infinitely high tower, with the (mass-less) guy wire attached at the top, what would be the minimum tension required, and why?

Mike

Great problem!



We know that the moment generated by a guywire (anchored at an unknown height x) about the base of the flagpole must equal the moment generated by a 500 lb side force applied at the top of the pole.  The moment equation that must be satisfied is:

where

where T is the tension force in the guywire.

Solving for T yields:


We can find a stationary point (which will either be a minimum or maximum) of this function by differentiating T with respect to x:


By setting this equal to zero, we can solve for x:


However, this equation does not have a solution for x in the finite interval [0,50].  By inspection, we can see that as x goes to infinity, this equation will approach 0.  Thus, x=50 is the optimal value of the attachment point.  This can be verified by numerically evaluating results of the T equation listed above for discrete values of x in [0,50]:



I think your intrepid mathematicians were a little off in their tension estimate- for a guywire anchored 9 feet away to generate a 500 lbs side force at an attachment point 50 feet up, the tension should be 500/sin(atan(9/50))=2822 lbs.

The reason the numbers are off is because I transferred the numbers incorrectly from the original problem - it was 600 pounds of force at 50 ft, not 500.