As an engineer you must design a flagpole that will withstand the forces of nature.   Figure 1 shows two of the important forces:  the force of the wind acting on the flag (F₁) and the force of the wind acting on the flag (F₁) and the force of the wind acting on the combination of the flag and flagpole (F₂).

Figure 1

Given:

Elevation of site - 4000 ft
Ambient temperature - 45 degrees F
Air density in lb/ft³ at 45 degrees F - 0.0679 lb per ft³
Wind velocity - 60 miles per hour, or 5280 feet per minute, or 88 feet per second
Density of concrete - 130 lb/ft³

EQUATIONS TO USE

Surface area for object:  length times width (l x w) in same units

Moment:  force in pounds times lever arms in feet

Volume: length times width times height in same units

DEFINITIONS

Center of mass.
Point at which a force acts upon a body.  The flagpole is one body with a length and width.  The flag is another body, and it, too, has length and width.  The flag has width because of the wind playing on the fabric, causing it to ripple.  The width of the ripple is the width of  the flag (as seen from above, not on the flat surface).  The large area of the flag is parallel to the direction of the wind flow.

A wind force acts at the middle of the area of each body.  The force that acts upon the flag (F₁ in Figure 1) is different from the force that acts upon the flagpole (F₂ in Figure 1).  These forces are given in the problem.   (Engineers have formulas for computing the size of these forces, but the formulas are too complicated for us to use here.)

Moment of force.
A force applied to a body may produce rotation about some axis.  You can easily see this if you push against an open door.  Try pushing a door at the handle, at the center, and right at the hinge.  The further out from the hinge you push, the less force you need to move the door.  The effectiveness of a force in setting a body in rotation is known as the moment of force.  It is measured by the product of the force times the length of the lever arm (here, the distance outward from the hinge.)

In order for a body to remain stationary, the moments must be equal and opposite.

Fulcrum.
The support or point of support on which a lever turns in raising or moving something.  On a door that's open and free to swing, the fulcrum is the hinge.

THE PROBLEM:

Figure 1 shows the two forces acting upon their separate centers of area.  The moment equals the forces that are acting upon the center of their areas times the distance from the base to the center of each area.  There are two parts to the problem:   The first is to determine the moment of force of the flag and flagpole together.   The second is to make sure that the moment of force of the base is the same as that of the flag and flagpole, in order to balance the two forces.

STEP 1.

Figure the moment, or force of rotation, of the flag and flagpole, which must also equal the moment of the base.  Thus M₁ must equal M₂, as shows in Figure 2.

Figure 2

Thus, them moment of force of the flag and flagpole, (28 feet x F₁) + (15ft. x F₂), 22,832 ft-lb, must also be the moment of force of the base.

STEP 2.

We know that the moment of force of the base (M₂) must be 22,832 ft-lb.  We are given two measures of the base, the length and width, which give us the area of the base of the block.  We need to find how deep the block must be in order to balance the moment of force of the flagpole.  The moment of force of the base will be the total weight of the base times the distance from the fulcrum to the base's center.

Figure 3

Remember that the density of concrete is 130 lb per cubic foot.  The weight of the base will be the volume of the base times the density of concrete in pounds per cubic foot.

Therefore, the base must be at least 5.5 feet deep in order to balance a flagpole as described.

(note: lb = pounds; ft = feet)