A 20.0-m tall hollow aluminum flagpole is equivalent in strength to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex?

To determine the amount of flex or deflection at the top of the flagpole, we can use the concept of torque.

The torque applied to the flagpole by the wind is proportional to the force exerted at the top and the distance from the axis of rotation. In this case, the axis of rotation is at the base of the flagpole.

First, let's calculate the torque applied by the wind. The force of 900 N exerted at the top creates a torque, which we'll call Torque_wind.

Torque_wind = Force * Distance

The distance from the axis of rotation to the top of the flagpole is the height of the flagpole, which is 20.0 m. So we have:

Torque_wind = 900 N * 20.0 m = 18000 N·m

Next, we need to calculate the torque required to bend a solid cylinder with a diameter of 4.00 cm. The torque required to bend a solid cylinder can be calculated using the equation:

Torque_solid_cylinder = (π/32) * (diameter)^4

First, let's convert the diameter to meters:

diameter = 4.00 cm = 0.04 m

Now we can calculate the torque:

Torque_solid_cylinder = (π/32) * (0.04 m)^4 = 1.963 x 10^-5 N·m

So, the torque applied by the wind (Torque_wind) is much greater than the torque required to bend a solid cylinder (Torque_solid_cylinder).

Since the flagpole is hollow, it can withstand greater torque before bending than a solid cylinder. Therefore, the top of the pole will only flex a small amount.

To calculate the flex or deflection at the top of the pole, we can use the concept of Young's modulus. Young's modulus measures the stiffness or rigidity of a material.

The flex or deflection can be calculated using the formula:

Flex = (Torque_wind * Length) / (Young's modulus * Moment of Inertia)

In this case, since the flagpole is hollow, its moment of inertia will be different compared to a solid cylinder. The moment of inertia for a hollow cylinder can be calculated using the formula:

Moment of Inertia = (π/4) * (outer_radius^4 - inner_radius^4)

Without knowing the specific dimensions of the flagpole, such as the outer and inner radii, we cannot accurately determine the flex or deflection at the top of the pole.

To find the specific dimensions and calculate the flex, you would need to know the thickness of the flagpole walls and the outer and inner radii. With this information, you can calculate the flex using the formulas mentioned above.