The maximum height of a cylindrical column is limited by the compressive strength of the material; if the compressive stress at the bottom were to exceed the compressive strength of the material, the column would be crushed under its own weight. (a) For a cylindrical column of height \(h\) and radius \(r,\) made of material of density \(\rho,\) calculate the compressive stress at the bottom of the column. (b) since the answer to part (a) is independent of the radius \(r,\) there is an absolute limit to the height of a cylindrical column, regardless of how wide it is. For marble, which has a density of $2.7 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\( and a compressive strength of \)2.0 \times 10^{8} \mathrm{Pa},$ find the maximum height of a cylindrical column. (c) Is this limit a practical concern in the construction of marble columns? Might it limit the height of a beanstalk?

Step by step solution

01

Calculate the compressive stress at the bottom of the column

To calculate the compressive stress, we need to find the force acting on the bottom of the column due to the weight of the material above it. Then, divide this force by the cross-sectional area of the column. The volume of the column \(V\) can be calculated as \(V = \pi r^2 h\). Then, we can calculate the mass of the column \(m\) using the density formula \(m = V \rho\). Now, we can calculate the weight of the column \(W\) using the formula \(W = mg\), where \(g\) is the acceleration due to gravity. Finally, we can calculate the compressive stress \(\sigma\) by dividing the weight by the cross-sectional area: \(\sigma = W / A\). The cross-sectional area of the column \(A = \pi r^2\).

02

Calculate the compressive stress at the bottom of the column using the equations from Step 1

Now we can plug in the equations from Step 1 to calculate the compressive stress: \(\sigma = \frac{W}{A} = \frac{mg}{\pi r^2} = \frac{(\pi r^2 h \rho)g}{\pi r^2} = h \rho g\) We found that the compressive stress at the bottom of the column is independent of the radius, and only depends on the height, density, and acceleration due to gravity.

03

Find the maximum height of a marble column

For a marble column, we have the density \(\rho = 2.7 \times 10^3 \, \mathrm{kg/m^3}\) and compressive strength \(\sigma_\mathrm{max} = 2.0 \times 10^8 \, \mathrm{Pa}\). To find the maximum height, we need to solve the equation: \(\sigma_\mathrm{max} = h_\mathrm{max} \rho g\) Rearrange the equation to solve for \(h_\mathrm{max}\): \(h_\mathrm{max} = \frac{\sigma_\mathrm{max}}{\rho g}\) Plug in the values for \(\rho\), \(g\), and \(\sigma_\mathrm{max}\), to find the maximum height: \(h_\mathrm{max} = \frac{2.0 \times 10^8 \, \mathrm{Pa}}{(2.7 \times 10^3 \, \mathrm{kg/m^3})(9.81 \, \mathrm{m/s^2})} \approx 7338.8 \, \mathrm{m}\)

04

Discuss if the limit is a practical concern in the construction of marble columns and if it might limit the height of a beanstalk

The maximum height of a marble column we found is 7338.8 meters, which is much taller than any existing or planned building. It means that the compressive strength limit is not a practical concern when constructing marble columns. However, other factors such as stability, wind, and earthquakes might still be concerns in practice. As for a beanstalk, although plants are not built with marble, it could be interesting to consider whether such a height limit could apply. In a beanstalk, support and water transportation would be important factors, so it is not directly comparable to a marble column. However, it can be noted that the extreme height found is an indication that the compressive strength might not be the main limiting factor for a beanstalk's height either.

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