Question

A 3.0-m-tall, 50-cm-diameter concrete column supports a 200,000 kg load. By how much is the column compressed?

Short answer

The column is compressed by$1mm$.

Step-by-step solution

Given information

The radius of the concrete column is $r=\frac{50cm}{2}=25cm=0.25m$

Mass of the load is $m=200,000kg$

The length of the concrete column is$L=3.0m$

Compression produced in the column 

The gravitational force on the load produces tensile stress $\raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$, where $A$is the cross-sectional area of the concrete column and $F$is the gravitational force on the load.

Young's modulus given by $Y=\frac{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$A$}\right.}}{{\displaystyle \raisebox{1ex}{$∆L$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}}$.

The compression produced in the column is

$$\begin{gathered} ∆L=\frac{F}{A}\frac{L}{Y} \\ =\frac{mg}{{\mathrm{\pi r}}^2}\frac{L}{Y} \\ =\frac{200000\times 9.8}{\mathrm{\pi }{\left(0.25\right)}^2}\frac{3.0}{3\times {10}^{10}} \\ =1mm \end{gathered}$$

Therefore, the column is compressed by$1mm$.