Question

Four identical hollow cylindrical columns, support a big structure of mass $M$ . The inner and outer radii of a column are $R_{1}$ and $R_{2}$ respectively. Assuming the load distribution to be uniform, the compressional strain of each column is (where $Y$ is Young?? modulus of the column)

• A. $\frac{Mg}{\pi\left(R_{2}^{2}-R_{1}^{2}\right)Y}$
• B. $\frac{Mg}{4\pi\left(R_{2}^{2}-R_{1}^{2}\right)Y}$
• C. $\frac{Mg}{\pi\left(R_{1}^{2}-R_{2}^{2}\right)Y}$
• D. $\frac{Mg}{4\pi\left(R_{1}^{2}-R_{2}^{2}\right)Y}$

Answer 1

Answer: (B)

$\frac{Mg}{4\pi\left(R_{2}^{2}-R_{1}^{2}\right)Y}$

Answer 2

Area of cross-section of each column, $A$ $=\pi\left(R_{2}^{2}-R_{1}^{2}\right)$ Since each column supports one-quarter of the load. $\therefore\quad$ $F=\frac{Mg}{4}$ Compressional strain of each column $=\frac{F}{AY}$ $=\frac{Mg}{4\pi\left(R_{2}^{2}-R_{1}^{2}\right)Y}$