Problem $4(9$ points$)$

$4\mathrm{.}1$ What is the purpose of a flexure test on materials like wood and cast iron? How is the modulus of rupture calculated in a flexure test? Why does cast iron typically exhibit brittle failure in flexure, while wood exhibits ductile behavior?

$4\mathrm{.}2$ A rectangular wood beam with a width of $100$ mm and depth of $50$ mm is tested in flexure under a simply supported span of $2$ meters. If the beam fails at a load of $1000$ kg applied at the center of the span, calculate the modulus of rupture for the wood. Repeat the calculation for a cast iron beam of similar dimensions, failing under a load of $5000$ kg $.$ Additionally, consider how the modulus of rupture could be increased without changing the material.