Prob. $2(2$ points$)$

A steel roller with a diameter of $15$ mm and a length of $20$ mm is under a load of $1150\frac{N}{m}m$ and is moving inside the steel ring with an internal diameter of $75$ mm $.($steel: $E=207$GPa,$v=0\mathrm{.}3)$ Assuming only pure rolling motion, use the equation given below,

Find:

$($a$)$ width of contact zone $(1$ point$)$

$($b$)$ maximum contact pressure $(1)$

Use:

spherical contact $($for circumscription$)$

$p_{\text{m}\text{a}\text{x}}=\frac{3F}{2a^2},a=\sqrt[3]{\frac{3}{8}\frac{m_1+m_2}{B}F},B=\frac{1}{2}(\frac{1}{r_1}+\frac{1}{r_2}),m=\frac{1-v^2}{E}$

cylindrical contact $($for circumscription$)$

$p_{\text{m}\text{a}\text{x}}=\frac{2F}{aL},a=\sqrt[\phantom{2}]{\frac{2}{}\frac{m_1+m_2}{B}\frac{F}{L}},B=\frac{1}{2}(\frac{1}{r_1}+\frac{1}{r_2}),m=\frac{1-v^2}{E}$