Problem $1(20$ Pts$)$

An aluminum$-$alloy cylindrical roller with the diameter of $1\mathrm{.}75$ in$.$ and length $2\mathrm{.}5$ in $.$ is sitting on the flat surface of a cast$-$iron block with the thickness of $5$ in$.$ as shown. It is assumed that the maximum shear stress induced by the contact is not to exceed $3,500.$ Determine the allowable contact force $F$ that can be applied on the roller if a factor of safety F$.$S$.=2\mathrm{.}20$ is maintained. Given the material properties: Aluminum alloy $($Material $1)$: $E_1=10\mathrm{.}4{10}^6,v_1=0\mathrm{.}333$; Cast iron $($Material $2)$: $E_2=14\mathrm{.}5{10}^6,v_2=0\mathrm{.}211.$