A cold$-$rolling process, using a two$-$high rolling mill of $250$ mm in radius, is performed on a steel plate of $200$ mm width and $20$ mm thickness. The plate is reduced in one$-$pass to a thickness of $17$ mm at a roll speed of $15$ rpm$.$ As the thickness is reduced, the width of the plate increases by $5\%($spreading$).$ The yield strength of the steel plate is $220$ MPa, and the tensile strength is $410$ MPa. The strength coefficient is $530$ MPa, and the strain$-$hardening exponent is $0\mathrm{.}26.$ Determine the followings:

a$.$ The minimum required coefficient of friction that would make this rolling operation possible $(4$ points$)$

b$.$ The exit velocity of the plate if the entrance speed of the plate is $11$ m$/$min

$(4$ points$)$

c$.$ The force for each roll

$(7$ points