A paper machine processes rolls of paper having a density of $984$k$($g$)/($m$^(3)).$

The paper roll is $1\mathrm{.}90-$m outside dia $($OD$)\backslash$times $0\mathrm{.}22-$m inside dia

$($ID$)\backslash$times $3\mathrm{.}23$m long and is on a simply supported, hollow, steel shaft with

S$\_($ut$)=400$MPa

Suppose, that the material reliability is $99\mathrm{.}9\%,$ the shaft operates at a

room temperature, the surface finish to be cold$-$rolled. Assume that the

paper roll provides the uniformly distributed load.

Part A

Find the shaft ID needed to obtain a dynamic safety factor of $2$ for a $10-$year life running three $8-$hour shifts per day if the shaft OD is $22$ cm and the roll turns at $50$ rpm $.$

Express your answer to three significant figures and include the appropriate units.

ID$\_($sh$)$

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