Problem $2.$ A stepped, solid round shaft is loaded as shown in the following figure. Find out $(1)$ the maximum deflection in $y-$direction and its location along the shaft, and $(2)$ the fundamental critical rotating speed for the shaft. Assume the Young's modulus of the shaft is $E=207$GPa, and all applied loads are in the $y-$direction and passing the axis of the shaft. Ignore the weight of the shaft in your calculation.

$(30$ marks$)$