$($Q$2)$ a: It is required to select a $619$ wire rope with $1570$ as a tensile designation for a hoist on the basis of long life. The weight of the hoist along with the material is $5$ kN $.$ It is to be raised from a depth of $100$ m $.$ The maximum speed of $5\frac{m}{s}$ is attained in $5$ s $.$ Determine the size of the wire rope and the sheave diameter for long life on the basis of the fatigue as a failure criterion. What is the factor of safety of this wire rope under static conditions?

$(5$points$)$