This is a design problem, but it is not open$-$ended because some of the design parameters have already been specified. A helical compression spring is to be designed for infinite life when subjected to a load associated with a cam follower, which fluctuates between $25$ and $50$ lb $.$ Steel spring wire have an $S_{ut}=180$ ksi and $S_{sy}=99$ksi. The number of active coils is $8.$ For a value of $C=7$ and using Gerber$-$Zimmerli criterion $($safety factor $=1),$

$($a$)$ Determine the wire diameter theoretically required if the steel wire is unpeened,

$($b$)$ Determine the wire diameter theoretically required if the steel wire is peened,

$($c$)$ Determine the natural frequency of the spring,

$($d$)$ If the cam rotates at $800$ rpm $,$ would the spring go in resonance?

$($e$)$ Zimmerli's strength data are for wire diameters less than $\frac{3}{8}.$ Is the wire diameter in $($a$)$ is greater than $\frac{3}{8},$ what changes to the design parameters would you make to redesign the spring?