$10-3$ A helical compression spring is wound using $2\mathrm{.}5-$mm$-$diameter music wire. The spring has an

outside diameter of $31$ mm with plain ground ends, and $14$ total coils.

$($a$)$ Estimate the spring rate. at this point in the course, just use elastic properties of carbon steel from table A$-5$

$($b$)$ Assume the shearing stress in the spring wire must not exceed $800$ MPa at spring closure and find:

$($i$)$ the maximum force the spring can exert at spring closure

$($ii$)$ the maximum allowable free length of the spring and the pitch of such a spring.

$($c$)$ Assume the spring is wound with the free length found in part $($b$)$ and that it will be acting against

stable, flat, parallel surfaces. Assume it will not be externally stabilized $($e$.$g$.$ with an internal rod

or by being placed within a hole$).$ Complete and answer the following:

$($i$)$ Determine if the spring manufactured as described in part $($b$)$ will be unconditionally stable

by comparing its free length with the maximum free length threshold for unconditional

stability.

If you determine that the spring is not unconditionally stable, answer the following:

$($ii$)$ How far can the spring be compressed before it is expected to buckle?

$($iii$)$ What is the maximum force that can be applied before the spring is expected to buckle?

$($d$)$ Assume that in service, the spring will only ever be compressed to a maximum of $90\%$ of the

deflection that you expect would cause the spring to buckle. Determine the "fractional overrun to

closure" $()$ for this maximum expected in$-$service spring deflection.