$12\mathrm{.}19$ A conical compression coil spring, as illustrated in Figure $12\mathrm{.}6,$ is made of $3-$mmdiameter steel wire and has an active coil diameter that varies from $20$ mm at the top to $45$ mm at the bottom. The pitch $($axial spacing between corresponding points on adjacent coils$)$ is $7$ mm throughout. There are four active coils. A force is applied to compress the spring, and the stresses always remain in the elastic range.

$($a$)$ As the force is increased, which coil deflects to zero pitch first?

$($b$)$ Calculate the magnitude of force required to cause the deflection in part a$.$

$($c$)$ Suppose the force is increased until the entire spring is compressed solid. Neatly sketch a force$-$deflection curve for the spring.

Figure $12\mathrm{.}6$

Conical compression coil spring.