A spring having 7 coils and a coil-to-coil diameter of 0.5 in. is to be made of colddrawn steel wire. When a tensile load of \(15 \mathrm{lb}_{\mathrm{f}}\) is applied the spring is to deflect no more than 0.60 in. The cold drawing operation will, of course, increase the shear yield strength of the wire, and it has been observed that \(\tau_{y}\) (in \(\mathrm{ksi}\) ) depends on wire diameter \(d\) (in in.) according to $$\tau_{y}=\frac{63}{d^{0.2}}$$ If the shear modulus for this steel is \(11.5 \times 10^{6} \mathrm{psi}\), calculate the minimum wire diameter required such that the spring will not plastically deform when subjected to the above load.

Hooke's Law is a fundamental principle in mechanics that relates the force required to extend or compress a spring to the distance it moves, given that the deformation is within the spring's elastic limit. This relationship is mathematically expressed as

\[ F = k \times \text{\(\backslash\)Delta} L \]
where \( F \) is the force applied to the spring, \( k \) represents the spring rate or stiffness, and \( \text{\(\backslash\)Delta} L \) denotes the displacement or change in length of the spring. The spring rate \( k \) is a constant that indicates how stiff the spring is—the higher the spring rate, the less it will stretch or compress under a given load.

Applying Hooke's Law to spring design, engineers can predict how a spring will behave under applied loads allowing for precise control over spring dynamics. In the case of the provided exercise, the calculation of the spring rate was the first step to guarantee that the spring would not deform past its elastic limit when the tensile load was applied. Proper utilization of Hooke's Law ensures that the mechanical systems dependent on springs perform reliably and within the design specifications.

The shear modulus, often symbolized as \( G \), is a property that measures the rigidity of a material when subjected to shear forces. It relates the shear stress to the shear strain, providing an indicator of the material's ability to resist shape changes when shear forces are applied. Mathematically, the shear modulus is defined as

\[ G = \frac{\text{shear stress}}{\text{shear strain}} \]
A higher shear modulus means the material is stiffer and less susceptible to deformation under shear forces. The shear modulus plays a pivotal role in spring design, as it impacts the spring's ability to retain its shape and function under transverse loads. It is particularly important when calculating the wire diameter of a spring, ensuring that it can withstand applied forces without permanent deformation.

In the exercise mentioned, knowing the shear modulus of the colddrawn steel is imperative. By combining this knowledge with the spring's dimensions and the intended load, the minimal wire diameter can be accurately determined. This ensures that the spring, when manufactured, will behave as intended without suffering from plastic deformation under the prescribed load conditions.

Plastic deformation refers to the irreversible change in the shape of a material when it is subjected to stresses beyond its elastic limit. This is in contrast to elastic deformation, where the material returns to its original shape after the stress is removed. The point at which a material goes from elastic to plastic deformation is called the yield point, and this property plays a crucial role in defining the durability and lifespan of a spring.

Making sure that a spring is designed to only experience elastic deformation within its operational load ensures that it can return to its initial shape and continue to perform its function consistently. In the exercise, it was essential to calculate the minimum wire diameter that would prevent plastic deformation even at its operational limit. This guarantees that the spring only experiences elastic deformation, extending its service life and reliability.

Understanding these concepts, engineers and designers can safeguard against unexpected failures and ensure the longevity of spring-equipped mechanical systems. When specifying the dimensions of a spring, such as in our exercise, considerations for the avoidance of plastic deformation are paramount to the success of the spring's application and its performance over time. The calculations involve taking into account the material's yield strength and ensuring the design remains within these constraints.