The spring constant (stiffness) of a helical spring depends on which combination of its physical properties and dimensions?

Introduction / Context:
Springs are widely used in mechanical systems to store and release energy, provide cushioning, and maintain forces. The spring constant, often represented by k, quantifies how stiff a spring is. Different springs can have very different values of k depending on how they are made. This question tests your understanding of which physical aspects of a spring influence its spring constant in a practical sense.

Given Data / Assumptions:
- We consider a typical helical spring, such as those used in scales or suspensions.- The options mention properties like length, thickness, diameter, and material.- We look for the most comprehensive and realistic combination that affects stiffness.

Concept / Approach:
The spring constant depends on both material properties and geometric factors. The material determines the elastic modulus, which is a measure of how resistant the material is to deformation. Geometric factors include the thickness (wire diameter) and the overall coil diameter and number of turns. A thicker wire usually makes the spring stiffer, while a larger coil diameter and longer spring tend to make it less stiff. Therefore, focusing only on length or only on material is not sufficient; thickness, diameter, and material all play important roles in determining the value of the spring constant.

Step-by-Step Solution:
1. Recognise that a spring made from a stronger material such as steel will have a different stiffness compared with the same geometry made from a softer material.2. Observe that a spring made from a thicker wire resists stretching more than one made from a thinner wire of the same material.3. Note that the diameter of the coils influences how much the spring bends under a given load, affecting stiffness.4. Understand that while length and number of turns matter, the option that mentions thickness, diameter, and material captures key influences on the spring constant.5. Compare the given options and identify that only the option including thickness, diameter, and material reflects the combined dependence.6. Select the option stating that the spring constant depends on its thickness, its diameter, and its material.

Verification / Alternative check:
Practical experience supports this view. If you try compressing two springs made of the same material but with different wire thicknesses, the one with the thicker wire is clearly harder to compress. Likewise, springs made from different materials but with identical shape also display different stiffness. Engineering formulas for helical springs explicitly include material modulus, wire diameter, coil diameter, and number of turns, reinforcing the idea that several parameters jointly determine the spring constant.

Why Other Options Are Wrong:
- Its length only: Length influences stiffness, but it is not the only factor; material and wire thickness also matter.- Its material only: Material properties are important, yet geometry cannot be ignored.- Its length and its diameter: This combination ignores the crucial role of material type and wire thickness, making it incomplete.

Common Pitfalls:
Many learners focus on just one parameter when thinking about stiffness, such as material or length. Real mechanical design always considers both material properties and geometric dimensions. Remember that springs are a good example where geometry and material work together to determine behaviour, which is why the most complete option that mentions several of these factors is the correct choice.

Final Answer:
The spring constant depends on the spring thickness, its diameter, and the material from which it is made.