Understanding Helical Spring Constant and Material Properties

This question delves into the mechanical behavior of helical springs, specifically focusing on the relationship between the spring's stiffness (spring constant) and the properties of the material it's made from. Let's break down both the Assertion and the Reason.

Analysis of Assertion (A)

Assertion (A) states: The spring constant of a helical spring is directly proportional to the modulus of rigidity (shear modulus) of the material from which it is made.

The spring constant, often denoted by $k$, quantifies how much force is needed to stretch or compress a spring by a certain distance. For a standard helical spring, the formula for the spring constant is derived from the principles of torsion.

A common formula for the spring constant ($k$) of a helical compression or tension spring is:

$k = \frac{Gd^4}{8nD^3}$

Where:

• $G$ is the modulus of rigidity (also known as the shear modulus) of the spring material.
• $d$ is the diameter of the spring wire.
• $n$ is the number of active coils in the spring.
• $D$ is the mean diameter of the spring coil.

From this formula, we can clearly see that $k$ is directly proportional to $G$. If the modulus of rigidity ($G$) increases, the spring constant ($k$) also increases, assuming other factors remain constant. Therefore, Assertion (A) is true.

Analysis of Reason (R)

Reason (R) states: When a helical spring is stretched or compressed axially, the primary deformation experienced by its wire is a torsional (twisting) shear.

Imagine applying a force to the ends of a helical spring to stretch or compress it. This axial force causes the coils to rotate relative to each other. This rotation results in the wire itself being subjected to a torque. This torque induces shear stresses and shear strains within the wire material. While there might be other minor stresses (like direct shear stress due to the axial load), the dominant effect causing the change in the spring's length is the twisting of the wire, which is a form of torsional shear deformation. Hence, Reason (R) is true.

Evaluating the Relationship Between Assertion and Reason

Now, we need to determine if Reason (R) correctly explains Assertion (A).

Assertion (A) states $k \propto G$. Reason (R) states that the deformation is primarily torsional shear. The modulus of rigidity ($G$) is the material property that measures its resistance to shear deformation. Since the primary deformation in a helical spring under axial load is torsional shear, it makes sense that the material's resistance to this specific type of deformation (its shear modulus $G$) directly influences how stiff the spring is (its spring constant $k$).

A material with a higher shear modulus ($G$) is stiffer in shear. This means it resists twisting more strongly. Consequently, a larger force is required to twist the wire by the same amount, leading to a higher spring constant ($k$). Thus, the fact that the wire experiences torsional shear (R) is the fundamental reason why the modulus of rigidity ($G$) affects the spring constant ($k$) (A).

Final Conclusion on Assertion and Reason

Both statements are factually correct:

• Assertion (A) is true: The spring constant ($k$) is indeed directly proportional to the shear modulus ($G$).
• Reason (R) is true: The deformation mechanism is primarily torsional shear.

Furthermore, Reason (R) provides the correct physical basis for Assertion (A). The resistance of the material to the specific type of deformation (torsional shear) dictates the spring's overall stiffness.

Therefore, the most appropriate answer is that both statements are true, and Reason (R) is the correct explanation of Assertion (A).