Problem 16 - University Physics with Modern Physics

Question

Given two springs of identical size and shape, one made of steel and the other made of aluminum, which has the higher spring constant? Why? Does the difference depend more on the shear modulus or the bulk modulus of the material?

Step-by-Step Solution

Answer

Answer: The shear modulus has a more significant impact on the spring constant because it determines the material's resistance to shearing deformation (twisting), which is a primary concern in spring systems.

Step 1: Understanding the spring constant

The spring constant (k) measures a spring's stiffness, determining how much force is needed to cause a certain displacement. It can be found using Hooke's Law: F = kx, where F is the force applied to the spring, and x is the displacement from the spring's resting position.

Step 2: The relationship between spring constant and material properties

The spring constant depends on the material's properties and the spring geometry. Generally, the spring constant can be written as k = (d^4 * G) / (8 * D^3 * N), where d is the wire diameter, G is the shear modulus, D is the mean coil diameter, and N is the number of active coils. In our case, since the springs are of identical size and shape, the factors related to geometry (d, D, and N) are equal. Therefore, we can conclude that k is directly proportional to G (shear modulus).

Step 3: Comparing the shear modulus of steel and aluminum

The shear modulus of steel is around 80 GPa, while the shear modulus of aluminum is around 26 GPa. Since the spring constant is directly proportional to the shear modulus, the spring made of steel has a higher spring constant than the one made of aluminum.

Step 4: Explaining the importance of shear modulus and bulk modulus

In this context, the shear modulus (G) is the dominant factor determining the spring constant because it describes how resistant a material is to shearing deformation (twisting). Bulk modulus (B) measures a material's resistance to uniform compression, which is not a primary concern in spring systems. Therefore, the difference in spring constants depends more on the shear modulus of the material than the bulk modulus.