What is the relationship between Young’s modulus (E) and the spring constant (k) for a rod of length L and cross-sectional area A?
- Ak = EA/LCorrect
- Bk = EL/A
- Ck = E/LA
- Dk = L/EA
Solution
Concept
Relationship between Young's modulus and spring constant
Formula
k = EA/L
Step-by-step solution
1. Axial stiffness k relates applied force to deformation in an elastic rod of length L and cross-sectional area A. 2. standard elastic formulation relates Hooke's law parameters directly to structural stiffness. Answer: A
Explanation
The axial stiffness or spring constant k is defined as the ratio of applied axial force to the resulting elongation. Using Hooke's law, elongation is given by FL/EA, which rearranges to F/δ = EA/L. Therefore, the spring constant is equal to EA/L, making option A correct.
Why the other options are wrong
B) Inverting L and A results in dimensionally incorrect units for spring stiffness. C) Dividing Young's modulus by the product of length and area does not yield force per unit displacement.
Shortcut
Axial stiffness k = EA/L; proportional to area A and inversely proportional to length L.
Common mistake
Inverting the ratio of length and cross-sectional area and selecting option B instead of k = EA/L.
Correct answer: A) k = EA/L